Submodular approximation: sampling-based algorithms and lower bounds

Submo dular Appro ximatio n: Sampling-based Algorithms and Lo w er Bounds ∗ Zo ya Svitkina † Lisa Fleisc her ‡ Ma y 31, 2010 Abstract W e int r oduce several generaliza tions o f clas sical computer science problems obtained by replacing s imp ler ob jective functions with g eneral submo dular functions. The new problems include submo dular lo ad balancing , whic h g eneralizes loa d balancing or minimum-mak espan scheduling, submo dular sparses t cut and submo dular balanced cut, which genera lize t heir re- sp ectiv e g raph cut pro blems, a s w ell as submo dular function minimization with a cardinality low er bound. W e establish upper and low er bounds for the approximabilit y of these pr oblems with a p olynomial n umber of queries to a function-v alue ora cle. The approximation guarantees for most of our algor ith ms are o f the or der of p n/ ln n . W e show that this is the inhere nt diﬃcult y of the problems by proving matching low er b ounds. W e also giv e an improv ed lower bo und for the problem of appr o ximating a monotone submo d- ular function everywhere. In addition, we prese nt an algo rithm for appr o ximating submodula r functions with sp ecial structure, whose guar a n tee is close to the low er bo und. Although quite restrictive, the class of functions with this s tructure includes the o nes that are used for low e r bo unds both b y us and in previous w o rk. This demonstrates that if there a re signiﬁcantly stronger lower b ounds for this problem, they r ely o n more genera l submo dular functions. 1 In tro duction A function f deﬁned on subsets of a ground set V is called submo dular if f o r all sub sets S, T ⊆ V , f ( S ) + f ( T ) ≥ f ( S ∪ T ) + f ( S ∩ T ). Submo dularit y is a d iscr ete analog of con v exity . It also sh ares some nice prop erties with conca ve functions, as it captur es decreasing m arginal returns. S ubmod - ular functions generalize cut functions of graphs and rank fun ctions of m a tr ic es and matroids, and arise in a v ariet y of applications in cl u ding facilit y lo cation, assignmen t, s cheduling, and n etw ork design. In this pap er, we introd uce and s tudy s everal generalizat ions of classical computer s c ience problems. These new problems h a ve a general sub modu la r function in th e ir ob jectiv es, in place of m u ch simpler functions in th e ob jectiv es of their classical coun terparts. The problems include submo dular lo ad b alancing , wh ic h generalizes load balancing or minim um-makespan scheduling, and submo dular minimization with c ar dinality lower b ound , whic h generalize s the min imum knapsac k ∗ This work supp orted in part by NS F gran t CCF-072886 9. A preliminary version of th is pap er has app eared in the Pro ceedings of the 49th Annual IEEE Symp osium on F oundations of Computer S cience. † Department of Computin g Science, Universit y of Alb erta, Canada. ‡ Department of Computer Science, Dartmouth, US A. 1 problem. In these tw o problems, the size of a collection of items, instead of b eing just a su m of their individual sizes, is now a su bmod ular function. Two other n ew problems are submo dular sp arsest cut and submo dular b alanc e d cut , whic h generalize their resp ectiv e graph cut problems. Here, a general submo dular function replaces the graph cut function, whic h itself is a w ell-kno wn sp ecia l case of a submo dular function. T he last problem that we study is app r oximating a submo dular function everywher e . All of these problems are deﬁned on a set V of n elemen ts w ith a nonnegativ e submo dular fun ction f : 2 V → R ≥ 0 . S ince the amount of information necessary to conv ey a general submo dular f u nctio n ma y b e exp o nen tial in n , w e rely on v alue-oracle access to f to dev elop algorithms with runn ing time p olynomial in n . A value or acle for f is a b lack b o x that, give n a subset S , returns the v alue f ( S ). Th e follo wing are f o r mal deﬁnitions of the problems. Submo dular Sparsest C ut (SSC): Giv en a set of unordered p a ir s {{ u i , v i } | u i , v i ∈ V } , eac h with a d e m a n d d i > 0, ﬁnd a s ubset S ⊆ V min imiz in g f ( S ) / P i : | S ∩{ u i ,v i }| =1 d i . The denominator is the amoun t of d ema n d separated by the “cut” ( S, ¯ S ) 1 . In uniform SS C , all pairs of no des ha ve demand equal to one, so th e ob jectiv e fun c tion is f ( S ) / | S || ¯ S | . An o th e r sp ecial case is the weighte d SSC pr o b lem, in whic h eac h elemen t v ∈ V has a non -n e gativ e w eigh t w ( v ), and the demand b et w een an y pair of elemen ts { u, v } is equ a l to the pro duct w ( u ) · w ( v ). Submo dular b -Balanced C ut (SBC) : Given a w eight fun ct ion w : V → R ≥ 0 , a cut ( S, ¯ S ) is called b -balanced (for b ≤ 1 2 ) if w ( S ) ≥ b · w ( V ) and w ( ¯ S ) ≥ b · w ( V ), where w ( S ) = P v ∈ S w ( v ). The goal of th e problem is to ﬁnd a b -balanced cut ( S, ¯ S ) that minimizes f ( S ). In the unweig hte d sp ecial case, the weig hts of all element s are equal to one. Submo dular Minimization with Ca rdina lit y Lo wer Bound (SML): F or a giv en W ≥ 0, ﬁnd a subset S ⊆ V w it h | S | ≥ W that minimizes f ( S ). A generalizatio n with 0-1 we ights w : V → { 0 , 1 } is to ﬁnd S with w ( S ) ≥ W minimizing f ( S ). Submo dular Load Balancing (SLB): T he uniform v er s io n is to ﬁ nd, giv en a monotone 2 submo dular function f and a p ositiv e int eger m , a partition of V in to m sets, V 1 , . . . , V m (some p ossibly empt y), so as to minimize max i f ( V i ). T h e non-u nifor m v ersion is to ﬁnd, for m mon otone submo dular functions f 1 , . . . , f m on V , a partition V 1 , . . . , V m that minimizes max i f i ( V i ). Appro ximating a Submodula r F unction Everywhere: Pro duce a function ˆ f (not nec- essarily sub modu lar) that for all S ⊆ V satisﬁes ˆ f ( S ) ≤ f ( S ) ≤ γ ( n ) ˆ f ( S ), with appro xim ation ratio γ ( n ) ≥ 1 as small as p ossible. W e also consid e r the sp ecial case of monotone t wo- partition functions, whic h we deﬁne as f ollo w s. A sub modu la r function f on a ground set V is a two-p artition (2P) function if there is a set R ⊆ V su c h that for all sets S , the v alue of f ( S ) dep ends only on the sizes | S ∩ R | and | S ∩ ¯ R | . 1.1 Motiv ation Submo dular functions arise in a v ariety of context s , often in optimization s ettings. The pr o b- lems that we deﬁne in this pap er use subm odular fun ct ions to generalize some of the b est-studied problems in computer s cience. These generalizatio n s capture many v arian ts of their corresp onding classical problems. F or examp le, the submo dular sparsest and balanced cut pr o blems generalize not only graph cuts, but also hyp er grap h cuts. In addition, they m a y b e useful as subr ou tin es for solving other problems, in the same w ay t h at sparsest and balanced cu ts are used for appro ximating graph problems, suc h as the minimum cut linear arrangemen t, often as part of d ivi de-and-conquer 1 F or any set S ⊆ V , we use ¯ S to denote its complemen t set, V \ S . 2 A function f is monotone if f ( S ) ≤ f ( T ) whenever S ⊆ T . 2 sc hemes. The SML problem can mo del a scenario in wh ic h costs follo w economies of s cale, and a certain num b er of items h a s to b e b ough t at the minimum total cost. An example app lic ation of SLB is compr e s sing and storing ﬁ les on m u lt ip le hard dr iv es or serv ers in a load-balanced wa y . Here the size of a compressed collection of ﬁles ma y b e m u c h s m a ller th a n th e s u m of individual ﬁle sizes, and mo deling it by a monotone submo dular function is r easonable considering that the en tropy function is kno w n to b e m o n ot one and s ubmod ular [10]. 1.2 Related w ork Because of the relation of subm odularity t o cut fu nctio n s and matroid rank fun cti ons , and their ex- hibition o f decreasing marginal r e tu rns, t h ere has been substan tial in terest in optimization problems in volving submo dular functions. Find ing the set that has the minimum function v alue is a wel l- studied problem that w as ﬁrst sho wn to b e p olynomially solv able using the ell ip soi d metho d [1 5 , 16]. F ur ther r ese arch has yielded sev eral more combinatorial appr o ac hes [9, 20 – 22, 24, 32, 33, 35]. Submo dular functions arise in facilit y lo catio n and assignmen t problems, and this has spa w n ed in terest in the problem of ﬁnding the set with the maximum fun ct ion v alue. Since this is NP- hard, researc h has fo cused on appro ximation algorithms for maximiz in g mon otone or non-monotone submo dular functions, p erh a p s sub ject to cardinalit y or other constrain ts [3, 8, 25 – 27, 31, 36]. A general ap p roac h for deriving inappro ximabilit y results for s u c h m aximization problems is presented in [40]. Researc h on other optimization problems that in v olv e submo dular fun ct ions includ e s [4, 5, 18 , 38, 39, 41]. Zhao et al. [42] study a submo dular multiw a y p a rtition p roblem, whic h is similar to our SLB problem, except that the subsets are required to b e non-empt y and the ob jectiv e is th e sum of function v alues on the subsets, as opp osed to th e maxim um. Subsequent to the p ublicati on of the preliminary v ersion of this pap er, generaliza tions of other combinatorial p roblems to submo dular costs hav e b een deﬁ n ed, with upp er and lo w er b ounds deriv ed for them. These include the set co v er problem and its sp ecial cases v ertex cov er and edge co v er, studied in [23], as w ell as vertex co v er, shortest p at h , p erfect matc hing, and spann ing tree studied in [12]. In [12], extensions to the case of multiple agen ts (with d iﬀe r en t cost fu nctio n s) are also considered . Since it is impossib le to learn a general submo dular function exac tly without looking at the fun c- tion v alue on all (exp onen tially many) subsets [7], there h a s b een r ec ent in terest in appro ximating submo dular functions ev erywhere with a p olynomial num b er of v alue oracle queries. Go emans et al. [13] giv e an algorithm that app ro ximates an arbitrary monotone submo dular function to a facto r γ ( n ) = O ( √ n log n ), and ap p ro ximates a rank fu ncti on of a m a tr oid to a factor γ ( n ) = √ n + 1. A lo we r b ound of Ω  √ n ln n  for this problem on monotone fu nctio n s and an impro ved lo wer b ound of Ω  p n ln n  for non-monotone fu nctio n s we r e obtained in [13, 14 ]. These lo we r b ounds apply to all algorithms that m a ke a p olynomial num b er of v alue-oracle queries. All of the optimizat ion problems th a t we consider in this pap er are k n o wn to b e NP-hard ev en when the ob jectiv e function can b e expr essed compactly as a lin ear or graph-cut function. While there is an FPT AS for the minimum kn a p sac k p roblem [11], the b est approximat ion for load balancing on un iform mac hines is a PT AS [19], and on un r ela ted mac h ines the b est p ossible u pp er and lo wer b ounds are constants [29]. The b est app ro ximation known for the sparsest cut problem is O ( √ log n ) [1, 2 ], and the b al anced cut problem is approximable to a factor of O (log n ) [34]. F or the sp ecial case of SML on graphs , introdu ce d in [37], an O (log n ) appro ximation is p ossible using the recent results of R¨ ac k e [34]. 3 1.3 Our results and tec hniques W e establish upp er and lo wer b ounds for the appr o x im ab ility of the pr oblems li sted ab o ve. Surp ris- ingly , these factors are qu ite high. Whereas the corresp onding classical pr oblems are appro ximable to c ons t an t or log arithm ic factors, the guaran tees that we p ro v e for most of our algorithms are of the order of p n ln n . W e show that this is the inherent diﬃculty of these prob le m s by proving matc h ing (or, in some cases, almost matc hing) lo we r b ounds. Our low er b ounds are u nconditional , and rely on the diﬃculty of distinguishing d iﬀe ren t subm odular functions by p erforming only a p olynomial n u m b er of queries in the oracle mo del. T h e pr oofs are based on the tec h niques in [8, 13]. T o pro ve the upp er b ounds, w e pr ese n t randomized approxima tion algorithms which use their randomness for samp li ng subsets of the ground set of elements. W e sh o w that w it h r e lative ly high probability (in verse p olynomial) , a sample can b e ob tained s uc h that its o verlap with th e optimal set is sig- niﬁcan tly higher than exp ected. Using the samp les, the algorithms employ subm odular fun ct ion minimization to ﬁnd candidate solutions. Th is is done in suc h a wa y that if the sample do es indeed ha ve a large o v erlap w ith the optimal set, then the solution satisﬁes the algorithm’s guarantee . F or S SC and u niform SLB, w e sh o w that they can b e appr o x im ated to a Θ  p n ln n  factor. F or SBC, w e use the w eighte d SSC as a subroutine, which allo ws us to obtain a bicriteria appro ximation in a similar w a y as Leigh ton and Rao [28] d o for graph s. F or SML, we also consider b icriteria results. F or ρ ≥ 1 and 0 < σ ≤ 1, a ( ρ, σ )-appro x im ation for SML is an algorithm that outputs a set S suc h that f ( S ) ≤ ρB and w ( S ) ≥ σ W , w henev er the input instance conta in s a set U w it h f ( U ) ≤ B and w ( U ) ≥ W . W e presen t a lo wer b ound sho win g that there is no ( ρ, σ ) appr o ximation for any ρ and σ with ρ σ = o  p n ln n  . F or 0-1 we igh ts, w e obtain a  5 p n ln n , 1 2  appro ximation. This algorithm can b e used to ob tain an O ( √ n ln n ) approxima tion for n on-uniform S LB. W e brieﬂy note here that one can consider th e prob lem of minimizing a submo dular function with an upp er b ound on card in a lit y (i.e., minimize f ( S ) sub ject to | S | ≤ W ). F or this problem, a ( 1 α , 1 1 − α ) b icrite r ia approximati on is p ossible for an y 0 < α < 1, u s ing tec hniqu es in [17]. F or non-bicriteria algorithms, a hardness resu lt of Ω  p n ln n  follo ws by reduction from S ML, using the submo dular function ¯ f , deﬁned as ¯ f ( S ) = f ( ¯ S ), and a cardinalit y b ound W = n − W . F or app ro ximating monotone s ubmod ular fu nctio n s ev ery w here, our lo we r b ound is Ω  p n ln n  , whic h impro ves the b o und for monotone functions in [13, 14], and matc hes the lo w er b ound for arbitrary submo dular f unctions, also in [13, 14]. Our lo we r b ound pr oof f or this p roblem, as well as the earlier ones, use 2P functions, and thus still hold for this sp ecial case. W e sho w that monotone 2P fu nctio n s can b e appro ximated w ith in a factor O ( √ n ). Besides lea ving a relativ ely small gap b et w een the upp er and lo w er b ounds, this sho ws that if m uch stronger lo wer b ounds for the appr o ximation problem exist, they rely on more general submo dular functions. F or th e prob lems studied in this pap er, our lo w er b ounds show th e imp ossibilit y of constan t or ev en p olylogarithmic app ro ximations in the v alue oracle mo del. This means that in ord e r to obtain b etter results for sp eciﬁc applications, one has to r e sort to more restricted mo dels, av oiding the full generalit y of arbitrary subm odular functions. 2 Preliminaries In the analysis of our algorithms, we rep eatedly u se the facts that the su m of s ubmod ular fu nctio n s is submo dular, and that sub m odular functions can b e m inimize d in p olynomial time. F or example, this allo w s us to minimize (o v er T ⊆ V ) expr essio n s lik e f ( T ) − α · | T ∩ S | , where α is a constan t 4 and S is a ﬁxed subset of V . W e present our algorithms b y p ro viding a r andomize d r elaxe d de cision pr o c e dur e f o r eac h of the problems. Given an instance of a min imiz ation problem, a target v alue B , and a probabilit y p , this pro cedure either d ec lares that the p roblem is in fea sib le (outpu ts fail ) , or ﬁnds a solution to the instance w it h ob jectiv e v alue at most γ B , where γ is the approximati on factor. W e sa y that an instance is feasible if it has a solution with cost strictly less than B (we use strict inequalit y for tec hnical reasons; this can b e a v oided b y addin g a small v alue ε > 0 to B ). The guaran tee pro vid e d with eac h decision pro cedure is that for any feasible instance, it outputs a γ -appr o x im ate solution with probabilit y at least p . On an infeasible instance, either of t he t wo outco m es is allo wed. Randomized relaxed decision p r ocedures can b e turned into randomized app ro ximation algorithms b y ﬁnding u pp e r and lo wer b ounds for the optimum and p erforming binary searc h. Ou r algorithms run in time p olynomial in n and ln 1 1 − p . Let us sa y that an algorithm distinguishes tw o functions f 1 and f 2 if it pro duces diﬀerent o utput if giv en (an oracle for) f 1 as input than if giv en (an oracle for) f 2 . T he follo w ing result is used for obtaining all of our low er b ound s . Lemma 2.1 L et f 1 and f 2 b e two set functions, with f 2 , bu t not f 1 , p ar ametrize d by a string of r andom bits r . If for any set S , chosen without know le dge of r , the pr ob ability (over r ) that f 1 ( S ) 6 = f 2 ( S ) is n − ω ( 1) , then any algorithm that makes a p olynomial numb er of or acle queries has pr ob ability at most n − ω ( 1) of distinguishing f 1 and f 2 . Pro of. W e use r ea son in g similar to [8]. Consider ﬁ rst a d et erm inistic algorithm and the compu- tation path that it follo ws if it receiv es the v alues of f 1 as ans w ers to all its oracle queries. Note that this is a single compu ta tion p a th that do es not dep end on r , b ecause f 1 do es not dep end on r . On this path the algorithm mak es some p olynomial n u m b er of oracle qu eries, say n a . Usin g the union b ound, we kn ow that the p r obabilit y that f 1 and f 2 diﬀer on an y of these n a sets is at most n a · n − ω ( 1) = n − ω ( 1) . S o , with probability at least 1 − n − ω ( 1) , if give n either f 1 or f 2 as inp u t, the algorithm only queries sets for whic h f 1 = f 2 , and therefore sta ys on the same computation p a th, pro ducing the same answer in b oth cases. A r andomized algorithm can b e viewed as a distribu tion o ve r a set of deterministic algorithms. Since, by the discussion abov e, eac h of these deterministic algorithms has probabilit y at most n − ω ( 1) of distinguishing f 1 and f 2 , the randomized alg orithm as a whole also h as probabilit y at most n − ω ( 1) of distinguish ing these tw o functions.  The follo wing theorem ab o ut random samp ling is used for bou n ding probabilities in the analyses of our algorithms. W e use the constant c = 1 / (4 √ 2 π ) throughout the pap er. Theorem 2.2 Supp ose that m elements ar e sele cte d indep endently, with pr ob ability 0 < q < 1 e ach. Then for 0 ≤ ε < 1 − q q , the pr ob ability that exactly ⌈ q m (1 + ε ) ⌉ elements ar e sele cte d is at le ast cq · m − 3 2 · exp h − ε 2 q m 1 − q i . Pro of. Let λ = q m (1 + ε ). First we consider th e case that λ is integ er. F or con venience , let κ = q (1 + ε ), and n o te that κ < 1. Using an appro ximation that √ 2 π n  n e  n ≤ n ! ≤ 2 √ 2 π n  n e  n , whic h is d eriv ed from Stirling’s f o r m ula [6, p. 55], we obtain the b oun d  m mκ  = m ! ( mκ )!( m − mκ )! ≥ √ 2 π (2 √ 2 π ) 2 · √ m √ mκ √ m − mκ · ( m/e ) m ( mκ/e ) mκ (( m − mκ ) /e ) m − mκ 5 ≥ 1 4 √ 2 π · 1 √ m · 1 κ mκ (1 − κ ) m − mκ . Let X b e the num b er of elemen ts selected in the random exp erimen t. Then Pr[ X = mκ ] =  m mκ  q mκ (1 − q ) m − mκ ≥ c √ m · q mκ · (1 − q ) m − mκ κ mκ · (1 − κ ) m − mκ = c √ m ·  1 1 + ε  mκ ·  1 − q 1 − q (1 + ε )  m − mκ = c √ m · 1 (1 + ε ) mκ · 1  1 − εq 1 − q  m − mκ ≥ c √ m · exp  − εmκ + εq 1 − q m (1 − κ )  , where w e ha ve used the inequalit y that 1 + x ≤ e x for all x . The assumption that ε < 1 − q q ensures that the denominator 1 − q (1 + ε ) is p ositiv e. No w, the exp onen t of e is equal to − εq m (1 + ε ) + εq 1 − q m (1 − q − εq ) = − εq m − ε 2 q m + εq m − ε 2 q 2 m 1 − q = − ε 2 q m 1 − q . Noting that c · m − 1 2 ≥ cq · m − 3 2 concludes the p roof f or the case th a t λ is inte ger. If λ is fractional, th en ⌈ λ ⌉ = ⌊ λ ⌋ + 1. Then Pr[ X = ⌈ λ ⌉ ] Pr[ X = ⌊ λ ⌋ ] =  m ⌊ λ ⌋ +1  q ⌊ λ ⌋ +1 (1 − q ) m −⌊ λ ⌋− 1  m ⌊ λ ⌋  q ⌊ λ ⌋ (1 − q ) m −⌊ λ ⌋ = ( m − ⌊ λ ⌋ ) q ( ⌊ λ ⌋ + 1) (1 − q ) . (1) As ε ≥ 0, w e ha ve λ ≥ q m . No w consider the case that ⌊ λ ⌋ ≤ q m . As q m is th e exp ect ation of X , either ⌈ λ ⌉ or ⌊ λ ⌋ is the most lik ely v alue of X , ha ving probabilit y of at least 1 m +1 . I n the ﬁrst case, Pr[ X = ⌈ λ ⌉ ] ≥ 1 m +1 ≥ c m , and we are done. In the second case, usin g sequen tially (1), ⌊ λ ⌋ ≤ q m , and ⌊ λ ⌋ + 1 = ⌈ λ ⌉ ≤ m (wh ic h is implied b y κ < 1 ab o v e), we obtain the r esult: Pr[ X = ⌈ λ ⌉ ] ≥ 1 m + 1 · ( m − ⌊ λ ⌋ ) q ( ⌊ λ ⌋ + 1) (1 − q ) ≥ 1 m + 1 · mq ⌊ λ ⌋ + 1 ≥ cq m . The remaining case is that ⌊ λ ⌋ > q m . Deﬁn e ε ′ > 0 to b e such that q m (1 + ε ′ ) = ⌊ q m (1 + ε ) ⌋ = ⌊ λ ⌋ . Note that ε ′ ≤ ε . Applying the pro of that we u sed f o r int eger λ , we obtain that Pr[ X = ⌊ λ ⌋ ] ≥ c √ m · exp  − ε ′ 2 q m 1 − q  ≥ c √ m · exp  − ε 2 q m 1 − q  , where we also used monotonicit y of the exp onen tial f unction. Using the fact that ⌊ λ ⌋ ≤ m − 1, w e simplify equation (1) to obtain that Pr [ X = ⌈ λ ⌉ ] / Pr[ X = ⌊ λ ⌋ ] ≥ q m . T ogether with the ab o v e inequalit y , this gives the desired result.  6 3 Submo dular sparsest cut and submo dular balanced cut 3.1 Lo wer b ounds Let ε > 0 b e suc h that ε 2 = 1 n · ω (ln n ), let β = n 4 (1 + ε ), and let R b e a subs e t of V of size n 2 , w it h parameters suc h that n is ev en and β is an int eger. W e d eﬁne the follo wing t wo functions, and sho w that they are submo dular and hard to distinguish . Moreo v er, these functions are sym m et r ic 3 . f 1 ( S ) = min  | S | , n 2  − | S | 2 f 2 ( S ) = min  | S | , n 2 , β + | S ∩ R | , β + | S ∩ ¯ R |  − | S | 2 Lemma 3.1 F unctions f 1 and f 2 deﬁne d ab ove ar e nonne gative, submo dular, and symmetric. Pro of. The ﬁrst f unction can b e w ritte n as f 1 ( S ) = 1 2 min( | S | , | ¯ S | ), whic h mak es it easy to see that it is nonnegativ e and symmetric. It su ﬃce s to sho w that f ( S ) = min( | S | , n 2 ) is sub modu la r , since − | S | 2 is mo dular 4 . W e use an alternativ e deﬁnition of submo dularit y : f is submo dular if for all S ⊂ V and a, b ∈ V \ S , with a 6 = b , it holds that f ( S ∪ { a, b } ) − f ( S ∪ { b } ) ≤ f ( S ∪ { a } ) − f ( S ). The only w a y that this inequalit y can b e violated for our function is if f ( S ∪ { a, b } ) − f ( S ∪ { b } ) = 1 and f ( S ∪ { a } ) − f ( S ) = 0. But this is a con tradiction, since the second part implies that | S | ≥ n/ 2, and the ﬁ r st one implies that | S ∪ { b }| < n/ 2. T o see th a t f 2 ( S ) is nonnegativ e, w e note that β + | S ∩ R | − | S | 2 ≥ n 4 + | S ∩ R | − | S ∩ R | 2 − | S ∩ ¯ R | 2 ≥ 0, since | S ∩ ¯ R | ≤ n 2 . A similar calculation shows that β + | S ∩ ¯ R | − | S | 2 ≥ 0, and thus f 2 ( S ) ≥ 0 for all S . T o sho w symmetry , we use the fact that | R | = n 2 , and thus | S ∩ R | − | S | 2 = n 2 − | ¯ S ∩ R | − | S | 2 = | ¯ S | 2 − | ¯ S ∩ R | = − | ¯ S | 2 + | ¯ S | − | ¯ S ∩ R | = | ¯ S ∩ ¯ R | − | ¯ S | 2 . Analogously , | S ∩ ¯ R | − | S | 2 = | ¯ S ∩ R | − | ¯ S | 2 . T h us, we ha ve th a t f 2 ( S ) = min  | S | 2 , | ¯ S | 2 , β + | S ∩ R | − | S | 2 , β + | S ∩ ¯ R | − | S | 2  = min  | S | 2 , | ¯ S | 2 , β + | ¯ S ∩ ¯ R | − | ¯ S | 2 , β + | ¯ S ∩ R | − | ¯ S | 2  = f 2 ( ¯ S ) . F or su bmod ularit y of f 2 , w e fo cus only on f ( S ) = m in  | S | , n 2 , β + | S ∩ R | , β + | S ∩ ¯ R |  . Sup- p ose for the sak e of con tradiction that f o r some a, b ∈ V , we ha ve f ( S ∪ { a, b } ) − f ( S ∪ { b } ) = 1 but f ( S ∪ { a } ) − f ( S ) = 0. W e assume th a t a ∈ R (the case that a ∈ ¯ R is s imil ar). First consider the case th a t b is also in th e set R . In this ca se f ( S ∪ { a } ) = f ( S ∪ { b } ). The fact that the function v alue d oes not in crea se w hen a ∈ R is added to S means that the minimum is ac h ie ved b y one of the terms that do not dep end on | S ∩ R | , namely f ( S ) = min( n 2 , β + | S ∩ ¯ R | ). But then the minim u m would also n o t increase when the second elemen t of R is added, and we w ould hav e f ( S ∪ { a, b } ) = f ( S ∪ { b } ), cont r a dicting the assump tio n . The remaining case is that a ∈ R and b ∈ ¯ R . As b efore, f ( S ) = min( n 2 , β + | S ∩ ¯ R | ). But if f ( S ) = n 2 , then f ( S ∪ { a, b } ) = n 2 , whic h con tradicts our assump tio n s. So f ( S ) = β + | S ∩ ¯ R | . Now, 3 A function f is symmetric if f ( S ) = f ( ¯ S ) for all S . 4 A mo dular fun ctio n is one for which the submo dular ineq u al ity is satisﬁed with equality . 7 f ( S ∪ { b } ) increases from the addition of a ∈ R , whic h means th a t its minim u m is ac hiev ed by a term that dep ends on | S ∩ R | : f ( S ∪ { b } ) = min( | S | + 1 , β + | S ∩ R | ). Supp ose that f ( S ∪ { b } ) = | S | + 1. T h is means that | S | + 1 ≤ β + | ( S ∪ { b } ) ∩ ¯ R | = β + | S ∩ ¯ R | + 1. But w e also kno w that β + | S ∩ ¯ R | ≤ | S | (from th e f a ct that f ( S ) = β + | S ∩ ¯ R | ). T h us, | S | = β + | S ∩ ¯ R | an d f ( S ∪ { b } ) = β + | S ∩ ¯ R | + 1 = β + | ( S ∪ { b } ) ∩ ¯ R | . But this term do es not dep end on | S ∩ R | , so adding a ∈ R to S ∪ { b } wo u ld n ot change the function v alue, a contradict ion. Finally , su pp o s e that f ( S ∪ { b } ) = β + | S ∩ R | . As f ( S ) = β + | S ∩ ¯ R | , w e kn o w that β + | S ∩ ¯ R | ≤ | S | , and therefore β ≤ | S ∩ R | . So f ( S ∪ { b } ) = β + | S ∩ R | ≥ 2 β > n 2 , by the d eﬁnition of β . But this is a con tradiction, as the v alue of f is alwa ys at most n 2 .  T o giv e a lo wer b ound for SSC and S BC, we p ro v e th e follo wing r esu lt and then apply Lemma 2.1 to sho w that the fu ncti ons f 1 and f 2 ab o v e are hard to distinguish. Lemma 3.2 Fix an arbitr ary subset S ⊆ V , and then let R b e a r andom subset of V of size n 2 . Then the pr ob ability (over the choic e of R ) that f 1 ( S ) 6 = f 2 ( S ) is at most n − ω ( 1) . Pro of. W e note that f 1 ( S ) 6 = f 2 ( S ) if and only if min( β + | S ∩ R | , β + | S ∩ ¯ R | ) < min( | S | , n 2 ). This happ ens if either β + | S ∩ R | < min( | S | , n 2 ) or β + | S ∩ ¯ R | < min( | S | , n 2 ). The probabilities of these t wo ev ents are equal, so let us d enote one of them by p ( S ). If we sho w that p ( S ) = n − ω ( 1) , then the lemma follo ws by an application of the un io n b ound. First, we claim that p ( S ) is maximized wh en | S | = n 2 . F or this, supp ose that | S | ≥ n 2 . Then p ( S ) = Pr[ β + | S ∩ R | < n 2 ]. But this p robabilit y can only increase if an element is r e m o ved from S . Similarly , in the case that | S | ≤ n 2 , p ( S ) = P r[ β + | S ∩ R | < | S | ] = Pr[ β < | S ∩ ¯ R | ]. But th is probabilit y can only increase if an elemen t is added to S . F or a set S of size n 2 , p ( S ) = Pr[ β + | S ∩ R | < n 2 ] = Pr[ | S ∩ R | < n 4 (1 − ε )]. If instead of c ho osing R as a random s ubset of V of size n 2 , w e consider a set R ′ for w hic h eac h elemen t is c hosen indep endent ly with probabilit y 1 2 , then p ( S ) b ecomes p ( S ) = Pr h | S ∩ R ′ | < n 4 (1 − ε )    | R ′ | = n 2 i = Pr  | S ∩ R ′ | < n 4 (1 − ε ) ∧ | R ′ | = n 2  Pr  | R ′ | = n 2  ≤ ( n + 1) · Pr h | S ∩ R ′ | < n 4 (1 − ε ) i . This allo ws us to make a switc h to indep enden t v ariables, so that we can us e Chernoﬀ b ounds [30]. The exp ectatio n µ of | S ∩ R ′ | is equ al to | S | / 2 = n / 4, so Pr  | S ∩ R ′ | < (1 − ε ) µ  < e − µε 2 / 2 = e − ω ( l n n ) = n − ω ( 1) , remem b ering that ε 2 = 1 n · ω (ln n ). This gives p ( S ) ≤ ( n + 1) · n − ω ( 1) = n − ω ( 1) .  Corollary 3.3 Any algorithm that makes a p olynomial numb er of or acle queries has pr ob ability at most n − ω ( 1) of distinguishing the functions f 1 and f 2 . W e now use these results to establish the hardness of the SSC and SBC pr oblems. F or concrete- ness, assume that the outp ut of an appr o ximatio n algorithm for on e of th ese problems consists of a set S ⊆ V as well as the v alue of the ob j ective fun ct ion on this set. 8 Theorem 3.4 The uniform SSC and the unweighte d SBC pr oblems (with b alanc e b = Θ(1) ) c annot b e appr oximate d to a r atio o  p n ln n  in the or acle mo del with p olyno mial numb er of queries, even in the c ase of symmetric functions. Pro of. Supp ose for the sak e of con tradiction th a t there is a p olynomial-time γ -appro ximation algorithm for the uniform SS C pr o b lem, for some γ = o  p n ln n  , that succeeds with high pr obabilit y . W e set ε = 1 2 γ δ with some δ > 1 such that β = n 4 (1 + ε ) is inte ger. Th is satisﬁes ε 2 = 1 n · ω (ln n ). One feasible solution for the uniform SS C on f 2 is the set R , with ratio β − n/ 4 n 2 / 4 = ε n . So if the algorithm is giv en function f 2 as input, then with high probabilit y it h a s to output a set S with r a tio f 2 ( S ) / | S || ¯ S | ≤ γ ε n = 1 2 δn < 1 2 n . Ho wev er, for the function f 1 , the ratio of any s e t is 1 / 2 max( | S | , | ¯ S | ) > 1 2 n . So if the algorithm is giv en f 1 as input, its output v alue diﬀers from the case of f 2 . But this con trad icts Corollary 3.3. F or the lo wer b ound to the s u bmod ular balanced cut p roblem, we consider the same t wo func- tions f 1 and f 2 and unit we ights. Assu ming that there is a γ -approxima tion algorithm for S BC, w e set ε = 2 b δγ , with γ > 1 ensuring the in tegralit y of β . Th is satisﬁes ε 2 = 1 n · ω (ln n ) if γ = o  p n ln n  and b is a constant. Sin ce one feasible b -balanced cut on f 2 is the set R , wh ose function v alue is nε 4 , the algorithm outputs a b -balanced set S with f 2 ( S ) ≤ γ nε/ 4 = bn/ 2 δ < bn/ 2. How ev er, for an y b , the optimal b -balanced cut on f 1 is a set of size bn , whose function v alue is bn/ 2. Thus, give n f 1 , the algorithm w ould pr oduce a d iﬀe r en t output, leading to a con tradiction.  3.2 Algorithm for submodular sparsest cut Our algorithm for SSC uses a random set S to assign w eigh ts to n odes (see Algorithm 1). F or eac h demand pair separated by the set S , w e add a p ositiv e weig ht equal to its d e mand d i to the no de that is in S , and a negativ e weigh t of − d i to the no de that is outsid e of S . This b ia s e s the subsequent f u nctio n minimization to separate th e demand pairs that are on diﬀerent sides of S . Algorithm 1 Submo dular sparsest cut. Input: V , f , d , B , p 1: for 8 n 3 c ln( 1 1 − p ) iterations do 2: Cho ose a random set S by including eac h no de v ∈ V in dep en den tly w it h prob ab ility 1 2 3: for eac h v ∈ V , in it ialize a w eigh t w ( v ) = 0 4: for eac h pair { u i , v i } with |{ u i , v i } ∩ S | = 1 do 5: Let s i ∈ { u i , v i } ∩ S and t i ∈ { u i , v i } \ S ⊲ name the uniqu e no de in eac h set 6: Up date w eights w ( s i ) ← w ( s i ) + d i ; w ( t i ) ← w ( t i ) − d i 7: end for 8: Let α = 4 p n ln n · B 9: Let T b e a s u bset of V min imizi n g f ( T ) − α · P v ∈ T w ( v ) 10: if f ( T ) − α · P v ∈ T w ( v ) < 0, return T 11: end for 12: return fail Lemma 3.5 If for some set T ⊆ V , it hold s that f ( T ) − α · P v ∈ T w ( v ) < 0 , then f ( T ) P i : | T ∩{ u i ,v i }| =1 d i < α. 9 Pro of. W e hav e X v ∈ T w ( v ) = X i : s i ∈ T d i − X i : t i ∈ T d i = X i : s i ∈ T , t i / ∈ T d i − X i : t i ∈ T , s i / ∈ T d i ≤ X i : s i ∈ T , t i / ∈ T d i ≤ X i : | T ∩{ u i ,v i }| =1 d i No w using the assumption of the lemma we ha ve f ( T ) − α X i : | T ∩{ u i ,v i }| =1 d i ≤ f ( T ) − α X v ∈ T w ( v ) < 0 . (2) Since th e function f is non-negativ e, it must b e that P i : | T ∩{ u i ,v i }| =1 d i > 0. Rearranging the terms, w e get f ( T ) / P i : | T ∩{ u i ,v i }| =1 d i < α .  Assuming that the inpu t in sta n ce is feasible, let U ∗ b e a set w it h size m = | U ∗ | , separated demand D ∗ = P i : | U ∗ ∩{ u i ,v i }| =1 d i , and v alue f ( U ∗ ) /D ∗ < B . Lemma 3.6 In one iter ation of the outer lo op of A lgorithm 1, the pr ob ability that P v ∈ U ∗ w ( v ) ≥ D ∗ · 1 4 q ln n n is at le ast c 8 n 3 . Pro of. Let ε = q ln n n . W e denote b y A the eve nt that | U ∗ ∩ S | ≥ m 2 (1 + ε ), w h ere S is the random set c hosen by Algorithm 1, and b ound the ab o ve probabilit y by the follo wing p r oduct: Pr " X v ∈ U ∗ w ( v ) ≥ ε 4 D ∗ # ≥ Pr " X v ∈ U ∗ w ( v ) ≥ ε 4 D ∗    A # · Pr[ A ] . W e observ e th a t b y Th eorem 2.2, the probability of A is at least c 2 n − 5 / 2 . All th e pr ob ab ilities and exp ecta tions in the rest of the p roof are cond itioned on the ev ent A . Let u s n o w consider the exp ected v alue of P v ∈ U ∗ w ( v ). Fix a particular demand pair { u i , v i } that is separated by the optimal solution, and assume without loss of generalit y that u i ∈ U ∗ and v i / ∈ U ∗ . Let p u b e the p r obabilit y that u i ∈ S , and p v b e the probabilit y that v i ∈ S . Th en p u = | U ∗ ∩ S | | U ∗ | ≥ (1 + ε ) / 2, p v = 1 2 , and the t wo ev ents are ind epend en t. S o Pr[ u i = s i ] = Pr[ u i ∈ S ∧ v i / ∈ S ] = p u · (1 − p v ) ≥ (1 + ε ) / 4 , Pr[ u i = t i ] = Pr[ u i / ∈ S ∧ v i ∈ S ] = (1 − p u ) · p v ≤ (1 − ε ) / 4 . Then the exp ected contribution of this d ema nd p air to P v ∈ U ∗ w ( v ) is equal to Pr[ u i = s i ] · d i + Pr[ u i = t i ] · ( − d i ) ≥ d i · ε 2 . By linearit y of exp ect ation, E " X v ∈ U ∗ w ( v ) # ≥ D ∗ · ε 2 . W e no w use Mark o v’s inequalit y [30] to b ound the desired probabilit y . F or th is w e d eﬁne a non- negativ e random v ariable Y = D ∗ − P v ∈ U ∗ w ( v ). Then E[ Y ] ≤ (1 − ε/ 2) D ∗ . S o Pr " X v ∈ U ∗ w ( v ) ≤ ε 4 D ∗ # = Pr h Y ≥ (1 − ε 4 ) D ∗ i ≤ E[ Y ] (1 − ε/ 4) D ∗ ≤ 1 − ε/ 2 1 − ε/ 4 = 1 − ε 4 − ε ≤ 1 − ε 4 10 It follo ws that Pr " X v ∈ U ∗ w ( v ) ≥ ε 4 D ∗ # ≥ ε 4 = 1 4 r ln n n ≥ 1 4 √ n , concluding the pro of of the lemma.  Theorem 3.7 F or any fe asible instanc e of SSC pr oblem, Algor i thm 1 r eturns a solution of c ost at most 4 p n ln n · B , with pr ob ability at le ast p . Pro of. B y Lemma 3.6, the inequalit y P v ∈ U ∗ w ( v ) ≥ D ∗ · 1 4 q ln n n holds with probability at least c/ 8 n 3 in eac h iteration. Then the probab ility that it h olds in an y of the 8 n 3 c ln( 1 1 − p ) iterations is at least p . No w, assuming that it do es hold, the algorithm ﬁn ds a set T suc h that f ( T ) − α · X v ∈ T w ( v ) ≤ f ( U ∗ ) − α · X v ∈ U ∗ w ( v ) ≤ f ( U ∗ ) −  4 r n ln n · B  D ∗ · 1 4 r ln n n ! < 0 . Applying Lemma 3.5, we get that f ( T ) / P i : | T ∩{ u i ,v i }| =1 d i < α = 4 p n ln n · B , whic h m eans that T is the r equired appro ximate s o lu ti on .  3.3 Submodular balanced cut F or s u bmod ular b al anced cut, w e use as a subroutine the weigh ted SSC problem that can b e appro ximated to a factor γ = O  p n ln n  using Algorithm 1 . Th is allo ws us to obtain a bicriteria appro ximation for SBC in a similar wa y that Leight on and Rao [28] use their algorithm for sparsest cut on graphs to appro ximate balanced cut on graph s. Leigh ton and Rao present tw o versions of an algo r ithm f o r the balanced cut problem on graphs — one for undir ec ted graphs, and one for directed graphs. The algorithm for u ndirected graph s has a b etter balance guaran tee. W e describ e adaptations of these algorithms to the submo dular v ersion of the balanced cut problem. Our ﬁr st algorithm extends the one for u ndirected graphs, and it w orks for symmetric s ubmod ular fu nctio n s. F or a giv en b ′ ≤ 1 / 3, it ﬁnds a b ′ -balanced cut whose cost is within a factor O  γ b − b ′  of th e cost of any b -balanced cut, for b ′ < b ≤ 1 2 . Th e second algo r ithm w orks for arbitrary non-negativ e submo dular functions and pro duces a b ′ / 2-balanced cut of cost w ith in O  γ b − b ′  of any b -balanced cut, for any b ′ and b with b ′ < b ≤ 1 / 2. 3.3.1 Algorithm for symmetric functions The algorithm for SBC on symmetric f unctions (Algorithm 2) r ep eatedly ﬁnds approximat e w eigh ted submo dular sp arsest cuts ( S i , ¯ S i ) and collects their smaller sides into th e set T , unt il ( T , ¯ T ) b ec omes b ′ -balanced. The algorithm and analysis basically follo w Leighto n and Rao [28], with the main dif- ference b eing that instead of r e m o v in g parts of the graph, we set the wei ghts of th e corresp onding elemen ts to zero. Then the obtained sets S i are not necessarily disjoint. Theorem 3.8 If the system ( V , f , w ) , wher e f is a symmetric submo dular function, c ontains a b -b alanc e d cut of c ost B , then Algorithm 2 ﬁnds a b ′ -b alanc e d cut T with f ( T ) = O  B b − b ′ p n ln n  , for a give n b ′ < b , b ′ ≤ 1 3 . 11 Algorithm 2 Submo dular balanced cut for symm e tr ic f u nctio n s. Inpu t : V , f , w , b ′ ≤ 1 3 1: I n iti alize w ′ = w , i = 0, T = ∅ 2: w hile w ′ ( V ) > (1 − b ′ ) w ( V ) do 3: Let S b e a γ -approximat e we ighted SS C on V , f , and weigh ts w ′ 4: Let S i = argmin( w ′ ( S ) , w ′ ( ¯ S )); w ′ ( S i ) ← 0; T ← T ∪ S i ; i ← i + 1 5: end while 6: ret urn T Pro of. The algorithm terminates in O ( n ) iterations, since the weigh t of at least one n ew elemen t is set to zero on line 4 (otherwise the solution to SS C found on line 3 w ould ha ve inﬁn ite cost). No w we consider w ( T ). By th e termin a tion condition of the while lo op, we know that when it exits, w ′ ( V ) ≤ (1 − b ′ ) w ( V ), w hic h means th a t w ′ has b een set to zero for elemen ts of total wei gh t at least b ′ w ( V ). But those are exactly the elemen ts in T , so w ( T ) ≥ b ′ w ( V ). Now consider the last iteration of the lo op. A t th e b eginning of this iteration, we h a v e w ′ ( V ) > (1 − b ′ ) w ( V ), w hic h means that at the end of it w e h a v e w ′ ( V ) > 1 2 (1 − b ′ ) w ( V ), b ecause the weig ht of the smaller (according to w ′ ) of S or ¯ S is set to zero. But w ′ ( V ) at the end of the algorithm is exactly the w eight of ¯ T , whic h means that w ( ¯ T ) > 1 2 (1 − b ′ ) w ( V ) ≥ 1 3 w ( V ) ≥ b ′ w ( V ), using the assumption b ′ ≤ 1 / 3 twic e. So th e cut ( T , ¯ T ) is b ′ -balanced. Supp ose that U ∗ is a b -b a lanced cut with f ( U ∗ ) = B . In any iteratio n i of the while lo op, w e kno w that t wo inequ alities h o ld : w ′ ( U ∗ ) + w ′ ( ¯ U ∗ ) > (1 − b ′ ) w ( V ) (b y the lo op condition), and max( w ′ ( U ∗ ) , w ′ ( ¯ U ∗ )) ≤ (1 − b ) w ( V ) (by b -balance). Given th e se in equali ties, the min imum v alue that the pr oduct w ′ ( U ∗ ) · w ′ ( ¯ U ∗ ) can hav e is ( b − b ′ ) w ( V ) · (1 − b ) w ( V ). So with we ights w ′ , there is a solution to the SSC pr o b lem with v alue f ( U ∗ ) w ′ ( U ∗ ) w ′ ( ¯ U ∗ ) ≤ B ( b − b ′ ) w ( V ) · (1 − b ) w ( V ) , and the set S i found by the γ -appr o x im ation algorithm satisﬁes f ( S i ) w ′ ( S i ) w ′ ( ¯ S i ) ≤ γ B ( b − b ′ ) w ( V ) · (1 − b ) w ( V ) . Since in iteration i , w ′ ( S i ) = w ( S i \ S i − 1 j =0 S j ), w ′ ( ¯ S i ) ≤ w ( V ), and (1 − b ) ≥ 1 / 2, f ( S i ) ≤ w ( S i \ i − 1 [ j =0 S j ) 2 B γ ( b − b ′ ) w ( V ) . No w f ( T ) ≤ P i f ( S i ) ≤ w ( T ) · 2 B γ / ( b − b ′ ) w ( V ) = B · O ( γ b − b ′ ).  3.3.2 Algorithm for general functions The algo r ithm for general fu nctio n s (Algorithm 3) also rep e atedly ﬁnds weigh ted subm odular spars- est cuts ( S i , ¯ S i ), but it uses them to collect tw o sets: either it p uts S i in to T 1 , or it pu t s ¯ S i in to T 2 . T h us, th e v alues of f ( T 1 ) and ¯ f ( T 2 ) can b e b ounded using the guarantee of the S SC algorithm (where ¯ f ( S ) = f ( ¯ S )). 12 Algorithm 3 S ubmod ular b a lanced cut. Inp u t: V , f , w , b ′ 1: I n iti alize w ′ = w , i = 0, T 1 = T 2 = ∅ 2: w hile w ′ ( V ) > (1 − b ′ ) w ( V ) do 3: Let S i b e a γ -appr o ximate w eight ed SSC on V , f , and weig hts w ′ 4: if w ′ ( S i ) ≤ w ′ ( ¯ S i ) then set T 1 ← T 1 ∪ S i ; w ′ ( S i ) ← 0; i ← i + 1 5: else set T 2 ← T 2 ∪ ¯ S i ; w ′ ( ¯ S i ) ← 0; i ← i + 1 6: end while 7: if w ( T 1 ) ≥ w ( T 2 ) then ret urn T 1 else return ¯ T 2 Theorem 3.9 If the system ( V , f , w ) c ontains a b -b alanc e d cut of c ost B , then Algo rithm 3 ﬁnds a b ′ / 2 -b alanc e d cut T with f ( T ) = O  B b − b ′ p n ln n  , for a given b ′ < b . Pro of. When the wh ile lo op exits, w ′ ( V ) ≤ (1 − b ′ ) w ( V ), so the total wei ght of elemen ts in T 1 and T 2 (the ones for whic h w ′ has been set to zero) is at least b ′ w ( V ). So max( w ( T 1 ) , w ( T 2 )) ≥ b ′ w ( V ) / 2. A t the b eginning of the last iteration of the lo op, w ′ ( V ) > (1 − b ′ ) w ( V ). Since the weig ht of the smaller of S i and ¯ S i is set to zero, at the end of this iteration w ′ ( V ) > 1 2 (1 − b ′ ) w ( V ). Let T b e the set output by the algorithm. Since w ′ ( T ) = 0, we ha ve w ( ¯ T ) ≥ w ′ ( V ) > 1 2 (1 − b ′ ) w ( V ) ≥ b ′ / 2, using b ′ ≤ 1 / 2. Thus w e ha ve sho wn that Algorithm 3 outputs a b ′ / 2-balanced cut. The fun ction v alues can b e b ounded as f ( T 1 ) = B · O ( γ b − b ′ ) and ¯ f ( T 2 ) = B · O ( γ b − b ′ ) using a pro of similar to that of Theorem 3.8.  4 Submo dular minimization with card i n al it y lo w er b ound W e start with the lo w er b ound resu lt . Let R b e a random subset of V of size α = x √ n 5 , let β = x 2 5 , and x b e an y parameter satisfying x 2 = ω (ln n ) and such that α and β are in teger. W e use the follo wing t wo monotone s u bmod ular functions: f 3 ( S ) = m in ( | S | , α ) , f 4 ( S ) = m in  β + | S ∩ ¯ R | , | S | , α  . (3) Lemma 4.1 Any algo rithm that makes a p olynomial nu m b er of or acle queries has pr ob ability n − ω ( 1) of distinguishing the functions f 3 and f 4 ab ove. Pro of. By Lemma 2.1, it suﬃces to pr o ve that for any set S , the probabilit y th a t f 3 ( S ) 6 = f 4 ( S ) is at most n − ω ( 1) . It is easy to chec k (similarly to the pr oof of Lemma 3.2) th at Pr[ f 3 ( S ) 6 = f 4 ( S )] is maximized for sets S of size α . And f o r a s e t S with | S | = α , f 3 ( S ) 6 = f 4 ( S ) if and only if β + | S ∩ ¯ R | < | S | , or, equiv alen tly , | S ∩ R | > β . So we analyze the p robabilit y that | S ∩ R | > β . R is a random sub set of V of size α . Let us consider a diﬀerent set, R ′ , w hic h is obtained by indep endent ly in cl u ding eac h elemen t of V with p robabilit y α/n . The exp ected size of R ′ is α , and the pr o babilit y that | R ′ | = α is at least 1 / ( n + 1). Th en Pr [ | S ∩ R | > β ] = Pr  | S ∩ R ′ | > β   | R ′ | = α  ≤ ( n + 1) · Pr  | S ∩ R ′ | > β  , and it su ﬃce s to s h o w that Pr [ | S ∩ R ′ | > β ] = n − ω ( 1) . F or this, w e u se Ch e rnoﬀ b ounds. The exp ecta tion of | S ∩ R ′ | is µ = α | S | /n = α 2 /n = x 2 / 25. Then β = 5 µ . Let δ = 4. Then Pr  | S ∩ R ′ | > (1 + δ ) µ  <  e δ (1 + δ ) 1+ δ  µ =  e 4 5 5  x 2 25 ≤ 0 . 851 x 2 . 13 Since x 2 = ω (ln n ), w e get that this probabilit y is n − ω ( 1) .  Theorem 4.2 Ther e is no ( ρ, σ ) bic r iteria appr oximation algorithm f o r the SML pr oblem, even with monotone fu nc tions, for any ρ and σ with ρ σ = o  p n ln n  . Pro of. W e assu me that any algorithm for this p roblem outpu ts a set of elemen ts as well as the function v alue on this set. Su pp ose that a b ic riteria algorithm with ρ σ = o  p n ln n  exists. Let f 3 and f 4 b e th e t wo monotone functions in (3), w it h x = σ √ n δρ , where δ > 1 is a constant that ensures that α and β are in teger. Then x satisﬁes x 2 = ω (ln n ). Consider the outpu t of the algorithm when giv en f 4 as inpu t and W = α . The optimal solution in this case is the set R , w it h f ( R ) = β . S o the algorithm ﬁ nds an app ro ximate solution T with f 4 ( T ) ≤ ρβ and | T | ≥ σ α . Ho w eve r , w e sh o w that no set S with f 3 ( S ) ≤ ρβ and | S | ≥ σ α exists, whic h means th a t if the input is th e function f 3 , then the algo rithm p rodu c es a diﬀerent answ er, thus distinguish ing f 3 and f 4 . W e assum e for contradict ion that suc h a set S exists and consider t wo cases. First, supp ose | S | ≥ α . Then f 3 ( S ) ≤ ρβ = σ √ n δx x 2 5 = σα δ < α , since δ > 1 and b y deﬁnition σ ≤ 1. But this is a con tradiction b ecause f 3 ( S ) = α for all S with | S | ≥ α . Th e second case is | S | < α . Th e n we ha ve | S | ≥ σ α and f 3 ( S ) ≤ ρβ = σα δ ≤ | S | /δ , whic h is also a con tradiction b ecause | S | ≥ σ α > 0 and f 3 ( S ) = | S | f or | S | < α .  4.1 Algorithm for SML Our relaxed decision pr ocedure for the SML p roblem with we ights { 0 , 1 } (Algo rithm 4 ) builds up the solution out of m u lti ple sets that it ﬁn d s u s ing sub modular function minimization. If the weig h t requirement W is large r than half the tota l w eight w ( V ) , then collect in g sets whose ratio of fu nctio n v alue to w eigh t of new elemen ts is lo w (less th a n 2 B /W ), until a total weigh t of at least W / 2 is collect ed , ﬁnds the required approximat e solution. In the other case, if W is less than w ( V ) / 2, the algorithm lo oks for sets T i with lo w ratio of function v alue to the wei ght of new elemen ts in the in tersection of T i and a random set S i . These sets not only hav e small f ( T i ) /w ( T i ) ratio, but also ha ve b ounded function v alue f ( T i ). If suc h a set is found , then it is added to the solution. Theorem 4.3 Algorithm 4 is a (5 p n ln n , 1 2 ) bicriteria de cisi on pr o c e dur e for the SML pr oblem. That is, giv e n a fe asible i nst anc e, it outputs a set U with f ( U ) ≤ 5 p n ln n B and w ( U ) ≥ W / 2 with pr ob ability at le ast p . Pro of. Assume that the instance is feasible, and let U ∗ ⊆ V b e a set with w ( U ∗ ) ≥ W and f ( U ∗ ) < B . W e consider t w o cases, W ≥ w ( V ) / 2 and W < w ( V ) / 2, whic h the algorithm handles separately . First, assume that W ≥ w ( V ) / 2 and consider one of the iterations of th e while lo op on line 3. By the lo op cond it ion, w ( U i ) < W / 2, so w ( U ∗ \ U i ) > W / 2. As a result, for the set U ∗ , the expression on line 4 is negativ e: f ( U ∗ ) − 2 B W · w ( U ∗ \ U i ) < f ( U ∗ ) − B < 0 . Then f o r the set T i whic h minimizes this expr ession, it wo uld also b e negativ e, implying that w ( T i \ U i ) is p osit iv e, and so w ( U i ) increases in eac h iterati on. As a result, if the instance is 14 Algorithm 4 S M L. In put: V , f , w : V → { 0 , 1 } , W , B , p 1: I n iti alize U 0 = ∅ ; i = 0 2: if W ≥ w ( V ) / 2 then ⊲ case W ≥ w ( V ) 2 3: while w ( U i ) < W / 2 do 4: Let T i b e a subset of V minim izing f ( T ) − 2 B W · w ( T \ U i ) 5: if f ( T i ) < 2 B W · w ( T i \ U i ) then Let U i +1 = U i ∪ T i ; i = i + 1 else return fail 6: end while 7: return U = U i 8: end if 9: L et α = 2 B W p n ln n ⊲ case W < w ( V ) 2 10: w hile w ( U i ) < W / 2 do 11: Cho ose a random S i ⊆ V \ U i , including eac h elemen t with pr ob ab ility W w ( V ) 12: Let T i b e a subs et of V m in imizi n g f ( T ) − α · w ( T ∩ S i ) 13: if f ( T i ) ≤ α · w ( T i ∩ S i ) and f ( T i ) ≤ 4 B p n ln n then Let U i +1 = U i ∪ T i ; i = i + 1 14: if the num b er of iterations exceeds 3 n 9 / 2 c ln  n 1 − p  , ret urn fail 15: end while 16: return U = U i feasible, then after at most n iteratio n s of the lo op on line 3, a set U is found with w ( U ) ≥ W / 2. F or the f unction v alue, we ha ve f ( U ) ≤ X i f ( T i ) < 2 B W X i w ( T i \ U i ) ≤ 2 B W · w ( V ) ≤ 4 B b y our assump ti on ab out W . The second case is W < w ( V ) / 2. Assum ing Claim 4.4 b elo w, which is pro ved later, w e show that in eac h iteration of the wh ile lo op on line 10, w it h probabilit y at least c 3 n 7 / 2 , a new n on - empt y set T i is add ed to U . Th is implies that after 3 n 9 / 2 c ln  n 1 − p  iterations, the lo op successfully terminates with p robabilit y at least p . Claim 4.4 In e ach iter ation of the while lo op on line 10 of Algorith m 4, b oth of the fol lowing two ine qualities hold with pr ob ability at le ast c 3 n 7 / 2 . w ( U ∗ ∩ S i ) > B α = W 2 r ln n n and w ( ¯ U ∗ ∩ S i ) ≤ 1 . 5 W . (4) W e sh o w that if inequalities (4) hold, then the set T i found b y the algorithm on line 12 is non-empt y and satisﬁes the conditions on lin e 13, which means that new elemen ts are added to U . Since T i is a min imiz er of the expression on line 12, and using (4), f ( T i ) − α · w ( T i ∩ S i ) ≤ f ( U ∗ ) − α · w ( U ∗ ∩ S i ) < f ( U ∗ ) − B < 0 , whic h means that T i satisﬁes the ﬁr s t condition on line 13 and is n o n-empt y . Moreov er, from the same inequalit y and the second p art of (4) we ha v e f ( T i ) ≤ f ( U ∗ ) + α · ( w ( T i ∩ S i ) − w ( U ∗ ∩ S i )) ≤ B + α · w ( ¯ U ∗ ∩ S i ) ≤ B + 1 . 5 αW ≤ 4 B r n ln n , 15 whic h means that T i also satisﬁes th e s e cond condition on line 13. No w we analyze the function v alue of the set outpu t by the algo r ithm. Let T i b e the last s et added to U by the wh il e lo op, and consider the s et U i just b efore T i is added to it to p rod u ce U i +1 . By the lo op condition, w e ha ve w ( U i ) < W / 2. Then, by su bmod ularit y and condition on line 13, f ( U i ) ≤ i − 1 X j =0 f ( T j ) ≤ i − 1 X j =0 α · w ( T j ∩ S j ) ≤ α · w ( U i ) < α · W 2 = B r n ln n . So for the set U that the algorithm outp u ts, f ( U ) ≤ f ( U i ) + f ( T i ) ≤ 5 B p n ln n . An d b y the exiting condition of the while lo op, w ( U ) ≥ W / 2.  Pro of of Claim 4.4. Because the even ts corresp onding to the t w o inequalities are indep enden t, w e b ound their probabilities separately and then m u ltiply . T o b ound the probability of the ﬁrst one let m = w ( U ∗ \ U i ) b e the n umb er of elements of U ∗ with weigh t 1 that are in V \ U i . S ince w ( U ∗ ) ≥ W and w ( U i ) < W / 2 b y the condition of the loop, we ha v e m > W / 2. W e in vok e Theorem 2.2 with parameters m , q = W /w ( V ), and ε = w ( V ) 2 m q ln n n . T o ensur e th at ε < 1 − q q and this theorem can b e applied, we assume that n ≥ 9, so that p ln n / n < 1 / 2, and get 1 − q q − ε = w ( V ) W − 1 − w ( V ) 2 m r ln n n > w ( V ) 2 W − 1 > 0 . Th us the inequalit y w ( U ∗ ∩ S i ) ≥ ⌈ q m (1 + ε ) ⌉ > q mε = W 2 q ln n n holds with probability at least (simplifying u sing inequalities w ( V ) − W ≥ w ( V ) / 2, w ( V ) ≤ n , and 1 ≤ W < 2 m ) c q m − 3 2 exp  − ε 2 q m 1 − q  = c W w ( V ) m − 3 2 exp  − w ( V ) 3 W m ln n 4 m 2 n w ( V ) ( w ( V ) − W )  ≥ cn − 7 / 2 . F or the second in equali t y , we notice that the exp ectati on of w ( ¯ U ∗ ∩ S i ) is w ( ¯ U ∗ ) · W w ( V ) ≤ W . So by Mark o v’s in e q u al it y , the p r obabilit y that w ( ¯ U ∗ ∩ S i ) ≤ 1 . 5 W is at least 1 / 3.  5 Submo dular load balancing 5.1 Lo wer b ound W e giv e t wo monotone su bmod ular fun ctions that are hard to distinguish, bu t whose v alue of the optimal solution to the S LB pr o blem diﬀers by a large factor. Th ese fun ct ions are: f 5 ( S ) = m in ( | S | , α ) f 6 ( S ) = min X i min ( β , | S ∩ V i | ) , α ! . (5) Here { V i } is a random (unknown to the algorithm) partition of V into m equal-sized sets. W e set m = 5 √ n x , α = n m = x √ n 5 , β = x 2 5 , with an y parameter x satisfying x 2 = ω (ln n ), and v alues c hosen so that α and β are in teger. Lemma 5.1 Any algo rithm that makes a p olynomial nu m b er of or acle queries has pr ob ability n − ω ( 1) of distinguishing the functions f 5 and f 6 ab ove. 16 Pro of. By Lemma 2.1, it suﬃces to b ound the pr ob ab ility , ov er the random c hoice of th e sets V i , that f 5 ( S ) 6 = f 6 ( S ) for any one set S . S ince f 5 ≥ f 6 , this is the s a m e as Pr [ f 6 ( S ) − f 5 ( S ) < 0]. First, we sho w that this p r obabilit y is maximized when | S | = α . F or | S | ≥ α , Pr [ f 6 ( S ) − f 5 ( S ) < 0] = Pr " X i min ( β , | S ∩ V i | ) < α # , and sin ce the su m in this expression can only decrease if an elemen t is remo v ed from S , we ha v e that for | S | ≥ α , th is pr obabilit y is m aximized at | S | = α . F or | S | ≤ α , Pr [ f 6 ( S ) − f 5 ( S ) < 0] = Pr " min X i min ( β , | S ∩ V i | ) , α ! − | S | < 0 # = Pr " min X i min ( β , | S ∩ V i | ) − X i | S ∩ V i | , α − | S | ! < 0 # = Pr " X i min ( β − | S ∩ V i | , 0) < 0 # . Since the sum in this expression can only decrease if an elemen t is added to S , we h a v e that for | S | ≤ α , the probability is maximized at | S | = α . So su pp ose th at | S | = α . W e notice that if f o r all i , | S ∩ V i | ≤ β , then f 5 ( S ) = f 6 ( S ). Therefore, a necessary condition for the tw o fun cti ons to b e diﬀeren t is that | S ∩ V i | > β for some i . Since V 1 is a random subset of V of size α , we can use the same calculation as in the pro of of Lemma 4.1 to sho w that Pr [ | S ∩ V 1 | > β ] ≤ n − ω ( 1) . App lying the union b ound , w e get that the probabilit y that | S ∩ V i | > β for any i is also n − ω ( 1) .  Theorem 5.2 The SLB pr oblem is har d to appr oximate to a factor of o  p n ln n  . Pro of. S upp ose that ther e is a γ -app ro ximation algorithm for SLB, where γ = o  p n ln n  . Let x = √ n/δγ , where δ > 1 is such that α and β are intege r . This satisﬁes x 2 = ω (ln n ). No w consid e r runn in g the algorithm with the input function f 6 and size of p artit ion m . F or this input, partition { V i } constitutes the optimal s olution w h ose v alue is f 6 ( V i ) = β , so the algorithm returns a solution whose v alue is at most γ β = α/δ . Ho wev er, for the in p ut f 5 and m , any partition m ust con tain a set S with size | S | ≥ n/m = α (since th is is the a verag e size). F or this set, the function v alue is f 5 ( S ) = α > α/δ . This means that f o r f 5 the algo r it h m p rodu ce s a diﬀerent answ er than for f 6 , whic h con tradicts Lemma 5.1.  5.2 Algorithms for SLB W e note that the tec h nique of Svitkina and T ardos [37] used for min-max multiw a y cut can b e applied to the non-uniform SL B problem to obtain an O ( √ n log n ) appro ximation algorithm, using the appro x im ation algo rithm for the S ML pr o b lem presente d in Section 4 as a subr outine. Also, an O ( √ n log n ) ap p ro ximation for the non-uniform S LB app ears in [13]. In this section we p resen t t wo algorithms, with imp ro v ed appro ximation ratios, for the uniform SLB p r oblem. W e b egin b y presen ting a very simple algorithm that giv es a min( m,  n m  ) = O ( √ n ) appro ximation. Then we giv e a more complex algorithm that impro ves the app ro ximation ratio to 17 O  p n ln n  , thus m atching the low er b ound. O u r ﬁ rst algorithm s imply partitions the elemen ts into m sets of roughly equal size. Theorem 5.3 The algorithm that p artitions the elements into m arbitr ary sets of size at most  n m  e ach is a m in ( m,  n m  ) appr oximation for the SLB pr oblem. Pro of. L e t { U ∗ 1 , ..., U ∗ m } denote the optimal solution with v alue B , and let A b e the v alue of the solution { S 1 , ..., S m } found by the algorithm. W e exhibit t wo low er b ounds on B an d t wo u pp er b ounds on A , and then establish the approximat ion ratio by comparing these b ound s. F or the lo we r b ounds on B , w e claim that B ≥ max j ∈ V f ( { j } ) and B ≥ f ( V ) /m . F or the ﬁrst one, let j b e the element maximizing f ( { j } ), and let U ∗ i b e the set in the optimal solution that cont ains j . Then B ≥ f ( U ∗ i ) ≥ f ( { j } ) by monotonicit y . F or the second b ound, by submo dularit y we hav e th a t f ( V ) ≤ P i f ( U ∗ i ) ≤ mB . T o b ound A , we n ot ice that A ≤ f ( V ) (by monotonicit y), and that A ≤  n m  max j ∈ V f ( { j } ), since eac h set S i con tains at most  n m  elemen ts, and f ( S i ) ≤ P j ∈ S i f ( { j } ). Comparing with the lo wer b ounds on B , we get the resu lt .  Algorithm 5 S ubmod ular load balancing. Inp ut: V , m > p n ln n , monotone f , B , p 1: if for any v ∈ V , f ( { v } ) ≥ B , return fail 2: L et α = B m/ √ n ln n ; I nitia lize V ′ = V , i = 0 3: w hile | V ′ | > m p n ln n do 4: Cho ose a random S ⊆ V ′ , includ ing eac h elemen t in d epend en tly with probabilit y n m | V ′ | 5: if | S | ≤ 2 n m then 6: Let T ⊆ S b e a sub set minimizing f ( T ) − α · | T | 7: if f ( T ) − α · | T | < 0 then set T i = T ; i = i + 1 ; V ′ = V ′ \ T 8: end if 9: if the num b er of iterations exceeds 2 n 3 c ln( n 1 − p ), return fail 10: end while 11: Let T b e the collection of sets T i pro duced by the wh ile lo op 12: Partiti on T in to m group s T 1 , ..., T m , such that P i : T i ∈T j | T i | ≤ 3 n m for eac h T j 13: Let U 1 , ..., U m b e an y partition of V ′ with eac h set of size at most p n ln n 14: F or eac h j ∈ { 1 , ..., m } , let V j = U j ∪ S T i ∈T j T i 15: return { V 1 , ..., V m } F or th e more complex Algorithm 5, we assume that m > 2 p n ln n , b ecause for lo we r v alues of m the ab o ve simple algorithm giv es the desired app ro ximation. Also, the simp le algorithm has b etter guaran tee for all n ≤ e 16 , so when analyzing Algorithm 5, we can assume that n is suﬃcien tly large for certai n inequalities to hold, suc h as ln 3 n < n . The alg orithm ﬁ nds small disjoin t sets of ele men ts that hav e low ratio of function v alue to size. Once a suﬃcient n umb er of elemen ts is group ed in to suc h lo w-ratio sets, these sets are com b ined to form m ﬁnal sets of the partition, while adding a few remaining elemen ts. These ﬁnal sets ha ve roughly n/m elements eac h , so using submo dularity and the lo w r a tio pr o p erty , w e can b ound th e fun c tion v alue for eac h set in the p artit ion. First w e describ e ho w some of the steps of algorithm work. The lo op condition | V ′ | > m p n ln n and our assu m ptio ns m > 2 p n ln n and ln 3 n < n imply that th e p r obabilit y n m | V ′ | (used on line 4) is less than one. The p a rtition on line 13 can b e found b ecause at this p oin t, the size of V ′ is at most m p n ln n . F or the partitioning done on line 12, w e note th a t since eac h T i is a sub set of a sample set 18 S with | S | ≤ 2 n/m , it holds th a t | T i | ≤ 2 n/m . Also, th e total num b er of elemen ts contai n ed in all sets T i is at most n (since they are d isjoin t). So a simple greedy pro cedure that adds the sets T i to T j in arb itrary order, u n til the total num b er of elements is at least n/m , will pro duce at most m groups, eac h with at most 3 n/m elemen ts. Theorem 5.4 If gi v en a fe asible instanc e of the SLB pr oblem, Algorith m 5 outputs a solution of value at most 4 p n ln n · B with pr ob ability at le ast p . Pro of. By monotonicit y of f , the algorithm exits on line 1 only if the instance is infeasible. Assume th at the instance is feasible and let { U ∗ 1 , . . . , U ∗ m } d e note a solution with max j f ( U ∗ j ) < B . W e consider one iteration of the while lo op and sho w that with pr o b abilit y at least c 2 n 2 it ﬁnd s a set T ⊆ S satisfying f ( T ) − α · | T | < 0. Then the probability that the size of V ′ is reduced to m p n ln n after 2 n 3 c ln( n 1 − p ) iterations is at least p . Assume, without loss of generalit y , that U ∗ 1 is the s et that maximizes | U ∗ j ∩ V ′ | for this iteration of the loop. If we let n ′ = | V ′ | , then | U ∗ 1 ∩ V ′ | ≥ ⌈ n ′ /m ⌉ . Su pp ose the sample S found by the algorithm h a s size at most 2 n/m , and let t = | U ∗ 1 ∩ S | denote the size of the ov erlap of S and U ∗ 1 . By monotonicit y of f , w e kno w that f ( U ∗ 1 ∩ S ) ≤ f ( U ∗ 1 ) < B . Sin ce the algorithm ﬁnd s a set T ⊆ S minimizing the exp ression f ( T ) − α | T | , we kno w that the v alue of this expression for T is at most that for U ∗ 1 ∩ S : f ( T ) − α | T | ≤ f ( U ∗ 1 ∩ S ) − α | U ∗ 1 ∩ S | < B − B mt √ n ln n . In ord er to ha v e f ( T ) − α | T | < 0, w e need t ≥ √ n ln n m . Next w e show that the even t that b oth t ≥ √ n ln n m and | S | ≤ 2 n/m happ ens with pr o b abilit y at least c 2 n 2 . Let x = √ n ln n m . T o b ound th e p robabilit y th a t t ≥ x , w e fo cus on an arbitrary ﬁxed sub set of U ∗ 1 ∩ V ′ of size ⌈ n ′ /m ⌉ (which is p ossible b ecause | U ∗ 1 ∩ V ′ | ≥ ⌈ n ′ /m ⌉ ), and compute the probability that exactly ⌈ x ⌉ elemen ts from this subset mak e it in to the sample S . In particular, this is the probabilit y that sampling ⌈ n ′ /m ⌉ items indep endently , with probabilit y n/mn ′ eac h, pro duces a sample of size ⌈ x ⌉ . W e note that x ∈ (1 , n ′ /m ), so ⌈ x ⌉ is a v alid sample size. These b ounds follo w b ecause inside the while lo op, m < n ′ p ln n/n ≤ √ n ln n , so x > 1. Also, n ′ /m > p n/ ln n > ln n > √ n ln n/m by the lo op condition and our assu mptions on n and m , so x < n ′ /m . Let γ , δ ∈ [1 , 2) b e such that γ · n ′ /m = ⌈ n ′ /m ⌉ and δ · x = ⌈ x ⌉ . W e u se an appro xim ation derived fr o m S ti r ling’s form u la as in the p roof of Theorem 2.2. Pr[ t = ⌈ x ⌉ ] ≥  γ n ′ /m δ x  ·  n mn ′  δx ·  1 − n mn ′  γ n ′ m − δx ≥ c √ n ·  γ n ′ m  γ n ′ m ·  n mn ′  δx ·  1 − n mn ′  γ n ′ m − δx  γ n ′ m δ √ n ln n γ n ′  δx ·  γ n ′ m  1 − δ √ n ln n γ n ′  γ n ′ m − δx = c √ n ·  n mn ′ γ n ′ δ √ n ln n  δx   1 − n mn ′ 1 − δ √ n l n n γ n ′   γ n ′ m − δx (6) 19 ≥ c √ n ·  γ δ m r n ln n  δ √ n ln n m , where the last inequalit y comes from observin g that our assump ti on of m > 2 p n ln n , together with γ /δ < 2, imply that the last term on line (6) is greater than 1. If we tak e a deriv ativ e of this b oun d with resp ect to m , whic h is ∂ ∂ m   c √ n ·  γ δ m r n ln n  δ √ n ln n m   = − c δ √ ln n m 2 ·  γ δ m r n ln n  δ √ n ln n m ·  ln  γ δ m r n ln n  + 1  , and set it to zero, we ﬁnd that the b ound is minimized when m = e γ δ p n ln n . Su b stituting th is v alue, Pr[ t = ⌈ x ⌉ ] ≥ c · n − δ 2 e γ − 1 2 ≥ c · n − 4 e − 1 2 ≥ c · n − 2 . T o b ound the second probabilit y , that | S | ≤ 2 n/m , we note that E [ | S | ] = n/m and use Chern oﬀ b ound as we ll as the lo op condition that implies m < n ′ q ln n n ≤ √ n ln n . Pr h | S | > 2 n m i <  e 4  n m ≤  e 4  √ n ln n If n is suﬃcien tly large that  e 4  √ n ln n ≤ c 2 n 2 , we can use the union b ound to get Pr h t ≥ x and | S | ≤ 2 n m i ≥ c n 2 − c 2 n 2 = c 2 n 2 . This establishes that on f easible in sta n ce s , the algorithm su cc essf ully terminates with p robabilit y at least p . Let us no w consider the fun c tion v alue on an y of the s e ts V j output by th e algorithm. By sub modu la r it y , f ( V j ) ≤ X v ∈ U j f ( { v } ) + X T i ∈T j f ( T i ) . F or eac h T i w e k n o w that f ( T i ) < α · | T i | , and by the c hec k p erformed on line 1, we ha v e f ( { v } ) < B for eac h v ∈ V . Using this and th e b ounds on set sizes, f ( V j ) ≤ B r n ln n + α X T i ∈T j | T i | ≤ B r n ln n + α · 3 n m = B ·  r n ln n + m √ n ln n 3 n m  = 4 r n ln n · B .  6 Appro ximating submo dular fu nctio n s ev erywhere W e present a lo we r b ound for the pr o b lem of appro ximating submo dular fu nctio n s ev erywhere, whic h holds eve n for the s pecial case of monotone fun ctions. W e use the same fun c tions (3) as f o r the SML lo we r b ound in Section 4. Theorem 6.1 Any algorithm tha t makes a p olynomial numb er of or acle queries c annot appr oxi- mate monotone sub m o dular functions to a factor o  p n ln n  . 20 Pro of. Supp ose that there is a γ -approximat ion algo r it hm f o r the problem, with γ = o  p n ln n  , whic h make s a p olynomial num b er of oracle queries. Let x = √ n/δγ , wh ic h satisﬁes x 2 = ω (ln n ). By Lemma 4.1, with high probabilit y this algorithm pro duces the same outp ut (sa y ˆ f ) if giv en as input either f 3 or f 4 . Thus, by the algorithm’s guaran tee, ˆ f is sim ultaneously a γ -appro ximation for b oth f 3 and f 4 . F or the set R used in f 4 , th is guaran tee implies that f 3 ( R ) ≤ γ ˆ f ( R ) ≤ γ f 4 ( R ). Since f 3 ( R ) = α and f 4 ( R ) = β , w e hav e that γ ≥ α/β = √ n/x = 2 γ , wh ic h is a contradict ion.  6.1 Appro ximating monotone tw o-partition submodular functions Recall that a 2P fu n ct ion is one for whic h th e re is a set R ⊆ V su c h that the v alue of f ( S ) dep ends only on | S ∩ R | and | S ∩ ¯ R | . Our algorithm for approximat in g m o n ot on e 2P f unctions everywhere (Algorithm 6) uses the follo w ing observ ation. Lemma 6.2 Given two sets S and T such that | S | = | T | , bu t f ( S ) 6 = f ( T ) , a 2P f unction c an b e found exactly using a p olynom ial numb er of or acle queries. Pro of. This is d one by inferr ing what the set R is. Using S and T , w e ﬁ nd tw o sets which diﬀer b y exact ly one elemen t and ha ve d iﬀeren t fun c tion v alues. Fix an ordering of the elemen ts of S , { s 1 , ..., s k } , and an ordering of element s of T , { t 1 , ..., t k } , suc h that the elemen ts of S ∩ T app ear last in b oth orderings, and in the same sequence. Let S 0 = S , and S i b e the s e t S with the ﬁrst i elemen ts replaced b y the ﬁrst i elemen ts of T : S i = { t 1 , ..., t i , s i +1 , ..., s k } . Ev aluate f on eac h of the sets S i in ord er, until the ﬁ rst time that f ( S i − 1 ) 6 = f ( S i ). Su c h an i m us t exist since S k = T , and by assu m ption f ( T ) 6 = f ( S ). Let U = { t 1 , ..., t i − 1 , s i +1 , ..., s k } , so that S i − 1 = U ∪ { s i } and S i = U ∪ { t i } . The fact that f ( U ∪ { s i } ) 6 = f ( U ∪ { t i } ) tells us that either s i ∈ R and t i / ∈ R , or vice v ersa. Without loss of generalit y , we assu me th e former ( s ince the names of R and ¯ R can b e in terc h an ged). No w all elements in V \ U can b e classiﬁed as b elonging or n o t b elonging to R . In particular, if for some elemen t j ∈ ¯ U , f ( U ∪ { j } ) = f ( U ∪ { s i } ), then j ∈ R ; otherwise f ( U ∪ { j } ) = f ( U ∪ { t i } ), and j / ∈ R . T o test an elemen t u ∈ U , ev aluate f ( U − { u } + { s i , t i } ). This is the set S i − 1 with elemen t u replaced by t i . If u ∈ ¯ R , then replacing one element from ¯ R b y another will ha ve n o eﬀect on the function v alue, and it will b e equal to f ( S i − 1 ). If u ∈ R , the we ha ve replaced an elemen t from R by an element from ¯ R , and w e kno w that this changes th e fun ct ion v alue to f ( S i ). So all elemen ts of V can b e tested for their memb e r ship in R , and then all f unction v alues can b e obtained by querying sets W with all p ossible v alues of | W ∩ R | and | W ∩ ¯ R | .  Algorithm 6 App r o ximating a monotone 2P fun c tion ev erywh er e. In p ut: V , f , p 1: Q uery v alues of f ( ∅ ), f ( V ), and f ( { j } ) for eac h j ∈ V 2: F or eac h i ∈ { 2 , ..., n − 1 } , in d epend en tly generate n 10 ln  4 n 1 − p  random sets by including eac h elemen t of V into eac h set with probability i n . Q uery the function v alue for eac h of these sets. 3: I f the p r evio u s t wo steps pro duce an y tw o sets S 1 and S 2 with | S 1 | = | S 2 | and f ( S 1 ) 6 = f ( S 2 ), then ﬁnd the function exactly , as describ ed in Lemma 6.2. 4: E lse, let j ∈ V b e an arbitrary elemen t, and output ˆ f ( S ) =      f ( ∅ ) if S = ∅ f ( { j } ) if 1 ≤ | S | ≤ 2 √ n f ( V ) 2 √ n if | S | > 2 √ n 21 Theorem 6.3 With pr ob ability at le ast p , the function ˆ f r eturne d by Algorithm 6 satisﬁes ˆ f ( S ) ≤ f ( S ) ≤ 2 √ n · ˆ f ( S ) for al l sets S ⊆ V . Pro of. If the algorithm ﬁnds t wo s e ts S 1 and S 2 suc h that | S 1 | = | S 2 | and f ( S 1 ) 6 = f ( S 2 ) during the sampling stage (steps 1 and 2), then the correctness of the output is imp lied by Lemm a 6.2. If it do es n ot ﬁnd suc h sets, then it outp uts th e f unction ˆ f sh o wn in step 4. It ob viously satisﬁes the inequalit y for the case that S = ∅ . F or th e case th a t 1 ≤ | S | ≤ 2 √ n , w e observ e that if the algorithm reac hes step 4, it m u s t b e that the v alue of f is identi cal for all singleton sets, i.e. f ( { j } ) = f ( { j ′ } ) for all j, j ′ ∈ V . No w, f ( S ) ≥ f ( { j } ) = ˆ f ( S ) b y m o n ot on icity . Also, b y submo dularit y , f ( S ) ≤ P j ∈ S f ( { j } ) = | S | · ˆ f ( S ) ≤ 2 √ n · ˆ f ( S ), establishing the correctness for the case that | S | ≤ 2 √ n . F or the last case, | S | > 2 √ n , th e inequalit y f ( S ) ≤ f ( V ) = 2 √ n · ˆ f ( S ) follo ws b y monotonicit y . F or the other one, ˆ f ( S ) ≤ f ( S ), we need an ad d iti onal nontrivial lemma. Since the 2P function f ( S ) dep ends only on t wo v alues, | S ∩ R | and | S ∩ ¯ R | , let us denote b y f ( k , l ) th e v alue of the fun c tion f on a set S with | S ∩ R | = k and | S ∩ ¯ R | = l . W e say t h at suc h a set S corresp onds to the pair ( k , l ). W e assume th at 0 < | R | < n , b ecause if | R | = 0 or | R | = n , then f ( S ) is a function that dep ends only on | S | , and it equally w ell can b e repr ese n ted as a 2P function with any other set ˆ R . F urth ermore, w e assume with ou t loss of generalit y that | R | ≤ | ¯ R | (otherwise in terchange R and ¯ R ), an d let K = | R | and L = | ¯ R | (which are n ot kn o wn to the algorithm). Lemma 6.4 F or any k and any l , f ( k , 0) ≥ k 2 n f ( V ) and f (0 , l ) ≥ l 2 n f ( V ) . Using this lemma to ﬁ nish the pro of, let k = | S ∩ R | and l = | S ∩ ¯ R | . W e observe that b y monotonicit y , f ( S ) ≥ f ( k , 0) and f ( S ) ≥ f (0 , l ). Moreo ver, since | S | = k + l ≥ 2 √ n , w e ha v e max( k , l ) ≥ √ n . So by Lemma 6.4, f ( S ) ≥ max( k ,l ) 2 n f ( V ) ≥ f ( V ) 2 √ n = ˆ f ( S ).  The pro of of Lemma 6.4 is inv olv ed, and we ﬁrst sk etc h the main ideas. W e call a pair ( k , l ) b alanc e d if k /l is close to K/L . Then, with signiﬁcan t probabilit y , the alg orithm samples sets corresp onding to all b al anced pairs. Since the algorithm chec ks for sets of the same size with diﬀeren t function v alues, w e can assume that if it pro ceeds to step 4, then for sets S corresp onding to balanced pairs, f ( S ) is a fun c tion F th at dep ends only on | S | . W e use s ubmo d ularit y to sho w that F is conca v e. Then w e decomp ose f ( k , 0) as P k i =1 [ f ( i, 0) − f ( i − 1 , 0)] and lo wer-boun d eac h term in this sum separately by comparing it to an increment f ( i, j ) − f ( i − 1 , j ) for some j with ( i, j ) balanced. Th en, using conca v ity of F , w e lo wer-b o u nd their sum. T o prov e Lemma 6.4, w e use a deﬁnition and sev eral preliminary lemmas. Deﬁnition 6.5 A p air of inte gers ( k , l ) with k ≤ K and l ≤ L is said to b e balanced i f it satisﬁes l · K L − 2 ≤ k ≤ l · K L + 2 . (7) In tu it ively , in a set corresp onding to a balanced p a ir , the n umb ers of elemen ts from R and ¯ R are pr op ortional to the sizes of the t wo sets (see Figure 1). Lemma 6.6 Supp ose that m ≤ n e lem ents ar e sele cte d indep endently with pr ob ability q ∈ [ 1 n , n − 1 n ] e ach, and let X denote the total numb er of sele cte d e lements. Then for any inte ger x ∈ [0 , m − 1] , 1 n 2 ≤ Pr[ X = x + 1] Pr[ X = x ] ≤ n 2 . 22 Pro of. Pr[ X = x + 1] Pr[ X = x ] =  m x +1  q x +1 (1 − q ) m − x − 1  m x  q x (1 − q ) m − x = ( m − x ) q ( x + 1)(1 − q ) , with the minimum v alue of 1 /m ( n − 1) ≥ 1 /n 2 ac hiev ed at x = m − 1 and q = 1 n , and the maxim um v alue of m ( n − 1) ≤ n 2 ac hiev ed at x = 0 and q = n − 1 n .  Lemma 6.7 If Algorithm 6 r e aches step 4, then with pr ob ability at le ast p , for al l b alanc e d ( k 1 , l 1 ) and ( k 2 , l 2 ) su c h that k 1 + l 1 = k 2 + l 2 , it holds that f ( k 1 , l 1 ) = f ( k 2 , l 2 ) . In other wor ds, f or al l b alanc e d p airs ( k , l ) , the value of f ( k , l ) dep ends only on k + l . Pro of. T he lemma follo ws if w e show that w it h probabilit y at least p , for eac h balanced ( k, l ) with k + l < n , the algorithm samples at least one set S corresp onding to ( k, l ). T his is b eca u se the algorithm veriﬁes that the fun c tion v alue for the sets that it samples dep ends only on the set size. 0 1 · · · K 2 k 0 1 2 3 4 5 L l · · · Figure 1: In the table of pairs ( k , l ), the s haded cell s corre- sp ond to balanced pairs, and the K -biased (dashed ) and L -biased (dotted) w alks are sho wn . So consid er a sp eciﬁc balanced p a ir ( k , l ) and one r a ndom set S generated by th e iteration i = k + l of step 2 of the algorithm. Th e probabilit y of sampling eac h elemen t in this iteration is q = i n = k + l K + L . Using (7) and its equiv alen t ( k − 2) L/K ≤ l ≤ ( k + 2) L/K , w e see that this probability satisﬁes the follo w in g: k K − 2 L K n ≤ q ≤ k K + 2 L K n and l L − 2 n ≤ q ≤ l L + 2 n . So the exp ect ed v alue of | S ∩ R | is q K ∈ [ k − 2 L/n, k + 2 L/n ] ⊆ [ k − 2 , k + 2]. Similarly , the exp ected v alue o f | S ∩ ¯ R | is q L ∈ [ l − 2 , l + 2]. Let µ k b e the most lik ely num b er of sampled elemen ts when indep endent ly s a m pling K elemen ts with probabilit y q eac h. Then µ k is equal to either ⌊ q K ⌋ or ⌈ q K ⌉ . F rom ab o v e considerations and b ecause k is an int eger, w e ha ve that µ k ∈ [ k − 2 , k + 2]. No w, since µ k is the m o s t lik ely v alue, we kno w that Pr[ | S ∩ R | = µ k ] ≥ 1 / ( K + 1) ≥ 1 /n . By Lemma 6.6 (w ith m = K ), Pr[ | S ∩ R | = k ] ≥ Pr [ | S ∩ R | = µ k ] · n − 2 ·| k − µ k | ≥ n − 5 . W e similarly deﬁ ne µ l , observe that µ l ∈ [ l − 2 , l + 2], and conclud e that Pr[ | S ∩ ¯ R | = l ] ≥ n − 5 . Since the t wo ev en ts are indep en- den t, th e probabilit y that b oth of them o ccur, and th u s th a t S corresp onds to ( k , l ), is at least n − 10 . W e observe that for an y i , there are at most four b al an ced pairs ( k , l ) suc h that k + l = i . This is b ecause if some pair ( k , l ) satisﬁes (7), then the pair ( k − 4 , l + 4) do esn’t satisfy it: k − 4 ≤  l K L + 2  − 4 = l K L − 2 < ( l + 4) K L − 2 . So there is a total of at most 4 n p ai rs ( k , l ) for w hic h we w ould like the algorithm to samp le th e ir corresp onding sets. S in ce the n u m b er of trials for eac h v alue of k + l is n 10 ln  4 n 1 − p  , the probability that a set corresp onding to any particular pair ( k , l ) is not sampled is at most  1 − n − 10  n 10 ln  4 n 1 − p  ≤ e − ln  4 n 1 − p  = 1 − p 4 n . 23 Since there are at most 4 n pairs of in terest, b y u nion b ound w e ha ve that the probability that at least one of them remains unsampled is at most (1 − p ).  Supp ose the condition in Lemma 6.7 holds. Let us deﬁne a fun ction F ( i ) to b e equal to f ( k , l ) suc h that k + l = i and ( k , l ) is balanced. F ( i ) is deﬁn ed for all i ∈ { 0 , ..., n } , since for any su ch i there is at least one balanced pair ( k , l ) with k + l = i . Lemma 6.8 F ( i ) is a non-de cr e asing c onc ave function. Pro of. Let ∆( i ) = F ( i + 1) − F ( i ). I t suﬃces to sh ow that the sequence of increments ∆( i ) is non-negativ e and non-increasing. F or an y i , we deﬁne a pair ( k i , l i ) =  iK n  ,  iL n  . It can b e v eriﬁed that all p a ir s ( k i , l i ) as we ll as ( k i + 1 , l i ) are balanced. F urth e r more, k i + l i = i (and consequen tly k i + 1 + l i = i + 1), so that f ( k i + 1 , l i ) − f ( k i , l i ) = ∆( i ). Also, b oth { k i } and { l i } are n o n-decreasing sequences. The decreasing marginal v alues of the sub modu lar f unction f imply that ∆( i + 1) = f ( k i +1 + 1 , l i +1 ) − f ( k i +1 , l i +1 ) ≤ f ( k i + 1 , l i ) − f ( k i , l i ) = ∆( i ), sho wing that ∆( i )’s are non-increasing. The m o n ot onicity of f implies th a t they are also n o n -nega tive.  W e next deﬁn e t w o sequences of pairs, ( k K i , l K i ) and ( k L i , l L i ), ranging from i = 0 to i = n , whic h w e call the K -bi ase d sequen c e (or wa lk) and the L - biase d sequence, resp ectiv ely (see Figure 1). The prop erties of these t w o sequences w ill b e u sed in th e remainder of th e pr oof. T he deﬁn it ions are indu ct iv e, with b oth sequences starting at (0 , 0 ). ( k K i +1 , l K i +1 ) = ( ( k K i + 1 , l K i ) if k K i ≤ l K i · K L ( k K i , l K i + 1) if k K i > l K i · K L ( k L i +1 , l L i +1 ) = ( ( k L i + 1 , l L i ) if k L i < l L i · K L ( k L i , l L i + 1) if k L i ≥ l L i · K L Let u s call the change from ( k i , l i ) to ( k i +1 , l i +1 ) in either of the tw o sequences a K -step if the ﬁrst comp onen t of the pair increases by one, and an L -step if the second comp onen t increases. T he only diﬀerence b et ween th e tw o sequences is that wh en equalit y k = l · K/L holds, we tak e a K -step in the case of the K -b ia sed sequence, and an L -step in the case of the L -b ia s ed sequence. F or b ot h sequences it holds that k K i + l K i = k L i + l L i = i , k K i and k L i range b et ween 0 and K , and l K i and l L i range b et ween 0 and L . Lemma 6.9 Al l p airs in the K -biase d and L -biase d se quenc es ar e b alanc e d. Pro of. The p roof is by ind uctio n , and it is the same for b oth sequences, so we denote either sequence by ( k i , l i ). The ﬁr st pair (0 , 0) is balance d . Now w e assume that the pair ( k i , l i ) is bala nced, and wo u ld like to sho w that the pair ( k i +1 , l i +1 ) is also balanced. Supp ose ( k i +1 , l i +1 ) = ( k i + 1 , l i ). Then it must b e that k i ≤ l i · K L . Then l i · K L − 2 ≤ k i ≤ k i + 1 ≤ l i · K L + 1 . If ( k i +1 , l i +1 ) = ( k i , l i + 1), then it m u st b e that k i ≥ l i · K L . Then ( l i + 1) · K L − 2 ≤ l i · K L ≤ k i ≤ l i · K L + 2 ≤ ( l i + 1) · K L + 2 , with the leftmost inequalit y follo w ing b ecause K/L ≤ 1.  24 Lemma 6.10 In the K - b ias e d se qu e nc e, e very K -step is fol lowe d by at most  L K  L -steps. In the L -biase d se quenc e, every L - step is fol lowe d by at most one K - step. Pro of. Supp ose th a t the K -biased sequence, after some p oint ( k , l ), tak es one K - s t ep follo w ed b y  L K  L -steps, reac hing th e p oin t ( k + 1 , l +  L K  ). Since th e step after ( k, l ) is a K -step, it must b e that k ≤ l K /L . So  l +  L K  · K L ≥ l · K L + 1 ≥ k + 1 , whic h means that th e next step in the K -b ia sed sequence will b e a K -step. Similarly , for the L -b ia s ed w alk, supp ose that f rom some p oin t ( k , l ), the sequence takes an L -step, follo wed by a K -step, reac hing the p oint ( k + 1 , l + 1). T hen k ≥ l K/L imp lies th a t ( l + 1) · K L = l · K L + K L ≤ l · K L + 1 ≤ k + 1 , and thus the next step is an L -step.  Pro of of Lemma 6.4. T o lo wer-bou n d the v alue of f ( k , 0), w e consider the K -biased w alk from (0 , 0) to a p oin t ( k, l ′ ) which is the last p oint b efore the K - step to ( k + 1 , · ). W e let f ( k, 0) = F (0) + P k j =1 δ ( j ), where δ ( j ) = f ( j, 0) − f ( j − 1 , 0). F or eac h K -step in the K -b iased walk, wher e k K i − 1 = j − 1 and k K i = j , let ∆ K ( j ) = f ( k K i , l K i ) − f ( k K i − 1 , l K i − 1 ) = f ( j, l K i ) − f ( j − 1 , l K i − 1 ). By submo dularit y of f it follo ws that ∆ K ( j ) ≤ δ ( j ). W e claim that P k j =1 ∆ K ( j ) ≥ [ f ( k , l ′ ) − F (0)] / (1 +  L K  ). In other wo r ds, at least 1 / (1 +  L K  ) fraction of the increase in F ( · ), as we p roceed in th e K -biased wa lk, is d ue to the K -steps. This follo ws from several observ ations. First, the K -biased walk starts with a K -step. Second, by Lemma 6.10, eac h K -step is follo wed by n o m ore than  L K  L -steps. And third, ∆ K ( j ) is a decreasing sequence (b y conca vit y of F ). F ur ther, by conca vit y of F , w e h a ve that f ( k , l ′ ) ≥ k + l ′ n F ( n ) . By deﬁnition of l ′ , w e ha v e l ′ ≥ k L/K . Also, 1 +  L K  ≤ 2( L/K + 1). Pu tting everything together, w e ha ve f ( k , 0) = F (0) + k X j =1 δ ( j ) ≥ F (0) + k X j =1 ∆ K ( j ) ≥ F (0) + f ( k , l ′ ) − F (0) 1 +  L K  ≥ f ( k , l ′ ) 1 +  L K  ≥ k + l ′ n F ( n ) 2( L/K + 1) ≥ k ( L/K + 1) n F ( n ) 2( L/K + 1) = k 2 n F ( n ) T o b ound f (0 , l ), we consider the L -biased walk from (0 , 0) to ( k ′ , l ) for some k ′ . Because of conca vit y of F , the L -steps in the w alk accoun t for at least half the increase in f , yielding f (0 , l ) ≥ 1 2 f ( k ′ , l ). Also, f ( k ′ , l ) ≥ k ′ + l n F ( n ) ≥ l n F ( n ) . 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Submodular approximation: sampling-based algorithms and lower bounds

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