Local approximation algorithms for a class of 0/1 max-min linear programs

Local Approximation Algorithms for a Class of 0/1 Max-Min Linear Prog rams Patrik Flor ´ een, Marja Hassinen, P etteri Kas ki, an d Ju kka Suo mela Helsinki Institute for Information T echnolog y HIIT , Department o f Co mputer Science, University of He lsinki, P .O. Box 68, FI-00014 Un i versity of He lsinki, Finland { firstname.lastname } @cs.helsinki.fi Abstract — W e study the applicability of distributed, local algorithms to 0/1 max-min LPs where the objective is to ma ximise min k P v c kv x v subject to P v a iv x v ≤ 1 for each i and x v ≥ 0 fo r e ach v . Her e c kv ∈ { 0 , 1 } , a iv ∈ { 0 , 1 } , and the supp ort sets V i = { v : a iv > 0 } an d V k = { v : c kv > 0 } h a ve boun ded size; in particular , we study the case | V k | ≤ 2 . Each agent v is responsible fo r choosing the value of x v based on infor mation within its constant-size neighbourhood; the communication network is the hyperg raph where the sets V k and V i constitute th e hyperedges. W e present a local approximation algorithm which achiev es an approximation ratio arbitrarily close to the theoretical lower bound presented in prior work. I . I N T RO D U C T I O N T o motiv ate the pro blem setting studied in this p aper, consider th e toy network de picted in Fig. 1. There ar e sev en customers, k 1 , k 2 , . . . , k 7 , who are served b y fi ve access points, i 1 , i 2 , . . . , i 5 . The cu stomers and access po ints are conn ected by th e 1 4 n umbered links. k 1 k 2 k 3 k 4 k 5 k 6 k 7 i 1 i 2 i 3 i 4 i 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Fig. 1. An example of a data communication network. Now , suppose that we want to provide a maxim um f air share of ban dwidth to e ach customer, sub ject to th e con straint that each access point can hand le at mo st 1 unit of network traffic. Put otherwise, we want to maximise the minimum band width av a ilable to a customer . In forma lly pr ecise terms, we want to solve th e following optimisation problem , where the variables x 1 , x 2 , . . . , x 14 de- termine the amou nt of network traffic allocated to eac h link : maximise ω = min { ( k 1 ) x 1 + x 2 , ( k 2 ) x 3 + x 4 , . . . , ( k 7 ) x 13 + x 14 } subject to x 1 + x 3 + x 5 ≤ 1 , ( i 1 ) x 2 + x 9 ≤ 1 , ( i 2 ) x 4 + x 7 + x 11 ≤ 1 , ( i 3 ) x 6 + x 8 + x 13 ≤ 1 , ( i 4 ) x 10 + x 12 + x 14 ≤ 1 , ( i 5 ) x 1 , x 2 , . . . , x 14 ≥ 0 . (1) An optimal solution is x 1 = x 7 = 2 / 7 , x 2 = x 8 = 3 / 7 , x 3 = x 6 = x 11 = 0 , x 4 = x 5 = x 12 = 5 / 7 , x 9 = x 13 = 4 / 7 , and x 10 = x 14 = 1 / 7 , gu aranteeing the ban dwidth ω = 5 / 7 to each custome r . This is the best possible fair bandwid th allocation for our toy network. Moreover , it can be argued that such an allocation is n ot co mpletely trivial to find with heuristic tec hniques, even in the toy network. So far so go od, b ut of co urse no one would seriously suggest a similar app roach fo r o ptimising a rea l-world network. For one, any realistic network is several orders of magnitud e larger, and, what is more, und er constant change . In particular, it is not f easible to pu t tog ether a snap shot of the relevant topolog y o f the e ntire network , such as Fig. 1, for purp oses of o ptimisation. Nev ertheless, a disciplined global optimisation appr oach, such as (1 ), provid es an un equiv ocal ben chmark fo r the design of distributed algorith ms. Id eally , after each ch ange in top ology , the entire network sho uld imm ediately converge to a g lobal op timum. Of cour se, this ideal is unattainab le if the n odes ar e on ly aware of their local neighbou rhood s in the network. But not comple tely so: in certain cases local informa tion does suffice to pr ovably appr oximate the global optimum . In this work we present a novel distributed algorith m for linear m ax–min optimisation pro blems such as (1). The algorithm is both an appr oximation algorith m, with a prov- able appro ximation guaran tee, and a local algorithm , with a constant local horizon r which is in depend ent of the size o f the network (see Section I-A). In practical terms, this implies all o f the following. • The algorithm conv erges in r time units and recovers from a topo logy chan ge in r time units. • Whenever the network – o r any part of it – ha s remained stable for r time units, the algorithm pr ovides a provable approx imation g uarantee f or th at part. • A topo logy change only affects those parts of the network that are within r h ops fro m a no de that loses or gains neighbo urs. The re st of the network stays in its cu rrent configur ation, which is feasible and app roximately opti- mal bo th bef ore and after th e topology chan ge. A. Local Alg orithms W e say that a distributed algorith m h as the loc al h orizon r if a topolo gy chan ge a t n ode v affects only those network nodes whic h are within r hop s from node v . In other words, the outp ut o f nod e u is a function of in put available in its radius r neig hbour hood. Distributed alg orithms wh ere the loc al h orizon r is constant are called local algo rithms or d istributed co nstant-time algo- rithms. Naturally th e lo cal setting is very restrictive; th ere are fundam ental obstacles which p revent us from solving problems by usin g a lo cal algorith m [1], [2]. Howe ver , a few p ositi ve results are known [3], [4], [5], [6 ], [ 7], [8], [9], [10]. Our work presents a new example of such positive r esults. If we assume that so me aux iliary inform ation – such as the coordin ates of the nod es – is available, we can design local algorithm s fo r a wider rang e of prob lems [11], [12], [13]. In the present work no such assumptions are necessary . B. Max-Min P acking Pr oblem Formally , th e problem setting that we study is a 0/1-version of the max-min packing pr ob lem [1 0], d efined as follows. Let V , I and K be disjoint index sets; we say that each v ∈ V is an agent , each k ∈ K is a bene ficiary party , and each i ∈ I is a resour ce . W e assume that one un it of activity by v benefits the party k b y c kv ∈ { 0 , 1 } units and consumes a iv ∈ { 0 , 1 } units of the resource i ; the obje cti ve is to set the acti vities to provide a fair share of b enefit f or e ach party . Assuming th at the acti vity o f agen t v is x v units, the ob jectiv e is to maximise ω = min k ∈ K X v ∈ V c kv x v subject to X v ∈ V a iv x v ≤ 1 ∀ i ∈ I , x v ≥ 0 ∀ v ∈ V . (2) W e assume that the suppo rt sets defined for all i ∈ I , k ∈ K , and v ∈ V by V i = { v ∈ V : a iv > 0 } , V k = { v ∈ V : c kv > 0 } , I v = { i ∈ I : a iv > 0 } , K v = { k ∈ K : c kv > 0 } have b ound ed size. T hat is, we focu s on instances of (2) such that | I v | ≤ ∆ I V , | K v | ≤ ∆ K V , | V i | ≤ ∆ V I and | V k | ≤ ∆ V K for some c onstants ∆ I V , ∆ K V , ∆ V I and ∆ V K . T o avoid uninter esting degenerate cases, we assume that I v , V i and V k are no nempty . Example 1: Th e pr oblem in stance (1) is of the for m ( 2). There is one agent f or each link. Customers k 1 , k 2 , . . . , k 7 are ben eficiary p arties and acce ss poin ts i 1 , i 2 , . . . , i 5 are resources. W e have ∆ V K = 2 and ∆ V I = 3 . C. Lo cal, Distributed Setting The model of distributed co mputation a ssumed in this work is as follows. Each agent v ∈ V is an ind ependen t computatio nal entity; all agen ts execute the same deterministic algorithm . Ag ent v co ntrols the associated variable x v . The com munication between the agents is constrain ed by the communica tion graph , a hy pergraph H = ( V , E ) , where the vertices V a re the agen ts and the h ypered ges are d efined by E = { V i : i ∈ I } ∪ { V k : k ∈ K } . T wo agents can commu- nicate directly with each other if they a re adjacent in H . Let d H ( u, v ) be the sho rtest-path distance (numbe r of hyp eredges, hop cou nt) b etween u ∈ V an d v ∈ V in H , an d let B H ( v , r ) = { u ∈ V : d H ( u, v ) ≤ r } , r = 1 , 2 , . . . , be the set of nodes within distance at most r from n ode v in hypergrap h H . Each agent v ∈ V has the following local in put: the identifier of v , the h yperedg es V i for which v ∈ V i , and the hypere dges V k for which v ∈ V k . The hy peredg es are given as sets of identifiers. The algo rithm executed by the ag ents has the local horizon r if, for every agent v ∈ V , th e value set to x v is a fun ction of th e lo cal inpu t of the ag ents in B H ( v , r ) . Thus, each agent v executing an algorithm with a loca l horizon r is com pletely oblivious to the ne twork beyond B H ( v , r + 1) . In p articular, two distinct agents u, v ∈ V may have the same identifier if d H ( u, v ) ≥ 2 r + 3 . Thus, without loss of generality w e may assume that the local input of each agent h as a size (in bits) that dep ends on ly on r , ∆ I V , ∆ K V , ∆ V I , and ∆ V K , but n ot on the size of the network. Example 2: Fig. 2 shows the h ypergraph H for the p roblem instance (1). The structur e o f H closely reflects the structure of the network shown in Fig. 1: two agents are able to commun icate with ea ch other if they sha re the sam e access point or the same customer . i 1 i 2 i 3 i 4 i 5 k 1 k 2 k 3 k 4 k 5 k 6 k 7 1 3 5 2 4 6 7 8 9 11 13 10 12 14 Fig. 2. Hyper graph H for instan ce (1). D. Prior W ork and Contributions This paper con tributes to work in pro gress aimed at a complete characterisatio n of the local appro ximability of the max-min p acking pro blem. Here we pr ovide th e answer for the case of 0/1 coefficients and ∆ V K = 2 : Theor em 1: Let ∆ V I ≥ 2 be given. For any ǫ > 0 , there is a local algorith m for 0 /1 m ax-min packin g pro blem (2) with the approx imation r atio ∆ V I / 2 + ǫ , assumin g ∆ V K = 2 . This uppe r bo und is tig ht; by prior work [10, Corollary 2] we know that for a giv en ∆ V I ≥ 2 , th ere is n o local app roxi- mation algorithm f or (2) with an a pprox imation ratio less th an ∆ V I / 2 , and this holds ev en if ∆ V K = 2 . The safe algorithm [3], [10] ac hiev es the ap proxim ation ratio ∆ V I for (2). Our algorith m imp roves this by a factor o f 2 . The proo f of The orem 1 is structured as f ollows. First, Section II p resents a sim ple modification o f (2) w hich reduces the size of each co nstraint to 2 , that is, we arrive at an instan ce with ∆ V I = ∆ V K = 2 . The rest of this work, starting from Section III, presents a lo- cal appr o ximation scheme fo r th e special case ∆ V I = ∆ V K = 2 . A local appro ximation schem e is a family of local algorithm s such that for any ǫ > 0 there is a local algorith m wh ich achieves th e appr oximation ratio 1 + ǫ . The local ap proxim ation sch eme and the reductio n of Sec- tion II constitute the proof of Theorem 1. W e are able to achieve an appr oximation ratio arb itrarily close to the lower bound ∆ V I / 2 , in spite o f the crude r eduction th at we used in Section I I to bring ∆ V I down to 2 . I I . R E D U C I N G T H E S I Z E O F C O N S T R A I N T S W e first wishfully assume that fo r any ǫ ′ > 0 there is a local approx imation alg orithm which achieves the approx imation ratio 1 + ǫ ′ for th e spe cial case ∆ V I = ∆ V K = 2 . In this section, we show that this assump tion directly implies our ma in result, Theorem 1. Fix an ǫ > 0 and a b ound ∆ V I > 2 . Given an in stance of (2), we replace each co nstraint wh ich in volves more than 2 variables by several constrain ts wh ich inv olve exactly 2 variables each. In precise terms, co nsider i ∈ I such that | V i | > 2 . Let n = | V i | . Remove constraint i from the instance. Add  n 2  distinct constra ints x u + x v ≤ 1 wh ere u, v ∈ V i , u 6 = v . For example, the c onstraint x 1 + x 2 + x 3 ≤ 1 is replaced by the set of constraints x 1 + x 2 ≤ 1 , x 1 + x 3 ≤ 1 , and x 2 + x 3 ≤ 1 . This can be do ne by a local algo rithm. The set o f feasible solutions d iffers b etween the mod ified instance and the original instance. However , the utility o f a solution, ω ( x ) = min k P v c kv x v , is the same in both instances. Once we have constru cted the modified instance, we ap- ply the local approx imation scheme to solve it within the approx imation ratio 1 + ǫ ′ where ǫ ′ = 2 ǫ/ ∆ V I ; let x ′ be the solution. W e form a solution x of the original instance b y setting x v = 2 x ′ v / ∆ V I . First, we show that x is a feasible solu tion of the o riginal instance. As x ′ satisfies all con straints of size at most 2 , so does x . Now consid er a constrain t i in th e o riginal instance with m ore th an 2 variables. Ad d u p all new co nstraints which replace i in the m odified instance to obtain ( n − 1) x ′ 1 + ( n − 1 ) x ′ 2 + · · · + ( n − 1) x ′ n ≤  n 2  which im plies x 1 + x 2 + · · · + x n ≤ n/ ∆ V I ≤ 1 . Second, we show that x is a (∆ V I / 2 + ǫ ) -ap proxim ate solution o f the origin al instance. Le t x ∗ be an op timal solution of the orig inal instance. Now x ∗ is also a fe asible solution of the modified instance, and ω ( x ∗ ) is a lower bo und for the optimum value of th e mo dified instance. Th erefore ω ( x ′ ) ≥ ω ( x ∗ ) / (1 + ǫ ′ ) . By the choic e of x , we conc lude that ω ( x ) = 2 ω ( x ′ ) ∆ V I ≥ ω ( x ∗ ) ∆ V I / 2 + ǫ . I I I . P R E S E N TA T I O N A S A G R A P H W e pro ceed to show that there in deed is a local approx ima- tion scheme for the special case ∆ V I = ∆ V K = 2 . T o simplify the d iscussion, we presen t the proble m in stance a s a n und i- rected multigr aph G , where b oth ed ges and vertices are two- coloured . This a llows us to describe th e alg orithm in graph- theoretic term s. Example 3: W e use the following instance of (2) to illu s- trate the presentation as a grap h. T he ben eficiary parties are K = { k 1 , k 2 } and the constraints ar e I = { i 1 , i 2 , i 3 , i 4 } . The objective is to maximise ω = min { ( k 1 ) x 1 , ( k 2 ) x 2 + x 3 } subject to x 1 + x 2 ≤ 1 , ( i 1 ) x 1 + x 3 ≤ 1 , ( i 2 ) x 3 + x 4 ≤ 1 , ( i 3 ) x 4 + x 5 ≤ 1 , ( i 4 ) x 1 , x 2 , . . . , x 5 ≥ 0 . (3) The h ypergrap h H is illustrated in Fig. 3a; solid lines ar e hypere dges V i and dashed lines are hyper edges V k . An optimal solution with ω = 2 / 3 is x 1 = 2 / 3 , x 2 = x 3 = 1 / 3 , and x 4 = x 5 = 0 . A. Remove Non-Con tributing Agents W e have assumed that I v , V i and V k are nonem pty fo r each v ∈ V , i ∈ I and k ∈ K . W e can make a f urther assumption that K v is non empty for each v . If this is not the case f or some v , we can simp ly choose x v = 0 and r emove the ag ent v fro m the prob lem instance. If such change s make V i empty for some i , we can remove the red undan t constraint i . Th ese modification s can be done by a loc al alg orithm; th is is step illustrated in Fig. 3b. B. Hyper edges of S ize 2 Only At this p oint, | V k | ∈ { 1 , 2 } fo r ea ch k ∈ K and | V i | ∈ { 1 , 2 } for each i ∈ I . If | V k | = 1 for some k , we add a new agen t v into V . The variable x v controlled by agen t v is forc ed to 0 b y adding the co nstraint x v = 0 . Now we can set c kv = 1 without chan ging the solution . Similar ly , if | V i | = 1 for some (a) (b) (c) (d) 5 4 3 2 1 3 2 1 6 3 2 1 7 6 3 2 1 7 i 1 i 2 i 3 k 1 k 2 Fig. 3. T ransforming the problem instance. i , we add a new agent v into V , we for ce x v = 0 , and we set a iv = 1 . After these chan ges, | V k | = 2 for each k ∈ K and | V i | = 2 for each i ∈ I . This simple structu re comes a t the co st of having som e n ew agents v for which we f orce x v = 0 ; we also allo w K v = ∅ or I v = ∅ for such ag ents. Example 4: Th is step is illustrated in Fig. 3c. W e have transform ed (3) in to the following form : m aximise ω = min { x 1 + x 6 , x 2 + x 3 } subject to x 1 + x 2 ≤ 1 , x 1 + x 3 ≤ 1 , x 3 + x 7 ≤ 1 , x 1 , x 2 , x 3 ≥ 0 , and x 6 = x 7 = 0 . C. Construct the Graph Next we re present the mo dified problem instance a s an undirected multigraph G . Th e set of vertices of G is th e set of agents V ; the vertices v for wh ich we f orce x v = 0 are called 0 -vertices and the r emaining vertices are ca lled x -vertices. For each party k ∈ K , we have th e edge V k ; these are called K -edges. For each constraint i ∈ I , we have the ed ge V i ; these ar e called I -edges. There are no other edges. If there is an I -ed ge { u, v } , we say that u and v are I - a djacent . W e define K - adjacen t vertices ana logously . In o ther words, the vertices of G are colour ed with two colours, 0 and x , and the edges are colour ed with two colou rs, K and I . W e h av e encoded the origin al prob lem instance as a c oloured gra ph G . The graph G f or the instance (3) is illustra ted in Fig. 3d. Open cir cles are 0 - vertices a nd closed c ircles ar e x -vertices; solid lines are I -edges and d ashed lines are K -edges. I V . D E FI N I T I O N S Definition 1 : Let X, Y ∈ { K, I } . A ( v 0 , X , Y , v n ) - walk is a finite sequence of th e fo rm ( v 0 , e 1 , v 1 , e 2 , v 2 , . . . , e n , v n ) which satisfies all of the following: each v j is a vertex o f G ; each e j is an edg e o f the form { v j − 1 , v j } in the gra ph G ; the edges e j are alter nately K -ed ges and I -edges; e 1 is an X -edg e; and e n is a Y -edge. A ( v , X, Y , 0) - walk is a ( v , X, Y , u ) -walk where u is a 0 - vertex. A (0 , X, Y , 0) - walk is a ( v , X, Y , u ) - walk where v and u ar e a 0 -vertices. The K - len gth of a walk is the nu mber of K -edges in the walk. W e em phasise that (i) there can be rep eated ed ges and repeated vertices in walks; and ( ii) all walks throu ghou t this work are altern ating walks whe re K - edges and I -edge s alternate. Definition 2 : Let v b e an x -vertex and let X , Y ∈ { K, I } . W e write a ( v , X , Y , 0) for the minimum K -length of a ( v , X , Y , 0) -walk; if no ( v , X , Y , 0) -walk exists, then we defin e that a ( v , X , Y , 0) = ∞ . W e wr ite A ( v , X ) for the max imum K -length of a ( v , X , · , · ) -walk; if such walks with an arbitrarily large K -length exist, then we de fine that A ( v , X ) = ∞ . Example 5: Consid er the vertex 1 ∈ V in Fig. 3d. W e have a (1 , K , K , 0) = 1 , a (1 , K , I , 0) = ∞ , a (1 , I , K , 0) = 2 , a (1 , I , I , 0 ) = 1 , A (1 , K ) = 1 an d A (1 , I ) = 2 . Note that it is po ssible to have A (1 , K ) < a (1 , K , I , 0) . Fix a co nstant R ∈ { 1 , 2 , . . . } . W e define the b ounded versions of a and A by b ( v , X , Y , 0) = min { a ( v , X , Y , 0) , R } , B ( v , X ) = min { A ( v , X ) , R } for each X , Y ∈ { K, I } . V . L O C A L A L G O R I T H M Now we are read y to p resent the lo cal app roximatio n al- gorithm. Mo re accur ately , we pr esent a local appro ximation scheme, a family of algorithms parametrised by the constant R . The value of R determin es th e desired trade- off between th e local horizo n a nd the appr oximation r atio: the local ho rizon is 2 R and the appro ximation ratio is 1 + 1 / ( R − 1) . A lo cal algor ithm with any finite lo cal hor izon canno t determine the value o f a ( v , X , Y , 0) or A ( v, X ) in the ge neral case. However , assum ing that th e local horizo n is 2 R , th en each agen t v can determine locally w hether a ( v , X, K, 0) ≤ R or no t. Fu rthermo re, eac h a gent v can d etermine lo cally the value of b ( v , X, K, 0 ) and B ( v , X ) . It turn s out that this informa tion is sufficient in order to obta in an appro ximation algorithm . Our local alg orithm consists o f two step s. In th e first step, each x -vertex v determ ines whether a ( v , K, K , 0 ) ≤ R , whether a ( v , I , K, 0 ) ≤ R , and wha t a re the values of b ( v , I , K, 0 ) , b ( v , K , K , 0 ) , B ( v , I ) , and B ( v , K ) . T o imple- ment this step in a real-world distrib uted system, it is sufficient to pr opagate K -hop counters alon g altern ating walks in G for 2 R communic ation rounds. In th e second step, each x -vertex v p erform s the following local co mputation s. First, ch oose the value p v as f ollows. p v = b ( v , I , K, 0 ) if a ( v , K , K , 0 ) ≤ R, (4a) p v = min { b ( v, I , K, 0 ) , B ( v , K ) } otherwise . (4b) Choose th e value q v in an analo gous man ner . q v = b ( v , K, K, 0) if a ( v , I , K, 0) ≤ R, (5a) q v = min { b ( v , K, K , 0 ) , B ( v , I ) } otherwise . (5b) Finally , le t x v = p v / ( p v + q v ) . Th is value is the output of the vertex v . Example 6: In Fig . 3d, agent 1 ∈ V choo ses x 1 = 1 / 2 if R = 1 and x 1 = 2 / 3 if R ≥ 2 . W e now p roceed to show that the chosen values x v provide a f easible and near-optimal solutio n to (2). V I . A U X I L I A RY R E S U LT S W e begin with so me observations on the structure of G . First, each 0 - vertex is incid ent to exactly on e edge . Second , each x - vertex is incident to at least one K - edge and at least o ne I -edge. Giv en a n x -vertex v , we can construct both a ( v , K, · , · ) -walk and a ( v , I , · , · ) -walk with at least one edge, and we can extend such alter nating walks indefinitely until we meet a 0 - vertex. Lemma 2: For any x -vertex v , the lo cal algo rithm chooses p v ≥ 1 , q v ≥ 0 , and x v ≤ 1 . Pr oo f: Follo ws fr om the definitions. A. Bounds for the Optimum Now we give u pper bound s f or the o ptimum value of (2). Let x ∗ be an op timal solu tion and let ω ∗ be its o bjective v alue. Lemma 3: If there exists a ( v , I , K, u ) -walk of K -le ngth n , then x ∗ v − x ∗ u ≤ (1 − ω ∗ ) n . Pr oo f: If n = 1 , th en th ere is a vertex t , an I -ed ge { v , t } , and a K - edge { t, u } . Then x ∗ v + x ∗ t ≤ 1 and x ∗ t + x ∗ u ≥ ω ∗ , that is, x ∗ v − x ∗ u ≤ 1 − ω ∗ . T he claim follows by induction . Cor olla ry 4: If ther e exists a ( v, I , K, u ) -walk of K -leng th n , then ω ∗ ≤ 1 + 1 / n . Pr oo f: Follows f rom x ∗ u ≤ 1 , x ∗ v ≥ 0 , a nd the p revious lemma. Cor olla ry 5: If there exists a (0 , K , K , 0 ) - walk of K - length n , then ω ∗ ≤ 1 − 1 / n . Pr oo f: Th e case n = 1 is not possible so assume n > 1 . Then there is a ( v, I , K, u ) - walk of K -length n − 1 such tha t u is a 0 -vertex an d there is a K -edge between v and a 0 -vertex. Therefo re x ∗ u = 0 and x ∗ v ≥ ω ∗ . By Lem ma 3, (1 − ω ∗ )( n − 1) ≥ x ∗ v − x ∗ u = x ∗ v ≥ ω ∗ . The claim fo llows. Example 7: By Corollar y 5, ω ∗ = 2 / 3 in (3). B. Adjacent V ertices Lemma 6: If v and u are I -adjace nt x -vertices, then a ( v , I , K, 0) ≤ a ( u, K, K, 0) , b ( v , I , K, 0 ) ≤ b ( u, K , K , 0) , A ( v , I ) ≥ A ( u, K ) , B ( v , I ) ≥ B ( u, K ) . Pr oo f: Any g i ven ( u, K , Y , b ) -walk can be extended into a ( v , I , Y , b ) -walk by first taking the I -edge { v , u } . The K -length d oes not change. Lemma 7: If v and u are K -ad jacent x -vertices, then a ( v , K , K , 0) ≤ a ( u, I , K, 0) + 1 , b ( v , K , K, 0) ≤ b ( u , I , K, 0 ) + 1 , A ( v , K ) ≥ A ( u, I ) + 1 , B ( v , K ) ≥ B ( u, I ) . Pr oo f: Any giv en ( u, I , Y , b ) -walk can be extend ed into a ( v , K , Y , b ) -walk b y first taking the K -edg e { v , u } . Th e K -length inc reases by 1 . V I I . F E A S I B I L I T Y W e sh ow that th e values x v chosen by the loca l algorithm provide a feasible solu tion to (2). Consider an I -edg e { v , u } . W e ne ed to prove that x v + x u ≤ 1 . If v or u is a 0 -vertex, then th e claim h olds by Lem ma 2 ; we focu s o n the case that v a nd u are x -vertices. W e begin with the following lemma. Lemma 8: If v and u are I -ad jacent x -vertices, th en we have p v ≤ q u . Pr oo f: First, assume that a ( v , K, K , 0 ) ≤ R . In this case, Lemm a 6 im plies th at a ( u, I , K , 0) ≤ R . W e have p v = b ( v , I , K, 0 ) and q u = b ( u, K, K, 0) . W e ap ply Lemma 6 again to obtain p v ≤ q u . Second, assume that a ( v , K, K , 0) > R . In this case, Lemma 6 implies that b ( v , I , K , 0 ) ≤ b ( u, K , K , 0) an d B ( v , K ) ≤ B ( u, I ) . W e obtain p v = min { b ( v, I , K, 0 ) , B ( v , K ) } ≤ min { b ( u, K , K , 0 ) , B ( u, I ) } ≤ q u . W e con clude that the claim hold s in both cases. Cor olla ry 9: If v and u ar e I -adjacen t x -vertices, then x v + x u ≤ 1 . Pr oo f: By L emma 8, we h av e p v ≤ q u , an d by symmetr y , p u ≤ q v . Therefore x v + x u = p v p v + q v + p u p u + q u ≤ p v p v + p u + p u p u + p v = 1 . This completes th e pro of. V I I I . A P P R OX I M AT I O N G UA R A N T E E Next we show tha t the values x v chosen by the lo cal algorithm provide a near-optimal solutio n to (2). Con sider a K -edge { v , u } . W e show th at x v + x u ≥ αω ∗ where α = 1 − 1 /R . A. One x -vertex and One 0-vertex Let us first con sider the c ase wh ere v is an x -vertex and u is a 0 -vertex. Then we h av e q v ≤ b ( v , K, K, 0) = a ( v , K, K, 0) = 1 ≤ R and p v = b ( v , I , K, 0 ) . Lemma 10 : If a ( v , I , K, 0) ≤ R , then x v + x u ≥ ω ∗ . Pr oo f: W e have q v = 1 an d x v + x u = x v = 1 − 1 / n where n = p v + 1 . There exists a (0 , K, K , 0 ) - walk of K - length n , starting from u an d g oing throug h v . Corollary 5 implies ω ∗ ≤ 1 − 1 /n . Lemma 11 : If a ( v , I , K, 0) > R , then x v + x u ≥ αω ∗ . Pr oo f: W e have q v ≤ 1 , p v = R and x v + x u = x v ≥ 1 − 1 / R = α. In the optimal solution , x ∗ v ≤ 1 and x ∗ u = 0 . Therefo re ω ∗ ≤ 1 . Cor olla ry 12 : If x -vertex v and 0 -vertex u are K -adjacent, then x v + x u ≥ αω ∗ . Pr oo f: Apply Lemmata 10 and 11. B. T wo x -vertices Second, we conside r the case wher e both v and u are x - vertices. Ther e are se veral subcases to stud y . Lemma 13 : If a ( v , K, K , 0 ) ≤ R and a ( v , I , K, 0) ≤ R , then x v + x u ≥ ω ∗ . Pr oo f: Regardless of wh ether q u satisfies (5 a) or (5b), by L emma 7 q u ≤ b ( u, K , K , 0 ) ≤ b ( v , I , K , 0) + 1 = p v + 1 . If p u satisfies ( 4a), we have p u = b ( u, I , K , 0 ) ≥ b ( v , K , K , 0) − 1 = q v − 1 . Otherwise p u satisfies (4b). W e have R < a ( u, K, K , 0) ≤ a ( v , I , K, 0) + 1 ≤ R + 1 , th at is, a ( u, K, K , 0) = R + 1 . This implies A ( u, K ) ≥ R + 1 , B ( u, K ) = R and p u = min { b ( u, I , K , 0) , R } = b ( u, I , K , 0) ≥ b ( v , K, K, 0) − 1 = q v − 1 . In b oth cases we have p u ≥ q v − 1 . Th erefore x v + x u ≥ p v p v + q v + q v − 1 ( q v − 1) + ( p v + 1) = 1 − 1 p v + q v . As there exists a (0 , K , K , 0 ) -walk of K - length p v + q v , Corollary 5 imp lies that ω ∗ ≤ x v + x u . Lemma 14 : If a ( v , K, K , 0 ) ≤ R and a ( v , I , K, 0) > R , then x v + x u ≥ αω ∗ . Pr oo f: Regardless of wh ether q u satisfies (5 a) or (5b), we hav e q u ≤ R = b ( v , I , K, 0 ) = p v . As fo r p u , there are three c ases. First, if p u satisfies ( 4a), we have b y Lem ma 7 p u = b ( u, I , K, 0 ) ≥ b ( v , K , K , 0 ) − 1 ≥ q v − 1 . Second, if p u satisfies (4b) an d b ( u , I , K , 0) < B ( u , K ) , we have p u = b ( u, I , K, 0 ) ≥ b ( v , K , K , 0 ) − 1 ≥ q v − 1 . Third, if p u satisfies (4b) and b ( u, I , K , 0) ≥ B ( u, K ) , we have p u = B ( u, K ) ≥ B ( v , I ) ≥ q v > q v − 1 . In eac h case we have p u ≥ q v − 1 . Th erefore x v + x u = R R + q v + p u p u + q u ≥ R R + p u + 1 + p u p u + R ≥ 1 − 1 R = α. Because a ( v , K, K , 0) ≤ R , there exists a 0 - vertex inc ident to a K - edge and therefo re ω ∗ ≤ 1 . Lemma 15 : If a ( v , K, K, 0) > R , a ( v , I , K, 0 ) ≤ R , a ( u, K , K , 0) > R , an d a ( u , I , K , 0) ≤ R , then x v + x u ≥ ω ∗ . Pr oo f: By a ssumption, we have q v = b ( v , K, K, 0) = R , q u = b ( u, K , K , 0 ) = R. Lemma 7 implies R < a ( u, K , K , 0) ≤ a ( v , I , K, 0 ) + 1 ≤ R +1 ; therefore a ( v , I , K, 0) = R . Then there is a ( u, K, K , 0 ) - walk of K -leng th R + 1 , which implies A ( u, K ) > R . W e hav e b ( v , I , K, 0 ) = R and B ( u, K ) = R . Exchang ing th e roles of v a nd u , also b ( u, I , K , 0) = R and B ( v , K ) = R . T herefor e p v = p u = R . W e con clude tha t x v + x u = 1 / 2 + 1 / 2 . Because we have a ( v , I , K, 0) ≤ R , there exists a 0 -vertex incident to a K -edge and th erefore ω ∗ ≤ 1 . Lemma 16 : If a ( v , K, K, 0) > R , a ( v , I , K, 0 ) ≤ R , a ( u, K , K , 0) > R , an d a ( u , I , K , 0) > R , then x v + x u ≥ ω ∗ . Pr oo f: By a ssumption, we have q v = b ( v , K, K, 0) = R . As b ( u, K, K , 0 ) = R and b ( u, I , K , 0 ) = R , we h av e p u = B ( u , K ) and q u = B ( u, I ) . By the same argument as in the proof of Lemma 15, we ca n co nclude th at b ( v , I , K, 0) = R and B ( u, K ) = R . Therefo re p u = R and p v = B ( v , K ) . By L emma 7, p v = B ( v , K ) ≥ B ( u, I ) = q u . Therefo re x v + x u = p v p v + R + R R + q u ≥ q u q u + R + R R + q u = 1 . Again, th ere exists a 0 -vertex in cident to a K - edge an d therefor e ω ∗ ≤ 1 . Lemma 17 : If a ( v , K, K , 0 ) > R , a ( v , I , K , 0) > R , a ( u, K , K , 0) > R , a ( u, I , K, 0 ) > R , and A ( v , I ) ≥ R , then x v + x u ≥ αω ∗ . Pr oo f: By assumption, b ( v , K , K, 0) = R , b ( v , I , K, 0 ) = R , b ( u, K, K, 0) = R , an d b ( u, I , K, 0) = R . Therefo re p v = B ( v , K ) , q v = B ( v , I ) , p u = B ( u, K ) , and q u = B ( u, I ) . Lemma 7 imp lies p v = B ( v , K ) ≥ B ( u, I ) = q u , p u = B ( u, K ) ≥ B ( v , I ) = q v . Therefo re x v + x u ≥ q u q u + q v + q v q v + q u = 1 . As A ( v, I ) ≥ R , there exists a ( v , I , K, · ) -walk of K -length at least R . By Corollary 4, ω ∗ ≤ 1 + 1 /R . Th erefore x v + x u ≥ αω ∗ . Lemma 18 : If a ( v , K, K , 0 ) > R , a ( v , I , K , 0) > R , a ( u, K , K , 0) > R , a ( u, I , K, 0 ) > R , A ( v , I ) < R , and A ( u, I ) < R , then x v + x u ≥ ω ∗ . Pr oo f: By assump tion, b ( v , K, K , 0 ) = R , b ( v , I , K, 0 ) = R , b ( u, K, K, 0) = R , an d b ( u, I , K, 0) = R . Therefo re p v = B ( v , K ) , q v = B ( v , I ) = A ( v, I ) , p u = B ( u , K ) , and q u = B ( u , I ) = A ( u, I ) . By L emma 7, A ( v , K ) ≥ A ( u, I ) + 1 . As R ≥ A ( u, I ) + 1 , we also h av e B ( v , K ) ≥ A ( u, I ) + 1 . Therefor e p v = B ( v , K ) ≥ q u + 1 . By sy mmetry , p u ≥ q v + 1 . W e con clude that x v + x u ≥ q u + 1 ( q u + 1) + q v + q v + 1 ( q v + 1) + q u = 1 + 1 q u + q v + 1 . There is a ( · , K , K , · ) -walk of K -len gth A ( u, I ) + A ( v , I )+ 1 = q u + q v + 1 . Because a ( v , I , K, 0) > R and a ( u, I , K , 0) > R , both en dpoints of this walk are x -vertices. Hence, th e walk can b e extend ed into a ( · , I , K, · ) -walk o f the same K -length. By Co rollary 4, x v + x u ≥ ω ∗ . Cor olla ry 19 : If v an d u are K - adjacent x -vertices, th en x v + x u ≥ αω ∗ . Pr oo f: App ly L emmata 13 – 18 and th e symmetr y of v and u . A C K N O W L E D G E M E N T S W e than k V alentin Polishchuk for comments and discus- sions. This research was supp orted in part by the Acad emy of Finland, Grants 116 547 and 117499, and by Helsinki Gradu ate School in Compu ter Science and En gineering (Hecse). R E F E R E N C E S [1] N. L inial, “Locali ty in distribut ed graph algorithms, ” SIAM Journal on Computing , vol. 21, no. 1, pp. 193–201, 1992. 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