Solving Min-Max Problems with Applications to Games

Solving Min-Max Problems with Applications to Games Daniel Andersson ⋆ Department of Computer Science, U niversi ty of Aarhus, Den mark koda@daimi .au.dk Abstract. W e refine existing general netw ork optimization techniques, giv e new characterizations for the class of problems to which they can b e applied, and show that they can also b e used to solve v ari ous tw o- pla yer games in almost linear time. Among these is a new v aria nt of the net w ork interdictio n problem, where the interdictor wa nts to destro y high-capacity paths from the source to th e destination using a vertex- wise li mited budget of arc remo v als. W e also sho w that replacing the limit av erage in mean pay off games by the maxim u m weig ht results in a class of games amenable to these techniques. 1 Min-Max P roblems W e consider problems whose instances hav e t w o parts: a finite discrete part (e.g., a graph), the structur e , and a list of n ob jects from some to- tally ordered set K (e.g., r eal n umber s ), the c omp ar ables . W e adopt the standard ran d om access machine mo del augmen ted w ith the capabilit y of comparing any t w o of the giv en comparables in constan t time. W e will only consider problems wh ere the solution is alwa ys one of the compara- bles in the problem instance (in our mac h ine m o del, this is output in the form of an ind ex b et ween 1 and n ). A min-max pr oblem is suc h a problem w here for an y fixed structure A and num b er of comparables n , the resulting function f A,n : K n → K can b e computed b y a m in-max circuit. The or der e d version of a min -max problem is obtained by alwa ys su pplying as additional input a p ermuta- tion that n on-decreasingly ord ers the list of comparables. Clearly , if the ordered v er s ion can b e solv ed in time ξ ( A, n ), then b y completely sorting the comparables, the original problem can b e solv ed in O ( n log n + ξ ( A, n )) time and O ( n log n ) comparisons. Ho w ever, it turns out that when n log n dominates o v er ξ ( A, n ), it is p ossib le to impro v e b oth of these b ounds sim u ltaneously . In particular, w e ha ve the follo wing results. ⋆ Researc h supported by Center for A lgorithmic Game The ory , funded by The Carl s- b erg F oundation. Theorem 1. If the or der e d version of a min-max pr oblem with structur e A and n c omp ar ables c an b e solve d in time ξ ( A, n ) ≥ n , then the original pr oblem c an b e solve d i n O ( ξ ( A, n ) lo g ∗ n ) time and O ( n ) c omp arisons. Theorem 2. If the or der e d version of a min-max pr oblem with structur e A and n c omp ar ables c an b e solve d in time ξ ( A, n ) ≥ n , then the original pr oblem c an b e solve d in O ( ξ ( A, n )(1 + log ∗ ξ ( A, n ) − log ∗ ( ξ ( A, n ) /n ))) time. The alg orithms are presen ted in Sectio n 2. The b asic tec hnique em- plo yed was used b y Gab ow and T arjan [6] to find a b ottlenec k spanning tree in a digraph with n ve rtices and m edges in O ( m log ∗ n ) time. Pun- nen [12] ga ve a general f ormulation of the tec h nique and used it to s olve the b ottlenec k S teiner ab orescence problem within the s ame b ound . In this pap er, we pro vid e a m ore d etailed analysis, showing that the almost linear r u nning time can b e ac hieve d while still mainta ining the asymptotically optimal linear num b er of comparisons (Theorem 1), and w e main tain the distinction b et wee n the structure and the num b er of com- parables to obtain a b ound that is nev er worse than the sorting m etho d (Theorem 2). Also, recen t w ork b y Litma n et al.[9] sho ws that the functions com- putable by min-max circuits are exact ly the c ontinuous or d er statistics of Rice [13] — con tin u ous 1 functions whose output is alw a ys one of th eir in- puts — b y showing that they are b oth the set of fu nctions that comm ute with every monotone fun ction. This yields tw o additional c h aracteriza- tions of the class of min-max problems. While previous work has b een fo cused on net work optimization prob- lems, we demonstrate in S ection 3 that the metho d s can b e applied to differen t classes of tw o-pla y er games as well. 2 Algorithms W e are giv en an instance of a min-max pr ob lem with structur e A and comparables x 1 , . . . , x n . T o simplify the presenta tion, w e shall assume that all the giv en x i are distin ct. Supp ose that w e hav e an algorithm Or d ( A, ( x ′ 1 , . . . , x ′ n ) , I ) that solve s the instance ( A, ( x ′ 1 , . . . , x ′ n )) in time ξ ( A, n ) ≥ n when x ′ I [1] ≤ · · · ≤ x ′ I [ n ] . The idea is to partition the comparables in to groups to obtain a coarse ordered instance of the p roblem. Due to con tinuit y , s olving the coarse 1 in the order topology for K and the box top ology based on it for K n instance correctly iden tifies the group in whic h the answer to th e original instance can b e foun d, and we can recursive ly cont in u e the searc h within this group. W e fi rst consid er division int o only tw o group s in eac h iteration: 1. I := h 1 , 2 , 3 , . . . , n i 2. l o := 1 3. hi := n 4. While l o < hi (a) m := ⌊ ( hi + l o ) / 2 ⌋ (b) Rearrange I so that max j ∈ I [ l o...m ] x j < min j ∈ I [ m +1 ...hi ] x j . (c) F or j := 1 . . . l o − 1 : x ′ I [ j ] := x I [1] . (d) F or j := l o . . . m : x ′ I [ j ] := x I [ l o ] . (e) F or j := m + 1 . . . hi : x ′ I [ j ] := x I [ hi ] . (f ) F or j := hi + 1 . . . n : x ′ I [ j ] := x I [ n ] . (g) If Or d ( A, ( x ′ 1 , . . . , x ′ n ) , I ) ∈ I [ l o . . . m ], then hi := m . (h) Else l o := m + 1. 5. Return l o . Let n i b e th e v alue of hi − l o + 1 in th e i ’th iteration of the while-loop. Step 4b can b e p erformed in O ( n i ) time and comparisons by a linear time median findin g algorithm[3]. Thus, eac h iteration of the wh ile-loop tak es O ( ξ ( A, n )) time, and since n i is halve d ( ± 1) with eac h iterati on, the total time is O ( ξ ( A, n ) log n ) and the total num b er of comparisons is O ( n ). An easy w a y to imp ro ve this algorithm is to break the w h ile-loop as so on as we can afford to simply sort the comparables in th e remaining in terv al, i.e., when n i log n i ≤ n . This brings the n u m b er of iterations do wn to O (log log n ). T o get ev en closer to linear time, w e need to partition the comparables in to more than tw o groups. Recursiv e p artitioning around the med ians (i.e., a prematurely cancelled p erfect quic ksort) can construct 2 k groups in O ( k n i ) time. The num b er of groups will b e c hosen so as to b alance the w ork sp ent on partitioning and solving. If we allot O ( n/i 2 ) time for partitioning in the i ’th iteratio n, then w e can ensure th at n i +1 ≤ n i 2 − 2 n/ ( n i i 2 ) , and solving this recurr en ce (see App end ix) sho w s that the num b er of iterations drops to O (log ∗ n ). S in ce P ∞ i =1 1 i 2 con verge s, the total num b er of comparisons remains O ( n ). If we are willing to giv e up the O ( n ) b ound on comparisons , we can instead allot O ( ξ ( A, n )) time for p artitioning in eac h iteration and get O (1 + log ∗ ξ ( A, n ) − log ∗ ( ξ ( A, n ) /n )) iteratio ns (see App endix for details). Note that if ξ ( A, n ) = Ω ( n log log lo g · · · log n ) for some fix ed n umber of log’s, then this b ound is actually O (1). 3 Applications to Games 3.1 Simple Rec ursiv e Games Andersson et al.[1] in tro duce the class of simple r e cursive games , whic h are t wo -pla ye r zero-sum p erf ect information extensiv e form games where the game tree is r eplaced by a game gr aph . Infi nite pla y is inte rpreted as a zero pay off. Thus, they are s imilar (but incomparable) to Condon’s simple sto c h astic games [4], but with no mo ves of c h ance and an arbitrary n u m b er of different pay offs. Figure 1 sho ws an example game. P S f r a g r e p l a c e m e n t s Min Min Max Max start s − 1 1 2 3 4 5 6 7 8 9 P S f r a g r e p l a c e m e n t s Min Min Max Max start s − 1 1 2 3 4 5 6 7 8 9 Fig. 1. A simp le recursive game ( left ) and a solution ( right ). In [1], the problem of find in g a wea k solution (v alue and minimax strategies for a sp ecified starting p osition) is reduced to a min-max prob- lem, whic h is solv ed u s ing alg orithms analogous to those presen ted herein. Ho we v er, f or th e case of optimal n u m b er of comparisons, T horem 1 con- stitutes an imp ro vemen t o ver the p revious r esult. 3.2 Maxim um P ay off Games Me an p ayoff games [14] is a class of infin ite duration games pla yed on a w eighte d sink-free digraph ( V , E , w ). S tarting from a sp ecified no de, t wo pla yers tak e tur ns c h o osing outgoing arcs to create an infinite path with arcs e 0 , e 1 , e 2 , . . . , and pla y er Min pa ys to pla yer Max the limit av erage lim n →∞ 1 n X 0 ≤ i ≤ n w ( e i ) . (1) In disc ounte d p ayoff games , (1) is replaced with a discoun ted av erage P 0 ≤ i λ i w ( e i ), w here λ ∈ [0 , 1) is a parameter. Both classes are closely related to model c hec king for the mo dal µ -calculus, a nd they are inter- esting from a complexit y-theoretic p oint of view, since they give rise to problems that are among the few natural ones kno wn to b e in NP ∩ coNP but not kn o wn to b e in P . The b est kno wn upp er boun ds for solving them are randomized sub exp onential time [2]. P S f r a g r e p l a c e m e n t s Min Min Max Max s t a r t s − 1 1 2 3 4 5 6 7 8 9 P S f r a g r e p l a c e m e n t s Min Min Max Max s t a r t s − 1 1 2 3 4 5 6 7 8 9 Fig. 2. A maxim um pa yo ff game ( left ) and a solution ( right ). In terestingly , if w e c hange the ev aluation function for the stream from limit or discounte d a v erage to simply max 0 ≤ i w ( e i ), the complexit y of solving the games plummets. Finding the v alue of a game w ith a sp ecified starting no de s is no w a min -max p roblem, and the ordered v ersion ca n b e solv ed in linear time as follo ws: 1. V alue [ V ] := + ∞ 2. While there is an arc (a) Let e b e a max-w eigh t arc, and let v b e its tail. (b) If v belongs to Min and e is not its only outgoing arc, remo ve e . (c) Else, increase the w eigh t o f v ’s incoming arcs to w ( e ), set V alue [ v ] := w ( e ), remo v e v ’s outgoing arcs except e , and con tract e (its head determining the type of the resulting no d e). 3. Return V alue [ s ] 3.3 Widest P ath In terdiction Motiv at ed by military applica tions, McMasters and Mustin [10] defi n ed and stu died network inter diction pr oblems , where an interdictor w ith lim- ited resources wan ts to inhibit the usefulness of a net work. In particular, shortest p ath inter diction is a we ll-studied v arian t [7,8]. Phillips [11] con- sidered minimizing th e maxim um fl o w. In this pap er w e define and consider a new v ariant : widest p ath in- ter diction . W e are giv en a connected net wo rk ( V , E ) with arc capacities c : E → R + , a s ource s , a destination t , and a budget k ( v ) f or eac h ve rtex v . F rom eac h vertex v the in terdictor remo v es at most k ( v ) outgoing arcs, so that th e wid th of th e wid est path from s to t is minimized (the w idth of a p ath is the m inim u m capacit y along that path). W e could also, as in [8], allo w more general budget constraint s, sp ecified by a certain class of oracles, without affecting the asymptotic ru nning time of our algo rithms. This is a min-max p roblem. The ordered v ersion can b e solv ed in O ( | E | ) time as f ollo ws: 1. Width [ V ] := 0 2. Width [ t ] := + ∞ 3. While inde g ( t ) > 0 (a) Let e b e a max-capacit y incoming arc to t , and let v b e its tail. (b) If v ’s budget allo ws it, remo v e e (this is an optimal remo v al). (c) Else, ensure that arcs to v ha ve capacit y at most Width [ v ] := c ( e ), remo ve all outgoing arcs f rom v , and merge v with t . 4. Return Wi dth [ s ] T o imp lemen t the algorithm, we need a max-priorit y queue for the incoming arcs to t . Ho wev er, th e extracted v alues form a non-increasing sequence, so w e can fi rst replace the capacities with in tegers fr om 1 to | E | (using the gi v en p ermutatio n) and then use an arr ay of | E | buc k ets to sta y within linear time. By Th eorem 1, th e widest path in terdiction problem can b e solv ed in in O ( | E | log ∗ | E | ) time and O ( | E | ) comparisons, w hic h is also , consid er in g b ound s in | E | only , the b est known b oun d for the (uninterdicted) widest path problem in d ir ected graphs [6]. If w e ins tead consider a glob al bu dget, i.e., an y set of at most k arcs ma y b e r emo ve d, then w e can solve the w idest path in terdiction problem b y p erformin g a binary searc h for the smallest capacit y q suc h that remo v- ing all arcs of ca pacit y at most q yields a net w ork w ith arc connectivit y at most k . Using Dinic’s blocking flo w algorithm [5] to compute th e arc con- nectivit y , the total runn ing time b ecomes O ( | E | min ( | E | 1 / 2 , | V | 2 / 3 ) log | E | ). This is in stark co n tr ast to shortest path in terdiction with a global budget [8], w here the maximin path length is NP-hard to app ro ximate within a factor less than 2. Ac kno wledgemen t s. 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T op olo gy Appl. , 85:299–318, 1998. 14. Uri Zwic k and Mik e S. P aterson. The complexit y of mean pay off games on graphs. The or etic al Computer Scienc e , 158(1–2):343–35 9, 1996. App endix: Tw o Recurrences O ( n/i 2 ) Time for P artitioning First we consider th e r ecur rence n 1 = n n i +1 ≤ n i 2 − 2 n/ ( n i i 2 ) . Letting x i = n/n i w e get x 1 = 1 x i +1 ≥ x i 2 2 x i /i 2 . Lemma 1. F or i ≥ 4 , x i ≥ i 2 log 2 ( i + 1) . Pr o of. By ind uction. ⊓ ⊔ Lemma 2. F or i ≥ 4 , x i +2 ≥ 2 x i . Pr o of. By d efi nition and Lemma 1, we hav e for i ≥ 4, x i +2 ≥ x i +1 2 2 x i +1 ( i +1) 2 ≥ 2 2 ( i +1) 2 x i 2 2 x i /i 2 ≥ 2 x i . ⊓ ⊔ It follo ws from Lemma 2 that m in { i : n i ≤ 1 } = O (log ∗ n ). O ( ξ ( A, n )) Time for P artitioning W e let m = ξ ( A, n ) and consider the r ecurrence n 1 = n n i +1 ≤ n i 2 − m/n i . Letting x i = n/n i w e get x 1 = 1 x i +1 ≥ x i 2 x i m/n ≥ 2 x 1 m/n , and th us min { i : n i ≤ 1 } = O (log ∗ 2 m/n n ) = O (1 + log ∗ m − log ∗ ( m/n )).