An exponential lower bound for Cunningham’s rule

Abstract

In this paper we give an exponential lower bound for Cunningham's least recently considered (round-robin) rule as applied to parity games, Markhov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules.Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not.The paper is self-contained and requires no previous knowledge of parity games.

A Parity Game Tale of Two Counters

“…For some time, the focus was on variations of strategy iteration algorithms, where a suitable improvement rule could lead to a polynomial solution. Friedmann et al provided superpolynomial or exponential lower bounds for many of such rules [1,16,17,20,21,22,23], as well as Fearnley and Savani recently [15]. For a long time, the main attractor-based algorithm was the recursive algorithm by McNaughton [32] and Zielonka [34], for which Friedmann also showed an exponential lower bound [19] and later Gazda and Willemse [25] improved upon this lower bound.…”

The Complexity of Zadeh's Pivot Rule

“…In particular, it was not only applied for Markov decision processes but also parity games and other classes of games. Among other, all of the constructions presented in [Fea10a,Fri11c,FHZ11b,FHZ11a,AF17] implement this idea and even share the same design principle. Before discussing Friedmann's subexponential lower bound construction in detail, we thus briefly discuss this general design principle.…”

“…In particular, the chosen improvement rule, that is, the procedure deciding how to change the current strategy, highly influences the behavior of the algorithm. For both parity games and Markov decision processes, superpolynomial lower bounds were established for the most important and natural improvement rules [Fri09,Fea10a,Fri11a,Fri11c,FHZ11b,FHZ11a,AF17,DH19]. It is an open question whether there is an efficiently computable improvement rule guaranteeing a polynomial number of iterations in the worst case.…”

“…Lower Bound Proven via Dantzig's rule [Dan51] exponential [KM72] Klee-Minty cube Shadow vertex rule [GS55,Bor87] exponential [Mur80] Klee-Minty cube Lexicographic rule [DOW55] exponential [DS15] Klee-Minty cube L I rule [Jer73] exponential [Jer73] Klee-Minty cube Bland's rule [Bla77] exponential [AC78] Klee-Minty cube S E rule [FG92] exponential [GS79] Klee-Minty cube R E rule subexponential [FHZ11b] Markov decision process Cunningham's rule [Cun79] exponential [AF17] Markov decision process R T B [Kal91] subexponential [FHZ11b] Markov decision process Zadeh's L E rule [Zad80] exponential [DFH19] Markov decision process R F [Kal92, SW92, Kal97] subexponential [FHZ11b] Markov decision process Randomized Bland [Mat94] subexponential [Han12] Markov decision process Table . . : An overview over important pivot rules that were developed for the simplex algorithm.…”

“…Rather recently, Avis and Friedmann proved that the worst-case running time using this pivot rule is also exponential [AF17].…”