## The Complexity of Mean Payoff Games on Graphs (1996)

Venue: | Theoretical Computer Science |

Citations: | 95 - 3 self |

### BibTeX

@ARTICLE{Zwick96thecomplexity,

author = {Uri Zwick and Mike Paterson},

title = {The Complexity of Mean Payoff Games on Graphs},

journal = {Theoretical Computer Science},

year = {1996},

volume = {158},

pages = {343--359}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the complexity of finding the values and optimal strategies of mean payoff games on graphs, a family of perfect information games introduced by Ehrenfeucht and Mycielski and considered by Gurvich, Karzanov and Khachiyan. We describe a pseudo-polynomial time algorithm for the solution of such games, the decision problem for which is in NP " co-NP. Finally, we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by Condon. These games are also known to be in NP " co-NP, but no polynomial or pseudo-polynomial time algorithm is known for them. 1 Introduction Let G = (V; E)be a finite directed graph in which each vertex has at least one edge going out of it. Let w : E ! f\GammaW; : : : ; 0; : : : ; Wg be a function that assigns an integral weight to each edge of G. Ehrenfeucht and Mycielski [EM79] studied the following infinite two-person game played on such a graph. The game starts at a vertex a 0 2 V . The first player chooses an edge e...

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Citation Context ...ons made in this section are very similar to observations made by Gurvich, Karzanov and Khachiyan [GKK88] and by Karzanov and Lebedev [KL93]. An O(jV j \Delta jEj) algorithm of Karp [Kar78] (see also =-=[CLR90]-=-, p. 548) for finding the maximum (or minimum) mean weight cycle of a weighted graph G = (V; E) supplies, almost immediately, an efficient purely combinatorial algorithm for such special cases. Theore... |

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Citation Context ...e transition from state i to state j costs a ij , and serving a request of type t from state i costs b it . What, in the worst-case, is the average cost of serving a request? Borodin, Linial and Saks =-=[BLS92]-=- performed a competitive analysis of such systems, which they call on-line metrical task systems . If we look at the worst-case instead, we get a bipartite mean payoff game G = (V 1 ; V 2 ; E) played ... |

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Citation Context ...as well as in ~ P (pseudo-polynomial time), but is not yet known to be in P. This gives the MPG problem a rare status shared only by a few number-theoretic problems, such as primality [Pra75]. Condon =-=[Con92] has -=-recently studied the complexity of simple stochastic games (SSG's) introduced originally by Shapley [Sha53]. Condon shows that the decision problem corresponding to SSG's is also in NP " co-NP. W... |

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Citation Context ...number is read is lost. What is the average number of comparisons needed per input element in the worst case? Selection and sorting problems with limited storage were considered by Munro and Paterson =-=[MP80]-=-. They allowed several passes over the input stream however. The problem can again be formulated as a bipartite mean payoff game G = (V 1 ; V 2 ; E) played by the designer of a selection algorithm and... |

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Citation Context ...hus in NP"co-NP as well as in ~ P (pseudo-polynomial time), but is not yet known to be in P. This gives the MPG problem a rare status shared only by a few number-theoretic problems, such as prima=-=lity [Pra75]-=-. Condon [Con92] has recently studied the complexity of simple stochastic games (SSG's) introduced originally by Shapley [Sha53]. Condon shows that the decision problem corresponding to SSG's is also ... |

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Citation Context ...ound in O(jV j 2 ) time. 2 The maximum (or minimum) mean weight cycle in a graph G = (V; E) with relatively small weights can be found more efficiently using the scaling algorithms of Orlin and Ahuja =-=[OA92]-=- and Young and Tarjan and Orlin [YTO91]. If there are only two different edge weights then it can be found even faster, using an algorithm of Butkovic and Cuninghame-Green [BC92]. Could methods used b... |

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Citation Context ... (or minimum) mean weight cycle in a graph G = (V; E) with relatively small weights can be found more efficiently using the scaling algorithms of Orlin and Ahuja [OA92] and Young and Tarjan and Orlin =-=[YTO91]-=-. If there are only two different edge weights then it can be found even faster, using an algorithm of Butkovic and Cuninghame-Green [BC92]. Could methods used by Karp's algorithm, or by the other max... |

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Citation Context ... polynomial time algorithm for SSG's is yet known. Some exponential algorithms for the problem are described in [MC90]. A subexponential randomized algorithm for SSG's was recently obtained by Ludwig =-=[Lud95]. Con-=-don [Con92] actually shows containment in NP " co-NP of the decision problem that corresponds to SSG's of the following restricted form. The outdegree of each non-sink vertex is exactly two and t... |

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Citation Context ...e positional optimal strategies for the finite game. The positional optimal strategies of the finite game are also positional optimal strategies for the infinite game. Gurvich, Karzanov and Khachiyan =-=[GKK88]-=-, unaware of the work of Ehrenfeucht and Mycielski [EM79], considered a slightly wider class of mean payoff games which they refer to as cyclic games . A cyclic game is played on a directed graph G = ... |

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Citation Context ... before serving this request. The weight of all the edges of the form i ! (i; j) is 0. The weight of an edge (i; j) ! k is a ik + b kj . Consider finite-window on-line string matching algorithms (see =-=[CHPZ95]-=- for a definition). What, in the worst-case, is the average number of comparisons that an optimal algorithm has to perform per text character? The problem can be formulated as a bipartite mean payoff ... |

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Citation Context ...ositional optimal strategies thereby giving a constructive proof of their existence. Cyclic games with prohibitions , a further generalisation of cyclic games, were considered by Karzanov and Lebedev =-=[KL93]-=-. Generalizations of mean payoff games to n players, where n ? 2, have been considered by Alpern [Alp91]. Ehrenfeucht and Mycielski [EM79] give no efficient algorithm for finding optimal strategies fo... |

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Citation Context ...ession. The goal of the first player is again to maximise, and of the second to minimise, the long-term average weight of the path formed. Gurvich et al. [GKK88] note that a general theorem of Moulin =-=[Mou76]-=- on stationary optimal strategies in stochastic games implies that both players of a cyclic game have positional optimal strategies. The theorem of Moulin is proved non-constructively using a fixed po... |

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Citation Context ...us attention on the possibly easier problem of obtaining a polynomial time algorithm for MPG's. Various path-forming games, such as the many different versions of geography were studied by Bodlaender =-=[Bod93]-=-, Fraenkel and Simonson [FS93] and Fraenkel, Scheinerman and Ullman [FSU93]. Many of these games are PSPACE-complete. It is therefore somewhat surprising that the mean payoff games that we are conside... |

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Citation Context ...prohibitions , a further generalisation of cyclic games, were considered by Karzanov and Lebedev [KL93]. Generalizations of mean payoff games to n players, where n ? 2, have been considered by Alpern =-=[Alp91]-=-. Ehrenfeucht and Mycielski [EM79] give no efficient algorithm for finding optimal strategies for the finite and infinite games. Gurvich, Karzanov and Khachiyan [GKK88] give, as mentioned, an exponent... |

5 |
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Citation Context ...f a simple stochastic game have positional optimal strategies. Simple stochastic games were first studied by Shapley [Sha53]. Many variants of them have been studied since then (see Peters and Vrieze =-=[PV87]-=- for a survey). Condon [Con92] was the first to study simple stochastic games from a complexity theory point of view. She showed that the natural decision problem corresponding to SSG's (i.e., given a... |

4 |
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Citation Context ...e G and a rational number 0 ! ffs1, is the value of G at least ff?) is in NP " co-NP. No polynomial time algorithm for SSG's is yet known. Some exponential algorithms for the problem are describe=-=d in [MC90]. A s-=-ubexponential randomized algorithm for SSG's was recently obtained by Ludwig [Lud95]. Condon [Con92] actually shows containment in NP " co-NP of the decision problem that corresponds to SSG's of ... |

3 |
Algorithms to solve some classes of network minmax problems and their applications, Cybernetics (29
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Citation Context ...claim that there exists a polynomial time algorithm for finding values and optimal strategies of cyclic games. According to Karzanov (personal communication), this claim was made by mistake. Lozovanu =-=[Loz91]-=-,[Loz93] also considers cyclic games and claims a strongly polynomial time algorithm for them. He describes a simple reduction from cyclic games to simple acyclic games. Unfortunately, his reduction i... |

2 |
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Citation Context ...algorithm for MPG's. Various path-forming games, such as the many different versions of geography were studied by Bodlaender [Bod93], Fraenkel and Simonson [FS93] and Fraenkel, Scheinerman and Ullman =-=[FSU93]-=-. Many of these games are PSPACE-complete. It is therefore somewhat surprising that the mean payoff games that we are considering do have relatively efficient algorithms. Mean payoff games arise natur... |

2 |
Strongly polynomial algorithms for finding minimax paths in networks and solution of cyclic games
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(Show Context)
Citation Context ...at there exists a polynomial time algorithm for finding values and optimal strategies of cyclic games. According to Karzanov (personal communication), this claim was made by mistake. Lozovanu [Loz91],=-=[Loz93]-=- also considers cyclic games and claims a strongly polynomial time algorithm for them. He describes a simple reduction from cyclic games to simple acyclic games. Unfortunately, his reduction is not va... |

1 |
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(Show Context)
Citation Context ...ms of Orlin and Ahuja [OA92] and Young and Tarjan and Orlin [YTO91]. If there are only two different edge weights then it can be found even faster, using an algorithm of Butkovic and Cuninghame-Green =-=[BC92]-=-. Could methods used by Karp's algorithm, or by the other maximum mean weight cycle algorithms, be used to obtain a more efficient algorithm for the general case? Could scaling methods be used to spee... |