A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games

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We suggest the first strongly subexponential and purely combinatorial algorithm for solving the mean payoff games problem. It is based on iteratively improving the longest shortest distances to a sink in a possibly cyclic directed graph. We identify a new “controlled” version of the shortest paths problem. By selecting exactly one outgoing edge in each of the controlled vertices we want to make the shortest distances from all vertices to the unique sink as long as possible. The decision version of the problem (whether the shortest distance from a given vertex can be made bigger than a given bound?) belongs to the complexity class NP ∩ CO NP . Mean payoff games are easily reducible to this problem. We suggest an algorithm for computing longest shortest paths. Player M AX selects a strategy (one edge from each controlled vertex) and player M IN responds by evaluating shortest paths to the sink in the remaining graph. Then M AX locally changes choices in controlled vertices looking at attractive switches that seem to increase shortest paths lengths (under the current evaluation). We show that this is a monotonic strategy improvement, and every locally optimal strategy is globally optimal. This allows us to construct a randomized algorithm of complexity min ( poly · W , 2 O ( n log n ) ) , which is simultaneously pseudopolynomial ( W is the maximal absolute edge weight) and subexponential in the number of vertices n. All previous algorithms for mean payoff games were either exponential or pseudopolynomial (which is purely exponential for exponentially large edge weights).