Deterministic Sub-exponential Algorithm for Discounted-sum Games with Unary Weights
arxiv(2024)
摘要
Turn-based discounted-sum games are two-player zero-sum games played on
finite directed graphs. The vertices of the graph are partitioned between
player 1 and player 2. Plays are infinite walks on the graph where the next
vertex is decided by a player that owns the current vertex. Each edge is
assigned an integer weight and the payoff of a play is the discounted-sum of
the weights of the play. The goal of player 1 is to maximize the discounted-sum
payoff against the adversarial player 2. These games lie in NP and coNP and are
among the rare combinatorial problems that belong to this complexity class and
the existence of a polynomial-time algorithm is a major open question. Since
breaking the general exponential barrier has been a challenging problem, faster
parameterized algorithms have been considered. If the discount factor is
expressed in unary, then discounted-sum games can be solved in polynomial time.
However, if the discount factor is arbitrary (or expressed in binary), but the
weights are in unary, none of the existing approaches yield a sub-exponential
bound. Our main result is a new analysis technique for a classical algorithm
(namely, the strategy iteration algorithm) that present a new runtime bound
which is n^O ( W^1/4√(n) ), for game graphs with n vertices and
maximum absolute weight of at most W. In particular, our result yields a
deterministic sub-exponential bound for games with weights that are constant or
represented in unary.
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