一元权重下折扣和博弈的确定性次指数算法
Deterministic Sub-exponential Algorithm for Discounted-sum Games with Unary Weights
AI总结：
针对折扣因子任意、权重为一元表示的回合制折扣和博弈缺乏次指数算法的问题，通过新分析技术改进策略迭代算法，得到n顶点、最大权重W下$n^{O(W^{1/4}\sqrt{n})}$的确定性次指数运行时间界。
摘要：
回合制折扣和博弈是在有限有向图上进行的双人零和博弈。图的顶点被划分为玩家1和玩家2所有。博弈过程是图上的无限游走，下一个顶点由当前顶点的所属玩家决定。每条边都被赋予一个整数权重，一次博弈的收益为该博弈过程中所有边权重的折扣和。玩家1的目标是在对抗敌对玩家2时最大化折扣和收益。这类博弈属于NP和coNP类，是少数属于该复杂度类的组合问题之一，是否存在多项式时间算法是一个重大开放问题。由于突破通用指数下界一直是个难题，研究者们开始考虑更快的参数化算法。当折扣因子用一元表示时，折扣和博弈可以在多项式时间内求解。但如果折扣因子是任意的（或用二进制表示）、仅权重为一元表示，现有方法都无法得到次指数时间界。我们的主要成果是对经典算法（即策略迭代算法）提出了一种新的分析技术，给出了新的运行时间界：对于有n个顶点、最大绝对权重不超过W的博弈图，运行时间为$n^{O ( W^{1/4} \sqrt{n} )}$。特别地，我们的结果为权重为常数或用一元表示的博弈提供了确定性的次指数时间界。
英文摘要：
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 $n^{O ( W^{1/4} \sqrt{n} )}$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
