In this paper, we study algorithms for special cases of energy games, a class of turn-based games on graphs that show up in the quantitative analysis of reactive systems. In an energy game, the vertices of a weighted directed graph belong either to Alice or to Bob. A token is moved to a next vertex by the player controlling its current location, and its energy is changed by the weight of the edge. Given a fixed starting vertex and initial energy, Alice wins the game if the energy of the token remains nonnegative at every moment. If the energy goes below zero at some point, then Bob wins. The problem of determining the winner in an energy game lies in $\mathsf{NP} \cap \mathsf{coNP}$. It is a long standing open problem whether a polynomial time algorithm for this problem exists. We devise new algorithms for three special cases of the problem. The first two results focus on the single-player version, where either Alice or Bob controls the whole game graph. We develop an $\tilde{O}(n^\omega W^\omega)$ time algorithm for a game graph controlled by Alice, by providing a reduction to the All-Pairs Nonnegative Prefix Paths problem (APNP). Thus we study the APNP problem separately, for which we develop an $\tilde{O}(n^\omega W^\omega)$ time algorithm. For both problems, we improve over the state of the art of $\tilde O(mn)$ for small $W$. For the APNP problem, we also provide a conditional lower bound which states that there is no $O(n^{3-\epsilon})$ time algorithm for any $\epsilon > 0$, unless the APSP Hypothesis fails. For a game graph controlled by Bob, we obtain a near-linear time algorithm. Regarding our third result, we present a variant of the value iteration algorithm, and we prove that it gives an $O(mn)$ time algorithm for game graphs without negative cycles, which improves a previous upper bound.

翻译：本文研究能量博弈特殊情形的算法，这类图上的回合制博弈出现在反应系统的定量分析中。在能量博弈中，加权有向图的顶点属于Alice或Bob。由当前所在顶点的控制玩家将标记移动到下一个顶点，其能量随边的权重变化。给定固定起始顶点和初始能量，若标记能量始终非负则Alice获胜；若某时刻能量低于零则Bob获胜。确定能量博弈胜负的问题属于$\mathsf{NP} \cap \mathsf{coNP}$，是否存在该问题的多项式时间算法是长期未解决的开放问题。我们针对该问题的三个特殊情形设计了新算法。前两个结果关注单人版本，即Alice或Bob控制整个博弈图。通过归约到所有点对非负前缀路径问题(APNP)，我们为Alice控制的博弈图开发了$\tilde{O}(n^\omega W^\omega)$时间算法。为此我们单独研究APNP问题，并为其设计了$\tilde{O}(n^\omega W^\omega)$时间算法。对于小参数$W$，这两个问题均改进了现有最优的$\tilde O(mn)$复杂度。针对APNP问题，我们还给出了条件下界：除非APSP假设不成立，否则不存在$O(n^{3-\epsilon})$时间的算法（对任意$\epsilon > 0$）。对于Bob控制的博弈图，我们获得了近线性时间算法。第三个结果是提出了值迭代算法变体，并证明该算法能在$O(mn)$时间内处理无负环的博弈图，改进了先前上界。