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), where $W$ is the maximum weight and $\omega$ is the best exponent for matrix multiplication. 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.

翻译：本文研究能量博弈特例的算法，这类博弈是基于图的回合制博弈，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。玩家控制当前位置的令牌，将其移动到下一个顶点，令牌的能量根据边上的权重变化。给定固定的起始顶点和初始能量，如果令牌的能量在每一时刻都保持非负，则Alice获胜；若能量在某时刻降至零以下，则Bob获胜。确定能量博弈胜者的问题属于$\mathsf{NP} \cap \mathsf{coNP}$。是否存在该问题的多项式时间算法是一个长期未解决的公开问题。我们为该问题的三个特例设计了新算法。前两个结果聚焦于单人版本，即由Alice或Bob控制整个博弈图。我们通过归约到全源非负前缀路径问题（APNP），为Alice控制的博弈图开发了$\tilde{O}(n^\omega W^\omega)$时间算法，其中$W$是最大权重，$\omega$是矩阵乘法的最佳指数。为此，我们单独研究了APNP问题，并为其开发了$\tilde{O}(n^\omega W^\omega)$时间算法。对于这两个问题，当$W$较小时，我们改进了现有的$\tilde O(mn)$复杂度。对于APNP问题，我们还给出了一个条件下界：除非APSP假设不成立，否则不存在任何$\epsilon > 0$的$O(n^{3-\epsilon})$时间算法。对于Bob控制的博弈图，我们获得了近线性时间算法。关于第三个结果，我们提出了一种值迭代算法的变体，并证明对于无负环的博弈图，该算法实现了$O(mn)$时间。