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 NP intersection 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 absolute value of any edge 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, which improves a previous upper bound. The all-Bob algorithm is randomized, all other algorithms are deterministic.

翻译：本文研究了能量博弈特殊情况下的算法，能量博弈是一类基于图的回合制博弈，出现在反应式系统的定量分析中。在能量博弈中，加权有向图的顶点属于Alice或Bob。玩家控制当前所在顶点，将令牌移动到下一个顶点，其能量值根据边的权重发生变化。给定固定的起始顶点和初始能量，若令牌的能量始终保持非负，则Alice获胜；若某时刻能量降至零以下，则Bob获胜。确定能量博弈胜负的问题属于NP ∩ coNP。该问题是否存在多项式时间算法是一个长期未解决的开放问题。我们针对该问题的三种特殊情况提出了新算法。前两个结果聚焦于单玩家版本，即Alice或Bob控制整个博弈图。我们通过归约到全对非负前缀路径问题（APNP），为Alice控制的博弈图开发了$\tilde{O}(n^\omega W^\omega)$时间算法，其中$W$是边权绝对值的最大值，$\omega$是矩阵乘法的最佳指数。因此，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，我们改进了现有最优的$\tilde O(mn)$算法。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$O(n^{3-\epsilon})$时间算法（对任意$\epsilon > 0$）。对于Bob控制的博弈图，我们得到了近线性时间算法。第三个结果中，我们提出了值迭代算法的一种变体，并证明它能给出无负环博弈图的$O(mn)$时间算法，改进了先前上界。所有Bob算法是随机的，其他算法均为确定性的。