game algorithm designers

Abstract: Mean-Payoff Games (MPGs) form an interesting family of deterministic, infinite-duration, zero-sum, turn-based games played by two players on a finite weighted directed graph. The goal of the players is to minimize, respectively maximize, the $\textit{limiting average}$ of the costs of the edges on an infinite path constructed by the two players. The complexity of solving MPGs is a long-standing open question. The decision problem associated with these games is known to be in $\text{NP}\cap \text{co-NP}$, and even $\text{UP}\cap \text{co-UP}$, yet the problem is not known to be in $\text{P}$. This is one of the simplest combinatorial problems with such a status. A full solution of an MPG gives the $\textit{value}$ of each vertex of the game, along with a pair of $\textit{optimal strategies}$ for the two players. (Each player has a succinct $\textit{positional}$ optimal strategy.) The problem of finding optimal strategies for the players given the values of all vertices is known as the $\textit{strategy recovery}$ problem. We describe a simple reduction from the problem of solving MPGs to the strategy recovery problem for MPGs. This shows that the strategy recovery problem for MPGs is essentially as hard as solving MPGs from scratch, i.e., without knowing the values. The reduction uses simple $\textit{game transformations}$ and simple $\textit{self-reducibility}$ and $\textit{peeling}$ arguments. This answers a question of Andersson and Miltersen. Mamino obtained a similar result for $\textit{stochastic}$ games with a limiting average objective, but his technique does not extend to deterministic games such as MPGs. We go one step further and show that finding an optimal strategy for one of the players of an MPG remains hard even if an optimal strategy of the second player, and hence all the values, are known.

Abstract: We study games on finite graphs with labelled or weighted edges in which plays end as soon as the first loop is formed. The resulting play has a lasso shape: a path (head) followed by a cycle (loop). We consider two semantics for determining the winner: the loop semantics, where only the loop matters, and the lasso semantics, where the entire play is taken into account. We show that adopting the lasso semantics may have a dramatic impact on both the complexity of the game and the memory required to play optimally. Mean-payoff games with lasso semantics are no longer positionally determined, and the threshold problem – deciding whether the value of the game is at least 0 – is PSPACE-complete. In contrast, lasso parity games are equivalent to weak parity games, remain positionally determined and are solvable in polynomial time. Furthermore, we show that if a positionally determined winning objective is closed under cyclic permutation and both the objective and its complement are closed under concatenation, then it is equivalent to the parity objective. Finally, we consider mod-k games, where the winner depends on the length of the path modulo k >= 2. We show that these games, in both semantics, are PSPACE-complete. Alongside computational complexity, we analyze the memory required to play optimally. We establish that for any game positionally determined in loop semantics, memory of size 2^n is sufficient in lasso semantics. We complement this by showing exponential memory lower bounds for lasso mean-payoff and mod-k games.

Abstract: We show that Arc Kayles is PSPACE-complete. This solves a question raised by Schaefer in 1978.