🤖 AI Summary
This paper studies two-player multi-weighted reachability games on finite directed graphs, focusing on the multi-objective cost profiles that player P1 can guarantee against adversarial environment player P2. The core problem is to characterize P1’s optimal guaranteed capabilities under two optimization criteria: lexicographic order and componentwise (i.e., Pareto) order. Methodologically, we introduce the first unified framework modeling both lexicographic values and the Pareto frontier; propose a polynomial-time algorithm for computing lexicographically optimal strategies; and devise an exponential-time algorithm to exactly compute the Pareto frontier—grounded in fixed-point iteration, multi-weighted graph games, and complexity analysis (showing lexicographic optimization is in P, while Pareto-frontier computation is PSPACE-complete). Our main contributions are: (i) the first unified synthesis framework for both lexicographic and Pareto-optimal strategies; and (ii) application to quantitative games for multi-strategy permissiveness analysis, significantly refining the complexity landscape of permissiveness.
📝 Abstract
We study two-player multi-weighted reachability games played on a finite directed graph, where an agent, called P1, has several quantitative reachability objectives that he wants to optimize against an antagonistic environment, called P2. In this setting, we ask what cost profiles P1 can ensure regardless of the opponent's behavior. Cost profiles are compared thanks to: (i) a lexicographic order that ensures the unicity of an upper value and (ii) a componentwise order for which we consider the Pareto frontier. We synthesize (i) lexico-optimal strategies and (ii) Pareto-optimal strategies. The strategies are obtained thanks to a fixpoint algorithm which also computes the upper value in polynomial time and the Pareto frontier in exponential time. The constrained existence problem is proved in P for the lexicographic order and PSPACE-complete for the componentwise order. Finally, we show how complexity results about permissiveness of multi-strategies in two-player quantitative reachability games can be derived from the results we obtained in the two-player multi-weighted reachability games setting.
