Abstract
We study two-player multi-weighted reachability games played on a finite directed graph, where an agent, called \({{\mathrm{\mathcal {P}_1}}}\), has several quantitative reachability objectives that he wants to optimize against an antagonistic environment, called \({{\mathrm{\mathcal {P}_2}}}\). In this setting, we ask what cost profiles \({{\mathrm{\mathcal {P}_1}}}\) 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. Finally, the constrained existence problem is proved in \({{\mathrm{\textsc {PTime}}}}\) for the lexicographic order and \({{\mathrm{\textsc {PSpace}}}}\)-complete for the componentwise order.
T. Brihaye—Partly supported by the F.R.S.- FNRS under grant n\(^{\circ }\)T.0027.21.
A. Goeminne—F.R.S.-FNRS postdoctoral researcher.
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Notes
- 1.
Where the following notation is used: \({{\mathrm{\overline{\mathbb {N}}}}}= {{\mathrm{\mathbb {N}}}}\cup \{ \infty \}\).
- 2.
To lighten the notations, we omit the mention of \(\lesssim \) in subscript.
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Brihaye, T., Goeminne, A. (2023). Multi-weighted Reachability Games. In: Bournez, O., Formenti, E., Potapov, I. (eds) Reachability Problems. RP 2023. Lecture Notes in Computer Science, vol 14235. Springer, Cham. https://doi.org/10.1007/978-3-031-45286-4_7
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