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Register a new userA Polynomial Time Algorithm for Spatio-Temporal Security Games
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Soheil Behnezhad
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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An ever-important issue is protecting infrastructure and other valuable targets from a range of threats from vandalism to theft to piracy to terrorism. The defender can rarely afford the needed resources for a 100% protection. Thus, the key question is, how to provide the best protection using the limited available resources. We study a practically important class of security games that is played out in space and time, with targets and patrols moving on a real line. A central open question here is whether the Nash equilibrium (i.e., the minimax strategy of the defender) can be computed in polynomial time. We resolve this question in the affirmative. Our algorithm runs in time polynomial in the input size, and only polylogarithmic in the number of possible patrol locations (M). Further, we provide a continuous extension in which patrol locations can take arbitrary real values. Prior work obtained polynomial-time algorithms only under a substantial assumption, e.g., a constant number of rounds. Further, all these algorithms have running times polynomial in M, which can be very large.
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In a mean-payoff parity game, one of the two players aims both to achieve a qualitative parity objective and to minimize a quantitative long-term average of payoffs (aka. mean payoff). The game is zero-sum and hence the aim of the other player is to either foil the parity objective or to maximize the mean payoff. Our main technical result is a pseudo-quasi-polynomial algorithm for solving mean-payoff parity games. All algorithms for the problem that have been developed for over a decade have a pseudo-polynomial and an exponential factors in their running times; in the running time of our algorithm the latter is replaced with a quasi-polynomial one. By the results of Chatterjee and Doyen (2012) and of Schewe, Weinert, and Zimmermann (2018), our main technical result implies that there are pseudo-quasi-polynomial algorithms for solving parity energy games and for solving parity games with weights. Our main conceptual contributions are the definitions of strategy decompositions for both players, and a notion of progress measures for mean-payoff parity games that generalizes both parity and energy progress measures. The former provides normal forms for and succinct representations of winning strategies, and the latter enables the application to mean-payoff parity games of the order-theoretic machinery that underpins a recent quasi-polynomial algorithm for solving parity games.
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