Adding Negative Prices to Priced Timed Games

TL;DR

This paper explores priced timed games with both negative and positive prices, establishing undecidability results for various objectives and providing a pseudo-polynomial algorithm for a specific subclass, advancing understanding of their computational complexity.

Contribution

It extends the study of priced timed games by analyzing cases with negative prices, proving undecidability in several variants, and offering a pseudo-polynomial algorithm for a bi-valued subclass.

Findings

Undecidability results for multiple cost objectives in PTGs with negative prices.

Identification of a bi-valued price-rate subclass with a pseudo-polynomial solution.

Extension of complexity boundaries for priced timed games beyond nonnegative prices.

Abstract

Priced timed games (PTGs) are two-player zero-sum games played on the infinite graph of configurations of priced timed automata where two players take turns to choose transitions in order to optimize cost to reach target states. Bouyer et al. and Alur, Bernadsky, and Madhusudan independently proposed algorithms to solve PTGs with nonnegative prices under certain divergence restriction over prices. Brihaye, Bruyere, and Raskin later provided a justification for such a restriction by showing the undecidability of the optimal strategy synthesis problem in the absence of this divergence restriction. This problem for PTGs with one clock has long been conjectured to be in polynomial time, however the current best known algorithm, by Hansen, Ibsen-Jensen, and Miltersen, is exponential. We extend this picture by studying PTGs with both negative and positive prices. We refine the undecidability results for optimal strategy synthesis problem, and show undecidability for several variants of optimal reachability cost objectives including reachability cost, time-bounded reachability cost, and repeated reachability cost objectives. We also identify a subclass with bi-valued price-rates and give a pseudo-polynomial algorithm to partially answer the conjecture on the complexity of one-clock PTGs.