Reachability Games with Relaxed Energy Constraints

TL;DR

This paper investigates reachability games with energy constraints, exploring relaxations of upper bounds and their computational complexities, revealing new insights into memory requirements and problem hardness.

Contribution

It introduces relaxed energy constraints in reachability games, analyzing their computational complexity and memory requirements, which were not previously understood.

Findings

Weak upper bound relaxations require memory in one-player games.

Reachability with relaxed bounds is polynomial-time solvable in one-player scenarios.

Allowing bounded violations increases complexity to PSPACE or EXPTIME.

Abstract

We study games with reachability objectives under energy constraints. We first prove that under strict energy constraints (either only lower-bound constraint or interval constraint), those games are LOGSPACE-equivalent to energy games with the same energy constraints but without reachability objective (i.e., for infinite runs). We then consider two kinds of relaxations of the upper-bound constraints (while keeping the lower-bound constraint strict): in the first one, called weak upper bound, the upper bound is absorbing, in the sense that it allows receiving more energy when the upper bound is already reached, but the extra energy will not be stored; in the second one, we allow for temporary violations of the upper bound, imposing limits on the number or on the amount of violations. We prove that when considering weak upper bound, reachability objectives require memory, but can still be solved in polynomial-time for one-player arenas; we prove that they are in co-NP in the two-player setting. Allowing for bounded violations makes the problem PSPACE-complete for one-player arenas and EXPTIME-complete for two players.