Deterministic Sub-exponential Algorithm for Discounted-sum Games with Unary Weights

TL;DR

This work addresses the value computation problem for turn-based discounted-sum games, which lies in NP ∩ coNP and is open to sub-exponential deterministic algorithms when weights are unary. The authors introduce a novel analysis of the Strategy Iteration algorithm by tying strategy profiles to rational polynomials P(λ)/Q(λ) and deriving tight bounds on roots of integer-coefficient polynomials, both lower and upper. This framework yields a deterministic sub-exponential bound of $n^{\mathcal{O}(W^{1/4}\sqrt{n})}$ iterations for DiscVal, and in particular a sub-exponential algorithm for DiscVal-Wun when weights are unary or constant. The results illuminate the role of polynomial-root behavior in the complexity of discounted-sum games and offer practical implications for reactive-systems analysis where weights are small or unary-coded.

Abstract

Turn-based discounted-sum games are two-player zero-sum games played on finite directed graphs. The vertices of the graph are partitioned between player 1 and player 2. Plays are infinite walks on the graph where the next vertex is decided by a player that owns the current vertex. Each edge is assigned an integer weight and the payoff of a play is the discounted-sum of the weights of the play. The goal of player 1 is to maximize the discounted-sum payoff against the adversarial player 2. These games lie in NP and coNP and are among the rare combinatorial problems that belong to this complexity class and the existence of a polynomial-time algorithm is a major open question. Since breaking the general exponential barrier has been a challenging problem, faster parameterized algorithms have been considered. If the discount factor is expressed in unary, then discounted-sum games can be solved in polynomial time. However, if the discount factor is arbitrary (or expressed in binary), but the weights are in unary, none of the existing approaches yield a sub-exponential bound. Our main result is a new analysis technique for a classical algorithm (namely, the strategy iteration algorithm) that present a new runtime bound which is $n^{O ( W^{1/4} \sqrt{n} )}$, for game graphs with $n$ vertices and maximum absolute weight of at most $W$. In particular, our result yields a deterministic sub-exponential bound for games with weights that are constant or represented in unary.

Figures (4)

• Figure 1: All roots
• Figure 2: Roots around 1
• Figure 4: A turn-based game.
• Figure 5: A turn-based game with two cycles.

Theorems & Definitions (30)

• theorem 1: condon1992ComplexityStochasticGames
• theorem 2
• Remark 3.1: Implications
• theorem 3: Main result
• Remark 3.2: Implications
• theorem 4: borwein1999littlewood
• Remark 4.1
• theorem 5
• lemma 1
• proof