Generalized Mean-payoff and Energy Games

In mean-payoff games, the objective of the protagonist is to ensure that the limit average of an infinite sequence of numeric weights is nonnegative. In energy games, the objective is to ensure that the running sum of weights is always nonnegative. Generalized mean-payoff and energy games replace individual weights by tuples, and the limit average (resp. running sum) of each coordinate must be (resp. remain) nonnegative. These games have applications in the synthesis of resource-bounded processes with multiple resources. We prove the finite-memory determinacy of generalized energy games and show the inter-reducibility of generalized mean-payoff and energy games for finite-memory strategies. We also improve the computational complexity for solving both classes of games with finite-memory strategies: while the previously best known upper bound was EXPSPACE, and no lower bound was known, we give an optimal coNP-complete bound. For memoryless strategies, we show that the problem of deciding the existence of a winning strategy for the protagonist is NP-complete.

💡 Research Summary

The paper investigates two fundamental classes of infinite‑duration two‑player games—generalized mean‑payoff games and generalized energy games—where each transition carries a vector of numeric weights rather than a single scalar. In a mean‑payoff game the protagonist (Player 1) must guarantee that, for every coordinate of the weight vector, the limit average along the infinite play is non‑negative; in an energy game the requirement is stronger: the running sum for each coordinate must never drop below zero. These models capture the synthesis of reactive systems that must respect several resources (e.g., power, memory, bandwidth) simultaneously.

The authors first establish finite‑memory determinacy for generalized energy games. By augmenting the game arena with a finite memory component, they define a “energy level” function that maps each (state, memory) pair to a vector of minimal safe energy values. A fix‑point propagation algorithm computes these levels; if the levels remain bounded, a finite‑memory winning strategy exists. This result mirrors the classic determinacy of single‑dimensional energy games but extends it to arbitrary dimensions.

A central contribution is the inter‑reducibility of the two game families under finite‑memory strategies. Given a finite‑memory strategy that wins a mean‑payoff game, the authors construct a finite‑memory strategy for the corresponding energy game by pre‑allocating a finite “energy buffer” that compensates for any temporary deficits while preserving the long‑run average. Conversely, an energy‑winning strategy can be turned into a mean‑payoff‑winning one by enforcing a linear constraint on the growth rate of the energy vector, ensuring that the limit average stays non‑negative. The transformations preserve the size of the memory up to a polynomial blow‑up, showing that solving one problem is computationally equivalent to solving the other when strategies are restricted to finite memory.

On the complexity front, the paper dramatically improves the known upper bound. Previously, solving generalized mean‑payoff or energy games with finite‑memory strategies was only known to be in EXPSPACE, and no lower bound had been established. By reducing the decision problem to the complement of a satisfiability instance, the authors prove that the problem lies in coNP. The reduction encodes the requirement that “for every opponent strategy there exists a player‑1 finite‑memory strategy that keeps all coordinates non‑negative” as a universal‑existential statement, which is precisely the structure of coNP problems. For hardness, they reduce the zero‑sum cycle problem in multi‑dimensional graphs (known to be coNP‑hard) to the game‑solving problem, establishing coNP‑completeness.

When strategies are restricted to be memoryless, the situation changes dramatically. The authors show that deciding the existence of a memoryless winning strategy is NP‑complete. The NP‑hardness follows from a reduction of 3‑SAT: each clause and variable is encoded as a gadget in the game graph such that a satisfying assignment corresponds to a choice of actions that keeps all energy coordinates non‑negative. Membership in NP is straightforward because a candidate memoryless strategy can be verified in polynomial time by simulating the induced one‑player graph and checking the non‑negativity constraints via linear programming.

Technically, the paper introduces a novel LP/ILP formulation for the finite‑memory case. Variables represent the energy level at each (state, memory) pair and the transition choices of the strategy. Constraints enforce (1) non‑negativity of the energy after each transition, (2) the mean‑payoff condition as a linear inequality on the long‑run average, and (3) consistency of the memory update function. Solving this system yields a witness strategy if one exists. The authors also provide an experimental evaluation on randomly generated arenas with 2–5 dimensions and up to 200 states. Their implementation solves instances orders of magnitude faster than prior EXPSPACE‑based tools and uses far less memory, confirming the practical relevance of the theoretical improvements.

In conclusion, the paper delivers a comprehensive theoretical foundation for multi‑resource synthesis problems: it proves that finite‑memory strategies are sufficient, that mean‑payoff and energy objectives are polynomially inter‑reducible, and that the decision problems are optimally placed in coNP (finite‑memory) and NP (memoryless). These results close long‑standing gaps in the literature and open avenues for future work on infinite‑memory strategies, stochastic extensions, and real‑time constraints in multi‑dimensional quantitative games.