Hyperplane Separation Technique for Multidimensional Mean-Payoff Games

We consider both finite-state game graphs and recursive game graphs (or pushdown game graphs), that can model the control flow of sequential programs with recursion, with multi-dimensional mean-payoff objectives. In pushdown games two types of strategies are relevant: global strategies, that depend on the entire global history; and modular strategies, that have only local memory and thus do not depend on the context of invocation. We present solutions to several fundamental algorithmic questions and our main contributions are as follows: (1) We show that finite-state multi-dimensional mean-payoff games can be solved in polynomial time if the number of dimensions and the maximal absolute value of the weight is fixed; whereas if the number of dimensions is arbitrary, then problem is already known to be coNP-complete. (2) We show that pushdown graphs with multi-dimensional mean-payoff objectives can be solved in polynomial time. (3) For pushdown games under global strategies both single and multi-dimensional mean-payoff objectives problems are known to be undecidable, and we show that under modular strategies the multi-dimensional problem is also undecidable (whereas under modular strategies the single dimensional problem is NP-complete). We show that if the number of modules, the number of exits, and the maximal absolute value of the weight is fixed, then pushdown games under modular strategies with single dimensional mean-payoff objectives can be solved in polynomial time, and if either of the number of exits or the number of modules is not bounded, then the problem is NP-hard. (4) Finally we show that a fixed parameter tractable algorithm for finite-state multi-dimensional mean-payoff games or pushdown games under modular strategies with single-dimensional mean-payoff objectives would imply the solution of the long-standing open problem of fixed parameter tractability of parity games.

💡 Research Summary

The paper investigates mean‑payoff games with multidimensional objectives on two fundamental classes of graphs: finite‑state game graphs and pushdown (recursive) game graphs, which model the control flow of sequential programs with recursion. The authors distinguish two types of strategies for pushdown games: global strategies, which may depend on the entire execution history, and modular strategies, which are confined to local memory within each module and thus ignore the calling context. Their contributions can be summarized as follows.

1. Finite‑state multidimensional mean‑payoff games.
   When both the number of dimensions d and the maximal absolute weight W are fixed constants, the authors present a polynomial‑time algorithm. The key technique is a hyperplane‑separation approach: the set of feasible long‑run average vectors is characterized by a finite family of linear inequalities, each corresponding to a candidate separating hyperplane. Because d and W are bounded, the number of candidates is polynomially bounded, and each candidate can be checked by solving a linear program or a one‑dimensional mean‑payoff sub‑game. This yields a runtime polynomial in the size of the game graph. In contrast, if d is part of the input, the problem is already known to be coNP‑complete.

2. Pushdown graphs (no adversarial player).
   For pushdown graphs (i.e., one‑player settings) with multidimensional mean‑payoff objectives, the authors show that the problem remains in polynomial time even without any restriction on the number of dimensions. The algorithm reduces the pushdown system to a finite‑state system by summarizing the effect of each recursive call using its “summary” vector and then applies the hyperplane‑separation method. This demonstrates that recursion alone does not increase the computational difficulty for the one‑player case.

3. Pushdown games under global strategies.
   It is known that both single‑ and multi‑dimensional mean‑payoff objectives become undecidable when players may use global strategies. The paper confirms this undecidability for the multidimensional case as well.

4. Pushdown games under modular strategies.
   For modular strategies, the single‑dimensional problem is NP‑complete, while the multidimensional version is shown to be undecidable, extending the hardness landscape. The authors further identify tractable fragments: if the number of modules M, the number of exits per module E, and the maximal absolute weight W are all fixed, then the single‑dimensional modular game can be solved in polynomial time. If either M or E is unbounded, the problem becomes NP‑hard. The algorithm constructs a finite‑state abstraction where each module is replaced by a finite set of “summary” edges that capture all possible local strategies; the resulting game is then solved using known techniques for finite‑state mean‑payoff games.

5. Fixed‑parameter tractability (FPT) implications.
   The authors prove a meta‑result: an FPT algorithm for either (i) finite‑state multidimensional mean‑payoff games parameterized by the number of dimensions, or (ii) pushdown games with modular strategies and a single‑dimensional mean‑payoff objective parameterized by M, E, and W, would immediately yield an FPT algorithm for parity games. Since the FPT status of parity games is a long‑standing open problem, this result suggests that achieving FPT for the studied mean‑payoff problems is unlikely without a breakthrough in parity‑game algorithms.

Overall, the paper introduces a novel hyperplane‑separation technique that bridges multidimensional quantitative objectives with classic linear‑algebraic tools, provides a detailed complexity map for both finite‑state and recursive game settings, and connects the tractability of these quantitative games to fundamental open questions in algorithmic game theory. The results have practical relevance for the verification of resource‑aware recursive programs, where modular strategies correspond to locally implemented controllers, and they delineate precisely which parameter regimes admit efficient analysis.