Quasipolynomial-time solution of mean-payoff games with binary weights

Claims a deterministic algorithm computing the exact set of vertices from which Max can force nonnegative liminf mean payoff, for arbitrary signed binary weights, self-loops and parallel edges, in $2^{O((\log(L+2))^2)}$ bit operations; a reduction gives exact rational values and optimal positional strategies for both players, energy-game minimum credits, and tropical feasibility tests within the same bound. A randomized companion of the same date gives a Las Vegas algorithm with a polynomial-time certificate check. It does NOT give a polynomial-time algorithm, which the paper names as a further improvement. Conflicting claim: Truffet (arXiv:2603.26423, and a 2025 HAL note) claims a strongly polynomial algorithm via tropical optimization; the randomized companion gives a small example on which the printed rule of version 4 loses the optimum, and makes no claim about the 2025 note.

Claims a deterministic algorithm computing the exact set of vertices from which Max can force nonnegative liminf mean payoff, for arbitrary signed binary weights, self-loops and parallel edges, in $2^{O((\log(L+2))^2)}$ bit operations; a reduction gives exact rational values and optimal positional strategies for both players, energy-game minimum credits, and tropical feasibility tests within the same bound. A randomized companion of the same date gives a Las Vegas algorithm with a polynomial-time certificate check. It does NOT give a polynomial-time algorithm, which the paper names as a further improvement. Conflicting claim: Truffet (arXiv:2603.26423, and a 2025 HAL note) claims a strongly polynomial algorithm via tropical optimization; the randomized companion gives a small example on which the printed rule of version 4 loses the optimum, and makes no claim about the 2025 note.