Directed hypergraphs and applications
Discrete Applied Mathematics - Special issue: combinatorial structures and algorithms
The complexity of stochastic games
Information and Computation
Discrete Applied Mathematics
Communication: A strongly polynomial algorithm for solving two-sided linear systems in max-algebra
Discrete Applied Mathematics
Max-plus Algebraic Tools for Discrete Event Systems, Static Analysis, and Zero-Sum Games
FORMATS '09 Proceedings of the 7th International Conference on Formal Modeling and Analysis of Timed Systems
Mean-payoff games and propositional proofs
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Mean-payoff games and propositional proofs
Information and Computation
The complexity of integer bound propagation
Journal of Artificial Intelligence Research
Tropical linear-fractional programming and parametric mean payoff games
Journal of Symbolic Computation
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Let F be a conjunction of atoms of the form max (x ,y ) + k ≥ z , where x , y , z are variables and k is a constant value. Here we consider the satisfiability problem of such formulas (e.g., over the integers or rationals). This problem, which appears in unexpected forms in many applications, is easily shown to be in NP. However, decades of efforts (in several research communities, see below) have not produced any polynomial decision procedure nor an NP-hardness result for this --- apparently so simple --- problem. Here we develop several ingredients (small-model property and lattice structure of the model class, a polynomially tractable subclass and an inference system) which altogether allow us to prove the existence of small unsatisfiability certificates, and hence membership in NP intersection co-NP. As a by-product, we also obtain a weakly polynomial decision procedure. We show that the Max-atom problem is PTIME-equivalent to several other well-known --- and at first sight unrelated --- problems on hypergraphs and on Discrete Event Systems, problems for which the existence of PTIME algorithms is also open. Since there are few interesting problems in NP intersection co-NP that are not known to be polynomial, the Max-atom problem appears to be relevant.