The Max-Atom Problem and Its Relevance

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.