Hard problems in max-algebra, control theory, hypergraphs and other areas

We introduce the max-atom problem (MAP): solving (in Z) systems of inequations of the form max(x, y)+k z, where x, y, z are variables and k ∈ Z. Our initial motivation for MAP was reasoning on delays in circuits using SAT Modulo Theories [10], viewing MAP as a natural extension of Difference Logic, i.e., inequations of the form x+ k y. Here we show that MAP is PTIME-equivalent to several rather different well-known problems for which no PTIME algorithm has been found so far, in spite of decades of independent efforts. One is on solving two-sided linear max-plus systems (Section 3 of this paper) that arise in Control Theory when modeling Discrete Event Systems, and another one on shortest paths in directed weighted hypergraphs (Section 4).