Looking for all solutions of the Max Atom Problem (MAP)

Abstract

This present paper provides the absolutely necessary corrections to the previous work entitled {\it A polynomial Time Algorithm to Solve The Max-atom Problem} (arXiv:2106.08854v1). The max-atom-problem (MAP) deals with system of scalar inequalities (called atoms or max-atom) of the form: $x \leq a + \max(y,z)$. Where $a$ is a real number and $x,y$ and $z$ belong to the set of the variables of the whole MAP. A max-atom is said to be positive if its scalar $a$ is $\geq 0$ and stricly negative if its scalar $a <0$. A MAP will be said to be positive if all atoms are positive. In the case of non positive MAP we present a saturation principle for system of vectorial inequalities of the form $x \leq A x + b$ in the so-called $(\max,+)$-algebra assuming some properties on the matrix $A$. Then, we apply such principle to explore all non-trivial solutions (ie $\neq -\infty$). We deduce a strongly polynomial method to express all solutions of a non positive MAP. In the case a positive MAP which has always the vector $x^{1}=(0)$ as trivial solution we show that looking for all solutions requires the enumeration of all elementary circuits in a graph associated with the MAP. However, we propose a strongly polynomial method wich provides some non trivial solutions.