Resolving infeasibility in extremal algebras

Abstract

Two extremal algebras ℬ=(B⊕,⊗) based on a linearly ordered set (B, ⩽) are considered: in the maxmin algebra ⊖=max, ⊗ = min and in the maxgroup algebra ⊕ = max and ⊗ is a group operation. If a system A ⊗ x = b of linear equations over an extremal algebra is insolvable, then any subset of equations such that its omitting leads to a solvable subsystem is called a relieving set. We show that the problem of finding the minimum cardinality relieving set is NP-complete in the maxmin algebra already for bivalent systems, while it is polynomially solvable for bivalent systems in maxgroup algebra and also NP-complete for trivalent systems.