Permuted max-algebraic eigenvector problem is NP-complete

Abstract

Let a⊕b=max(a,b) and a⊗b=a+b for a,b∈R¯:=R∪{-∞} and extend these operations to matrices and vectors as in conventional linear algebra. The following eigenvector problem has been intensively studied in the past: Given A∈R¯n×n find all x∈R¯n,x≠(-∞,…,-∞)T (eigenvectors) such that A⊗x=λ⊗x for some λ∈R¯. The present paper deals with the permuted eigenvector problem: Given A∈R¯n×n and x∈R¯n, is it possible to permute the components of x so that it becomes a (max-algebraic) eigenvector of A? Using a polynomial transformation from BANDWIDTH we prove that the integer version of this problem is NP-complete.