Calculating eigenvectors in max-algebra by mutation-sunflower method

TL;DR

This paper addresses the computation of max-eigenvectors in the max-times algebra for irreducible nonnegative matrices by introducing the mutation-sunflower method. The approach constructs mutation-sunflower matrices $A^*_1,\dots,A^*_r$ with a single nonzero per row that preserve the max-eigenvalue $\mu(A)$, and proves that a basis for the principal max-eigencone $V_{\oplus}(A,\mu(A))$ can be formed from principal-sunflower max-eigenvectors corresponding to each connectivity component of the critical graph. It provides a concrete algorithm to generate this basis and discusses robustness to choices within components, along with special-case simplifications and comparisons to existing methods such as Karp’s algorithm and the power method. While the method is not polynomial in general, it offers practical advantages for sparse matrices or matrices with prescribed circuit structure. Overall, the work supplements current max-times techniques by yielding a targeted tool for extracting a basis of max-eigenvectors in structured irreducible matrices.

Abstract

In this article we introduce a new method, which we call a mutation-sunflower method, for calculating max-eigenvectors of a nonnegative irreducible $n\times n$ matrix $A$. Our method works in the general irreducible case, but it is in comparison with existing methods most effective for some special classes of matrices for example for sparse enough matrices. Our method reduces to solving max-eigenproblems for simple mutation-sunflower matrices that have exactly one positive entry in each row. We include some instructive examples.

Theorems & Definitions (2)

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