Finiteness conditions for graph algebras over tropical semirings

TL;DR

This paper explores conditions under which graph parameters valued in tropical semirings can be computed efficiently, extending classical results to broader algebraic structures and establishing that rank finiteness is a weak restriction.

Contribution

It introduces join matrices as a generalization of connection matrices for tropical-valued graph parameters and links rank finiteness to polynomial-time computability on certain graph classes.

Findings

Rank finiteness of join matrices implies polynomial-time computability on bounded clique-width graphs.

Graph parameters with fixed finite rank join matrices are computable in polynomial time.

There are uncountably many integer-valued graph parameters with fixed finite rank matrices, showing rank finiteness is a weak condition.

Abstract

Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lov{\'a}sz and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-width. In the case of graph parameters with values in arbitrary commutative semirings, this remains true for graph classes of bounded linear clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that definability of a graph parameter in Monadic Second Order Logic implies rank finiteness. We also show that there are uncountably many integer valued graph parameters with connection matrices or join matrices of fixed finite rank. This shows that rank finiteness is a much weaker assumption than any definability assumption.