Counting $4\times 4$ Matrix Partitions of Graphs

TL;DR

This paper establishes a complexity dichotomy for counting $4\times 4$ matrix partitions of graphs, showing the problem is either efficiently solvable or #P-complete, using computer-assisted proofs and conjecturing this extends to larger matrices.

Contribution

It provides a complete classification of the computational complexity for counting $4\times 4$ matrix partitions, combining computer-assisted proofs with manual analysis.

Findings

The problem is either in FP or #P-complete for $|D|=4$.

A computer program determines the complexity for most matrices.

The authors conjecture the dichotomy extends to larger matrices.

Abstract

Given a symmetric matrix $M\in \{0,1,*\}^{D\times D}$, an $M$-partition of a graph $G$ is a function from $V(G)$ to $D$ such that no edge of $G$ is mapped to a $0$ of $M$ and no non-edge to a $1$. We give a computer-assisted proof that, when $|D|=4$, the problem of counting the $M$-partitions of an input graph is either in FP or is #P-complete. Tractability is proved by reduction to the related problem of counting list $M$-partitions; intractability is shown using a gadget construction and interpolation. We use a computer program to determine which of the two cases holds for all but a small number of matrices, which we resolve manually to establish the dichotomy. We conjecture that the dichotomy also holds for $|D|>4$. More specifically, we conjecture that, for any symmetric matrix $M\in\{0,1,*\}^{D\times D}$, the complexity of counting $M$-partitions is the same as the related problem of counting list $M$-partitions.