Computational complexity of counting coincidences

TL;DR

This paper investigates the computational difficulty of determining when two combinatorial objects have the same count, establishing that the problem is highly complex and unlikely to be efficiently solvable.

Contribution

It proves the coincidence problem is outside the polynomial hierarchy for various combinatorial counting problems, providing different proofs and generalizations.

Findings

Coincidence problem is not in the polynomial hierarchy unless it collapses.

Different proof techniques are used for different cases.

The work extends to various combinatorial objects and conjectures further complexity results.

Abstract

Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the problem, with $2\times 1 \times 1$ and $2\times 2 \times 1$ boxes. We prove that in both cases the coincidence problem is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. While the conclusions are the same, the proofs are notably different and generalize in different directions. We proceed to explore the coincidence problem for counting independent sets and matchings in graphs, matroid bases, order ideals and linear extensions in posets, permutation patterns, and the Kronecker coefficients. We also make a number of conjectures for counting other combinatorial objects such as plane triangulations, contingency tables, standard Young tableaux, reduced factorizations and the Littlewood--Richardson coefficients.