Circuit partitions and #P-complete products of inner products

TL;DR

This paper introduces a natural #P-complete problem involving the expectation of products of inner products over graph vertices, revealing its computational complexity and connection to cycle partition polynomials.

Contribution

It establishes that the expectation of these inner product products is proportional to the cycle partition polynomial, proving #P-completeness for all k>1.

Findings

q(G;k) is proportional to G's cycle partition polynomial

The problem is #P-complete for any k>1

Connects graph polynomial computation with complex inner product expectations

Abstract

We present a simple, natural #P-complete problem. Let G be a directed graph, and let k be a positive integer. We define q(G;k) as follows. At each vertex v, we place a k-dimensional complex vector x_v. We take the product, over all edges (u,v), of the inner product <x_u,x_v>. Finally, q(G;k) is the expectation of this product, where the x_v are chosen uniformly and independently from all vectors of norm 1 (or, alternately, from the Gaussian distribution). We show that q(G;k) is proportional to G's cycle partition polynomial, and therefore that it is #P-complete for any k>1.