The complexity of approximating the matching polynomial in the complex plane

TL;DR

This paper investigates the computational complexity of approximating the matching polynomial on graphs with complex edge parameters, establishing hardness results and extending approximation techniques to complex values outside the negative real axis.

Contribution

It completes the complexity classification for approximating the matching polynomial on bounded degree graphs for all complex parameters, and extends correlation decay approximation methods to certain complex values.

Findings

Approximation is #P-hard for certain negative real parameters on bounded degree graphs.

Correlation decay methods do not extend to all complex parameters, especially on the negative real axis.

Approximation is feasible for complex parameters not on the negative real axis using geodesic distances.

Abstract

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter $\gamma$, where $\gamma$ takes arbitrary values in the complex plane. When $\gamma$ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of $\gamma$, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree $\Delta$ as long as $\gamma$ is not a negative real number less than or equal to $-1/(4(\Delta-1))$. Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all $\Delta\geq 3$ and all real $\gamma$ less than $-1/(4(\Delta-1))$, the problem of approximating the value of the matching polynomial on graphs of maximum degree $\Delta$ with edge parameter $\gamma$ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real $\gamma$ it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of $\gamma$ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value $\gamma$ that does not lie on the negative real axis. Our analysis accounts for complex values of $\gamma$ using geodesic distances in the complex plane in the metric defined by an appropriate density function.