Abstract
We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ 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 γ, 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 γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.
Highlights
We study the problem of approximating the matching polynomial of a graph
We explore whether the maximum degree parameter can be replaced by the connective constant
Srivastava, Štefankovič and Yin [13, Theorem 1.3] showed that, for fixed ∆, when γ is a positive real, the correlation decay method gives an FPTAS for approximating ZG(γ) on graphs G with connective constant at most ∆
Summary
We study the problem of approximating the matching polynomial of a graph. This polynomial has a parameter γ, called the edge activity. Our first theorem shows that, for all ∆ ≥ 3 and all rational numbers γ < −1/(4(∆ − 1)), it is #P-hard to approximate |ZG(γ)| on bipartite graphs of maximum degree ∆ within a constant factor. Srivastava, Štefankovič and Yin [13, Theorem 1.3] showed that, for fixed ∆, when γ is a positive real, the correlation decay method gives an FPTAS for approximating ZG(γ) on graphs G with connective constant at most ∆ (wit√hout any bound on the maximum degree of G). The algorithmic contribution of our paper is to show that, despite the hardness result of Theorem 4, correlation decay gives a good approximation algorithm for any complex value γ that does not lie on the negative real axis when the input graph has bounded connective constant. An alternative approach was given by Peters and Regts [12], again in the context of the hard-core model, where they showed contraction within the basin of an attracting fixpoint using the theory of complex dynamical systems
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