Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials

Abstract

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models.More specifically, we define a large class of graph polynomials C and show that if p ∈ C and there is a disk D centered at zero in the complex plane such that p(G) does not vanish on D for all bounded degree graphs G, then for each z in the interior of D there exists a deterministic polynomial-time approximation algorithm for evaluating p(G) at z. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures.Our work builds on a recent line of work initiated by Barvinok [2,3,4,5], which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.

A Deterministic Algorithm for Counting Colorings with 2-Delta Colors

“…1 These algorithms have made use of one of two main techniques: decay of correlations, which exploits decreasing in uence of the spins (colors) on distant vertices on the spin at a given vertex; and polynomial interpolation, which uses the absence of zeros of the partition function in a suitable region of the complex plane. Early examples of the decay of correlations approach include [1,2,40], while for early examples of the polynomial interpolation method, we refer to the monograph of Barvinok [3] (see also, e.g., [4,13,25,27,30,34] for more recent examples). Unfortunately, however, in the case of colorings on general bounded degree graphs, these techniques have so far lagged well behind the MCMC algorithms mentioned above.…”

Zeros of ferromagnetic 2-spin systems

“…In fact, as it has been observed in [PR17], the algorithm can be extended to a multivariate version of the partition fucntion easily. Let λ ∈ C V be a vector that speci es an external eld for each vertex.…”

“…In general, computing these coe cients naively will take quasipolynomial-time. However, Patel and Regts [PR17] have provided additional insights on how to compute these coe cients e ciently for a large family of graph polynomials in bounded degree graphs. As explained in [LSS19b], the idea of Patel and Regts [PR17] can be applied to the partition functions of spin systems in much more generality, which includes Z spin (G; λ) that we are interested in.…”

“…An illustration of comparing λ MCMC , λ c and λ is given in Figure 1. Our algorithm is based on a recent algorithmic technique developed by Barvinok [Bar16] and extended by Patel and Regts [PR17]. e idea is to view Z spin as a polynomial in λ, and turn zero-free regions of this polynomial in the complex plane into e cient approximation algorithms of the corresponding parameters.…”

Counting independent sets in unbalanced bipartite graphs

“…(2) For each cluster Γ ∈ C m , compute φ(H(Γ)) and γ∈Γ w γ . It is shown in [17], using ideas and tools from [15,24,3], that this algorithm can be implemented with running time O n · (n/ǫ) O(log(∆ L ∆ R )/η) , which for ∆ L , ∆ R fixed is polynomial in n and 1/ǫ.…”