Computing the Partition Function for Cliques in a Graph

Alexander Barvinok

doi:10.4086/toc.2015.v011a013

Abstract

We present a deterministic algorithm which, given a graph G with n vertices and an integer 1 0 is an absolute constant. This allows us to tell apart the graphs that do not have m-subsets of high density from the graphs that have sufficiently many m-subsets of high density, even when the probability to hit such a subset at random is exponentially small in m. ACM Classification: F.2.1, G.1.2, G.2.2, I.1.2 AMS Classification: 15A15, 68C25, 68W25, 60C05

Highlights

Introduction and main results1.1 Density of a subset in a graphLet G = (V, E) be an undirected graph with set V of vertices and set E of edges, without loops or multiple edges
Σ (S) = 0 if and only if S is an independent set, that is, no two vertices of S span an edge of G and σ (S) = 1 if and only if S is a clique, that is, every two vertices of S span an edge of G
Let W = be a set of real or complex numbers indexed by unordered pairs {i, j} where 1 ≤ i = j ≤ n and interpreted as a set of weights on the edges of the complete graph with vertices 1, . . . , n

Summary

Density of a subset in a graph

Let S ⊂ V be a subset of the set of vertices of G. We present a deterministic algorithm, which, given a graph G with n vertices and an integer 1 < m ≤ n computes in nO(lnm) time (within a relative error of 0.1, say) the sum Densitym(G) = ∑ exp {γmσ (S) − ε(m)} S⊂V |S|=m. If there are no m-subsets S with density σ or higher n Densitym(G) ≤ m exp {γmσ − ε(m)}. There are sufficiently many m-subsets S with density σ or higher, that is, if the probability to hit such a subset at random is at least 2 exp{−γ(σ − σ )m}, n Densitym(G) ≥ 2 m exp {γmσ − ε(m)}.

The partition function of cliques

The idea of the algorithm

Weighted enumeration of subsets

Comparison with results in the literature

The algorithm for approximating the partition function

Definitions