Searching for dense subsets in a graph via the partition function

TL;DR

This paper introduces a method to approximate a partition function related to dense subsets in graphs, enabling efficient analysis of such subsets with potential applications in graph theory and network analysis.

Contribution

The paper presents a quasi-polynomial time approximation algorithm for a partition function over m-subsets, advancing computational techniques for dense subgraph detection.

Findings

Approximation of the partition function within relative error in quasi-polynomial time.

Numerical experiments show larger gamma values are feasible for certain random graphs.

The approach is effective especially when the ratio n/m is large.

Abstract

For a set $S$ of vertices of a graph $G$, we define its density $0 \leq \sigma(S) \leq 1$ as the ratio of the number of edges of $G$ spanned by the vertices of $S$ to ${|S| \choose 2}$. We show that, given a graph $G$ with $n$ vertices and an integer $m$, the partition function $\sum_S \exp\{ \gamma m \sigma(S) \}$, where the sum is taken over all $m$-subsets $S$ of vertices and $0 < \gamma <1$ is fixed in advance, can be approximated within relative error $0 < \epsilon < 1$ in quasi-polynomial $n^{O(\ln m - \ln \epsilon)}$ time. We discuss numerical experiments and observe that for the random graph $G(n, 1/2)$ one can afford a much larger $\gamma$, provided the ratio $n/m$ is sufficiently large.