Searching for dense subsets in a graph via the partition function

Abstract

For a set $S$ of vertices of a graph $G$, we define its density $0\le \sigma (S)\le 1$ as the ratio of the number of edges of $G$ spanned by the vertices of $S$ to $\left(\genfrac{}{}{0ex}{}{|S|}{2}\right)$. 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.

22 pages

References in corpus (1)

• Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials