Computing the Volume of a Restricted Independent Set Polytope Deterministically

TL;DR

This paper presents the first deterministic quasi-polynomial time algorithm for approximating the volume of a restricted independent set polytope, using correlation decay and applicable to certain graph classes.

Contribution

It introduces a deterministic approximation algorithm for the volume of restricted independent set polytopes, extending the scope beyond randomized methods.

Findings

Algorithm works for graphs with maximum degree 3 when 0.488<α<0.5

Correlation decay method applied to discretized polytopes

Interpolation method fails even for simple cases

Abstract

We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex body have been known now for several decades, but the corresponding deterministic counterparts are not available, and our algorithm is the first of this kind. The class of polytopes for which our algorithm applies arises as linear programming relaxation of the independent set problem with the additional restriction that each variable takes value in the interval $[0,1-\alpha]$ for some $\alpha<1/2$. (We note that the $\alpha\ge 1/2$ case is trivial). We use the correlation decay method for this problem applied to its appropriate and natural discretization. The method works provided $\alpha> 1/2-O(1/\Delta^2)$, where $\Delta$ is the maximum degree of the graph. When $\Delta=3$ (the sparsest non-trivial case), our method works provided $0.488<\alpha<0.5$. Interestingly, the interpolation method, which is based on analyzing complex roots of the associated partition functions, fails even in the trivial case when the underlying graph is a singleton.