Independent Sets from an Algebraic Perspective

TL;DR

This paper explores the algebraic complexity of counting independent sets and antichains in graphs and posets, revealing computational hardness and providing algebraic tools like universal Gröbner bases for certain ideals.

Contribution

It establishes the computational difficulty of evaluating independence polynomials and Hilbert series, and introduces a universal Gröbner basis for a family of ideals related to posets.

Findings

Counting independent sets is #P-hard for bipartite Cohen-Macaulay graphs.

Evaluating Hilbert series of certain ideals is #P-hard.

Universal Gröbner bases are constructed for specific radical zero-dimensional ideals.

Abstract

In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen-Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P=P. Moreover, we present a family of radical zero-dimensional complete intersection ideals J_P associated to a finite poset P, for which we describe a universal Gr\"obner basis. This implies that the bottleneck in computing the dimension of the quotient by J_P (that is, the number of zeros of J_P) using Gr\"obner methods lies in the description of the standard monomials.