Independent Sets from an Algebraic Perspective

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 JP associated to a finite poset P , for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by JP (that is, the number of zeros of JP ) using Gröbner methods lies in the description of the standard monomials.