Representing polynomial of ST-CONNECTIVITY

TL;DR

This paper characterizes the polynomial representing the ST-CONNECTIVITY problem using lattice theory, providing explicit formulas for coefficients and analyzing the complexity of the polynomial's monomials.

Contribution

It links the coefficients of the representing polynomial to the M"obius function of a related lattice and computes the polynomial structure for acyclic quivers and grid graphs.

Findings

Coefficients are given by the M"obius function values.

Only monomials corresponding to unions of paths are non-zero.

Number of non-zero monomials in grid connectivity is exponential in n^2.

Abstract

We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional $n \times n$ grid connectivity problem is $2^{\Omega(n^2)}$.