On cardinality constrained cycle and path polytopes

TL;DR

This paper characterizes the convex hulls of specific cardinality-constrained paths and cycles in directed graphs using linear inequalities, providing polynomial-time separation algorithms and extending to undirected cases and parity-specific inequalities.

Contribution

It provides new integer characterizations and facet-defining inequalities for cardinality-constrained path and cycle polytopes, including polynomial-time separation algorithms.

Findings

Facet-defining inequalities for these polytopes are identified.

Separation problems for these inequalities are solvable in polynomial time.

Extensions to undirected and parity-specific cases are developed.

Abstract

Given a directed graph D = (N, A) and a sequence of positive integers 1 <= c_1 < c_2 < ... < c_m <= |N|, we consider those path and cycle polytopes that are defined as the convex hulls of simple paths and cycles of D of cardinality c_p for some p, respectively. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. These inequalities can simply be transformed into inequalities that characterize the integer points of the undirected counterparts of cardinality constrained path and cycle polytopes. Beyond we investigate some further inequalities, in particular inequalities that are specific to odd/even paths and cycles.