Polyhedral results and stronger Lagrangean bounds for stable spanning trees

Abstract

Given a graph $G=(V,E)$ and a set $C$ of unordered pairs of edges regarded as being in conflict, a stable spanning tree in $G$ is a set of edges $T$ inducing a spanning tree in $G$, such that for each $\left\lbrace e_i, e_j \right\rbrace \in C$, at most one of the edges $e_i$ and $e_j$ is in $T$. The existing work on Lagrangean algorithms to the NP-hard problem of finding minimum weight stable spanning trees is limited to relaxations with the integrality property. We exploit a new relaxation of this problem: fixed cardinality stable sets in the underlying conflict graph $H =(E,C)$. We find interesting properties of the corresponding polytope, and determine stronger dual bounds in a Lagrangean decomposition framework, optimizing over the spanning tree polytope of $G$ and the fixed cardinality stable set polytope of $H$ in the subproblems. This is equivalent to dualizing exponentially many subtour elimination constraints, while limiting the number of multipliers in the dual problem to $|E|$. It is also a proof of concept for combining Lagrangean relaxation with the power of MILP solvers over strongly NP-hard subproblems. We present encouraging computational results using a dual method that comprises the Volume Algorithm, initialized with multipliers determined by Lagrangean dual-ascent. In particular, the bound is within 5.5% of the optimum in 146 out of 200 benchmark instances; it actually matches the optimum in 75 cases. All of the implementation is made available in a free, open-source repository.