Polyhedral results and stronger Lagrangean bounds for stable spanning trees

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