A Polyhedral Intersection Theorem for Capacitated Spanning Trees

In a two-capacitated spanning tree of a complete graph with a distinguished root vertex v, every component of the induced subgraph on V\{v} has at most two vertices. We give a complete,non-redundant characterization of the polytope defined by the convex hull of the incidence vectors of two-capacitated spanning trees. This polytope is the intersection of the spanning tree polytope on the given graph and the matching polytope on the subgraph induced by removing the root node and its incident edges. This result is one of very few known cases in which the intersection of two integer polyhedra yields another integer polyhedron. We also give a complete polyhedral characterization of a related polytope, the 2-capacitated forest polytope

Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arbores- cence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very signi¯cant improvements over previous algorithms. Several open instances could be solved to optimalityNo keywords;

Capacitated Trees, Capacitated Routing, and Associated Polyhedra

We study the polyhedral structure of two related core combinatorial problems: the subtree cardinalityconstrained minimal spanning tree problem and the identical customer vehicle routing problem. For each of these problems, and for a forest relaxation of the minimal spanning tree problem, we introduce a number of new valid inequalities and specify conditions for ensuring when these inequalities are facets for the associated integer polyhedra. The inequalities are defined by one of several underlying support graphs: (i) a multistar, a "star" with a clique replacing the central vertex; (ii) a clique cluster, a collection of cliques intersecting at a single vertex, or more generally at a central" clique; and (iii) a ladybug, consisting of a multistar as a head and a clique as a body. We also consider packing (generalized subtour elimination) constraints, as well as several variants of our basic inequalities, such as partial multistars, whose satellite vertices need not be connected to all of the central vertices. Our development highlights the relationship between the capacitated tree and capacitated forest polytopes and a so-called path-partitioning polytope,and shows how to use monotone polytopes and a set of simple exchange arguments to prove that valid inequalities are facets