Matching Polyhedra Research Articles

On a Matching Arrangement of a Graph and $$LP$$-Orientations of a Matching Polyhedron

On a Matching Arrangement of a Graph and $$LP$$-Orientations of a Matching Polyhedron

On a Matching Arrangement of a Graph and $LP$-Orientations of a Matching Polyhedron

On a Matching Arrangement of a Graph and $LP$-Orientations of a Matching Polyhedron

Matching structure and the matching lattice

The matching polyhedron, i.e., the convex hull of (incidence vectors of) perfect matchings of a graph was characterized by Edmonds; this result is the key to a large part of polyhedral combinatorics and is used in many combinatorial algorithms. The linear hull of perfect matchings was characterized by Naddef, and by Edmonds, Lovász, and Pulleyblank. In this paper we describe the lattice generated by these vectors, i.e., the set of all integer linear combinations of perfect matchings. It turns out that the Petersen graph is, in a sense, the only difficult example. Our results also imply a characterization of the linear hull of perfect matchings over fields of characteristic different from 0. The main method is a decomposition theory developed by Kotzig, Lovász, and Plummer, which breaks down every graph into a number of graphs called bricks with very good matching properties. The number of Petersen graphs among these bricks will turn out to be an essential parameter of the matching lattice. Some refinements of the decomposition theory are also given. Among others, we show that the list of bricks obtained during the decomposition procedure is independent of the special choices made during the procedure.

Linear and combinatorial sharing problems

We consider sharing problems, i.e., the minimization of separable objectives ƒ(x) = max{ƒ j(x j) | j=1,…,n} on polyhedra from linear and combinatorial optimization. An algebraic approach enables a unifying treatment of quite different ‘share’ functions ƒ j as well as of real, rational and integral variables. We develop a dual (threshold) method based on lower bounds from knapsack sharing problems. Finiteness and efficiency is derived for several applications, in particular on submodular flow and matching polyhedra.

Reductions to 1–matching polyhedra

AbstractThe matching polyhedron theorem of Edmonds and Johnson, which gives the convex hull of capacitated perfect b‐matchings of a bidirected graph, is proved by reducing this matching problem to the ordinary perfect 1–matching problem, for which there exists a short inductive proof of the corresponding polyhedral theorem. The proof method makes it possible to deduce nestedness and discreteness properties of optimal dual solutions to the general matching problem from analogous properties of optimal dual solutions to the perfect 1–matching problem. In particular, the total dual half‐integrality of the inequality system for general matching is shown to follow from that for 1–matching. Applications considered include determining the convex hull of unions of disjoint circuits of a graph.

Short proofs on the matching polyhedron

A short proof of Edmonds' matching polyhedron theorem and the total dual integrality of the associated system of linear inequalities, proved first by W. H. Cunningham and A. B. Marsh ( Math. Programming Stud. 8 (1978), 50–72), is given.

Hamiltonicity and combinatorial polyhedra

We say that a polyhedron with 0–1 valued vertices is combinatorial if the midpoint of the line joining any pair of nonadjacent vertices is the midpoint of the line joining another pair of vertices. We show that the class of combinatorial polyhedra includes such well-known classes of polyhedra as matching polyhedra, matroid basis polyhedra, node packing or stable set polyhedra and permutation polyhedra. We show the graph of a combinatorial polyhedron is always either a hypercube (i.e., isomorphic to the convex hull of a k-dimension unit cube) or else is hamilton connected (every pair of nodes is the set of terminal nodes of a hamilton path). This implies several earlier results concerning special cases of combinatorial polyhedra.