We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we present results implying the polynomial-time solvability of the two problems. We also use our results to give the first strongly polynomial separation algorithm for the convex hull of matchable sets of a graph, and the first polynomial-time algorithm to compute the rank of a certain matrix of indeterminates. Our algorithmic results are based on polyhedral characterizations, and on the equivalence of separation and optimization.

This paper relates an axiomatic generalization of matroids, called a jump system, to polyhedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron but form a subset of them. However, it is shown that the convex hull of the set of points of a jump system is a bisubmodular polyhedron, and that the integral points of an integral bisubmodular polyhedron determine a (special) jump system. The authors prove addition and composition theorems for jump systems, which have several applications for delta-matroids and matroids.

We give bounds on total lengths of augmenting paths in standard implementations of the matroid partition and intersection algorithms, and indicate how these observations can be used to improve the running times in certain applications. For example, for the matroid intersection algorithm on two r by n matrices the running time is shown to be $O(nr^2 \log r)$. We also give improved versions of the two algorithms, when running times are measured in terms of calls to an independence oracle. For example, there is a matroid partition algorithm on $O(n)$n-element matroids using $O(n^{2.5} )$ independence tests.

In a nonnegative edge-weighted network, the weight of an edge represents the effort required by an attacker to destroy the edge, and the attacker derives a benefit for each new component created by destroying edges. The attacker may want to minimize over subsets of edges the difference between (or the ratio of) the effort incurred and the benefit received. This idea leads to the definition of the “strength” of the network, a measure of the resistance of the network to such attacks. Efficient algorithms for the optimal attack problem, the problem of computing the strength, and the problem of finding a minimum cost “reinforcement” to achieve a desired strength are given. These problems are also solved for a different model, in which the attacker wants to separate vertices from a fixed central vertex.

Abstract The minimum cut problem is a well‐solved special case of submodular function minimization. We show that it is in fact equivalent to minimizing a modular function over a ring family. One‐half of this equivalence follows from classical work of Rhys and Picard. We give a number of applications to testing membership in special kinds of matroid polyhedra.

Abstract The 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.