William H. Cunningham

Optimal attack and reinforcement of a network

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.

Converting Linear Programs to Network Problems

We describe an algorithm which converts a linear program min{cx ∣ Ax = b, x ≥ 0} to a network flow problem, using elementary row operations and nonzero variable-scaling, or shows that such a conversion is impossible. If A is in standard form, the computational effort required is bounded by Orn, where r is the number of rows and n is the number of nonzero entries of A. A method for determining whether a “binary matroid” is “graphic” plays an important role in the algorithm.

Compositions for perfect graphs

In this paper we introduce a new graph composition, called 2-amalgam, and we prove that the 2-amalgam of perfect graphs is perfect. This composition generalizes many of the operations known to preserve perfection, such as the clique identification, substitution, join and amalgam operations. We give polynominal-time algorithms to determined whether a general graph is decomposable with respect to the 2-amalgam or amalgam operations.

A Primal-Dual Algorithm for Submodular Flows

Previously the only polynomial-time solution algorithm to solve the optimal submodular flow problem introduced by Edmonds and Giles was based on the ellipsoid method. Here, modulo an efficient oracle for minimizing certain submodular functions, a polynomial time procedure is presented which uses only combinatorial steps like building auxiliary digraphs, finding augmenting paths. The minimizing oracle is currently available only via the ellipsoid method, in general; however in important special cases, such as network flows, matroid intersections, orientations, and directed cut coverings, the necessary oracle can be provided combinatorially.

Minimum cuts, modular functions, and matroid polyhedra

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.

Decomposition of submodular functions

A decomposition theory for submodular functions is described. Any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposable functions, and the latter are completely characterized. Applications include decompositions of hypergraphs based on edge and vertex connectivity, the decomposition of matroids based on three-connectivity, the Gomory—Hu decomposition of flow networks, and Fujishige’s decomposition of symmetric submodular functions. Efficient decomposition algorithms are also discussed.

On Matroid Connectivity

Abstract Three types of matroid connectivity, including Tuttes, are defined and shown to generalize corresponding notions of graph connectivity. A theorem of Tutte on cyclically 3-connected graphs, is generalized to matroids.

Computing the binding number of a graph

Abstract The binding number of a graph G = (V,E) is min(|N(A)|⧸|A|: O≠A⊆V,N(A)≠V), where N(A) is the set of neighbours of A. This invariant is shown to be computable in polynomial time.

On cycling in the network simplex method

There are well-known examples of cycling in the linear programming simplex method having basis size two and requiring only six pivots. We prove that any example having basis size two for the network simplex method requires at least ten pivots. We also present an example that achieves this lower bound. In addition, we show that an attractive variant of Cunninghams noncyling method does admit cycling.

b-MATCHING DEGREE-SEQUENCE POLYHEDRA

A capacitatedb-matching in a graph is an assignment of non-negative integers to edges, each at most a given capacity and the sum at each vertex at most a given bound. Its degree sequence is the vector whose components are the sums at each vertex. We give a linear-inequality description of the convex hull of degree sequences of capacitatedb-matchings of a graph. This result includes as special cases theorems of Balas-Pulleyblank on matchable sets and Koren on degree sequences of simple graphs. We also give a min-max separation theorem, and describe a connection with submodular functions.