
William H. CunninghamUniversity of Waterloo | UWaterloo · Department of Combinatorics & Optimization
William H. Cunningham
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71
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Publications (71)
IntroductionHeuristics for the TSPLower BoundsCutting PlanesBranch and Bound
Matchings and Alternating PathsMaximum MatchingMinimum-Weight Perfect MatchingsT-Joins and Postman ProblemsGeneral Matching ProblemsGeometric Duality and the Goemans-Williamson Algorithm
Network Flow ProblemsMaximum Flow ProblemsApplications of Maximum Flow and Minimum CutPush-Relabel Maximum Flow AlgorithmsMinimum Cuts in Undirected GraphsMulticommodity Flows
IntroductionWordsProblemsAlgorithms and Running TimeThe Class NpNp-CompletenessNp-Completeness of the Satisfiability ProblemNp-Completeness of Some Other ProblemsTuring Machines
Minimum-Cost Flow ProblemsPrimal Minimum-Cost Flow AlgorithmsDual Minimum-Cost Flow AlgorithmsDual Scaling Algorithms
Convex hullsPolytopesFacetsIntegral PolytopesTotal UnimodularityTotal Dual IntegralityCutting PlanesSeparation and Optimization
Two ProblemsMeasuring Running Times
We present a simplex method for the solution of the optimal submodular flow problem. Like the network simplex method for solving
the minimum cost network flow problem, this algorithm is purely combinatorial. It requires an oracle which can minimize a
submodular function. In general this oracle is available only via the ellipsoid algorithm, but in s...
An algorithm for finding an optimum weight perfect matching in a graph is described. It differs from Edmonds’ “blossom” algorithm
in that a perfect matching is at hand throughout the algorithm, and a feasible solution to the dual problem is obtained only
at termination. In many other respects, including its efficiency, it is similar to the blossom...
Let G = (V, E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A ⊆ V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the...
Consider an integer program max (c
t
x : Ax = b, x ≥ 0, x ∈ Z
n
) where A ∈ Z
m×n, b ∈ Z
m
, and c ∈ Z
n
. We show that the integer program can be solved in pseudo-polynomial time when A is non-negative and the column-matroid of A has constant branch-width.
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 polyhedr...
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 possib...
. Given an undirected graph G = (V; E) and three specified terminal nodes t1 ; t2 ; t3 , a 3-cut is a subset A of E such that no two terminals are in the same component of GnA. If a non-negative edge weight ce is specified for each e 2 E, the optimal 3-cut problem is to find a 3-cut of minimum total weight. This problem is NP-hard, and in fact, is...
Three types of matroid connectivity, including Tutte's, are defined and shown to generalize corresponding notions of graph connectivity. A theorem of Tutte on cyclically 3-connected graphs, is generalized to matroids.
We consider the convex hull of the even permutations on a set of n elements. We define a class of valid inequalities and prove that they induce a large class of distinct facets of the polytope. Using the inequalities, we characterize the polytope for n=4, and we confirm a conjecture of Brualdi and Liu that, unlike the convex hull of all permutation...
Perhaps the two most fundamental well-solved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones -- jump systems and path-matchings.
The optimal k-restricted 2-factor problem consists of finding, in a complete undirected graph K
n
, a minimum cost 2-factor (subgraph having degree 2 at every node) with all components having more than k nodes. The problem is a relaxation of the well-known symmetric travelling salesman problem, and is equivalent to it when
≤k≤n−1. We study the k-r...
Given an undirected graph G = (V,E) and three specified terminal nodes t
1,t
2,t
3, a 3-cut is a subset A of E such that no two terminals are in the same component of G\A. If a non-negative edge weight c
e is specified for each e ∈ E, the optimal 3-cut problem is to find a 3-cut of minimum total weight. This problem is NP-hard, and in fact, is max-...
if every principal submatrix has determinant 0 or ±1. Let A be a symmetric (0, 1)-matrix, with a zero diagonal. A PU-orientation of A is a skew-symmetric signing of A that is PU. If A′ is a PU-orientation of A, then, by a certain decomposition of A, we can construct every PU-orientation of A from A′. This construction is based on the fact that the...
We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we establish common generalizations of the main results on those two problems---polynomial-time solvability, min-max theorems, and totally dual integral polyhedral descriptions. New applications of these results includ...
We introduce a new class of valid inequalities for the symmetric travelling salesman polytope. The family is not of the common handle-tooth variety. We show that these inequalities are all facet-inducing and have Chv'atal rank 2. Key words: polyhedra, facets, travelling salesman problem AMS 1980 subject classification. Primary: 90C08 Copyright (C)...
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...
We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytope PG of a graph G. Each "wheel configuration" gives rise to two such inequalities. The simplest wheel configuration is an "odd" subdivision W of a wheel, and for these we give necessary and sufficient conditions for the wheel inequality to be facet-...
We characterize the class of integral square matrices M having the property that for every integral vector q the linear complementarity problem with data M; q has only integral basic solutions. These matrices, called principally unimodular matrices, are those for which every principal nonsingular submatrix is unimodular. As a consequence, we show t...
Given (0,1) variables and a set of clauses having at most two literals per clause, the max 2-sat problem is to find an assignment that maximizes the number (or weight) of the satisfied clauses. We implement a cutting plane algorithm that starts with a linear programming (l.p.) relaxation of max 2-sat, and that has separation subroutines for two fam...
Given a graph G, we introduce several classes of valid inequalities, called wheel inequalities, for the stable set polytope of G. Moreover, we show that the corresponding separation problems can be solved in polynomial time. Each wheel configuration generates two wheel inequalities. The most basic wheel configuration is a subdivision of a wheel. Mo...
We relate 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, we do show that the convex hull of the set of points of a jump syst...
The strength of a network having non-negative edge weights, is the minimum over subsets A of edges, of the weight of A divided by the number of additional components created by deleting A. We show that the strength can be computed in time O(n4).
A matchable set of a graph is a set of vertices joined in pairs by disjoint edges. Balas and Pulleyblank gave a linear-inequality description of the convex hull of matchable sets. We give a polynomial-time combinatorial algorithm for the separation problem for this polytope, and a min—max theorem characterizing the maximum violation by a given poin...
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-matching...
We introduce a new class of valid inequalities for the polytope of the symmetric travelling salesman problem. We also give complete characterizations of the polytope for 6 and 7 cities. For the latter case, the new inequalities are needed. These results are related to work of R. Z. Norman [“On the convex polyhedra of the symmetric travelling salesm...
The binding number of a graph G = (V,E) is min(|N(A)|⧸|A|: Ø≠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.
Edmonds and Johnson proved an integrality property of optimal dual solutions of matching problems, under the assumption that all the edge-costs are even integers. It is shown that the same conclusion holds if the costs are integers whose sum around any cycle is even. This result is a consequence of a form of the matching algorithm.
Given a simple graph G having vertex set V, it is obvious that for any spanning tree T, there is an edge of T whose fundamental cutset has size at most |V|— 1. We extend this result to matroids.Call a cocircuit of a matroid M short if its size is at most the rank of M. Then we prove that for any simple binary matroid M having no Fano minor, and for...
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...
LetP be the convex hull of perfect matchings of a graphG=(V, E). The dominant ofP is {y∈R
E
∶y≥x for somex∈P}. A theorem of Fulkerson implies that, ifG is bipartite, then the dominant ofP can be described by linear inequalities having {0, 1}-valued coefficients. However, this is far from true in general. Here
it is proved that, for every positive...
Given a (polymatroid) rank function f and its corresponding polymatroid P(f), we associate with each extreme point of P(f) a certain partial order. We show that this partial order is efficiently constructible, and that it characterizes all the orderings with which the greedy algorithm can be used to generate the given extreme point. We give several...
Earlier work of Bixby, Cunningham, and Topkis is extended to give a combinatorial algorithm for the problem of minimizing
a submodular function, for which the amount of work is bounded by a polynomial in the size of the underlying set and the largest
function value (not its length).
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 r...
Previously the only polynomial-time solution algorithm to solve the optimal submodular flow problem introduced by J. Edmonds and R. 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 auxilia...
Given a matroid M on E and a nonnegative real vector x=(xj:j∈E), a fundamental problem is to determine whether x is in the convex hull P of (incidence vectors of) independent sets of M. An algorithm is described for solving this problem for which the amount of computation is bounded by a polynomial in |E|, independently of x, allowing as steps test...
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...
We consider a class of two-commodity network flow problems with additional, highly structured linear constraints in place of the usual capacity constraints. This class of linear programs arose as subproblems in a mixed integer programming formulation of a location model, and was solved by Ramakrishnan using a special purpose primal simplex algorith...
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 decomposi...
The matching polyhedron theorem of J. Edmonds and E. L. 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...
We consider a composition for independence systems, and show how knowledge of polyhedral descriptions of the small systems provides such a description for the composed system. Applications to other compositions, including sums and substitutions, are treated. Theorems of Chvátal on graph substitution and Bixby on the composition of perfect graphs ar...
A composition for directed graphs which generalizes the substitution (or X-join) composition of graphs and digraphs, as well as the graph version of set-family composition, is described. It is proved that a general decomposition theory can be applied to the resulting digraph decomposition. A consequence is a theorem which asserts the uniqueness of...
A cocircuit of a matroid is separating if deleting it leaves a separable matroid. We give an effecient algorithm which finds a separating cocircuit or a Fano minor in a binary matroid, thus proving constructively a theorem of Tutte. Using this algorithm and a new recursive characterization of bond matroids, we give a new method for testing binary m...
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 n...
Given a finite undirected graph G and A ⊆ E(G) , G(A) denotes the subgraph of G having edge-set A and having no isolated vertices. For a partition { E 1 , E 2 } of E ( G ), W(G; E 1 ) denotes the set V(G(E 1 )) ⋂ V ( G ( E 2 )). We say that G is non-separable if it is connected and for every proper, non-empty subset A of E(G) , we have | W ( G ; A...
An example of cycling in the network simplex method is given and some restrictions on its occurrence are proved. An example of ″stalling″ (an exponentially long sequence of consecutive degenerate pivots without cycling) is also given, and two methods which prevent cycling are shown to admit stalling. Pivoting rules which prevent the occurrence of b...
A chord of a circuit C of a matroid M on E is a cell e ϵ S\C such that C spans e. Menger's theorem gives necessary and sufficient conditions for a cell of a graphic matroid to be a chord of some circuit. We extend this result to a large class of matroids and find all minimal counterexamples. The theorem is used to obtain results on disjoint paths a...
A characterization of the maximum-cardinality common independent sets of two matroids via an unbounded convex polyhedron is proved, confirming a conjecture of D.R. Fulkerson. A similar result, involving a bounded polyhedron, is the well-known matroid intersection polyhedron theorem of Jack Edmonds; Edmonds's theorem is used in the proof.
Simple combinatorial modifications are given which ensure finiteness in the primal simplex method for the transshipment problem and the upper-bounded primal simplex method for the minimum cost flow problem. The modifications involve keeping strongly feasible bases. An efficient algorithm is given for converting any feasible basis into a strongly fe...