Jean-Claude Picard

Minimum cuts and related problems

Abstract : The paper is concerned with an integer programming characterization of a cut in a network. This characterization provides a fundamental equivalence between directed pseudosummetric networks and undirected networks. It also identifies a class of problems which can be solved as minimum cut problems on a network. (Author)

On the structure of all minimum cuts in a network and applications

This paper presents a characterization of all minimum cuts, separating a source from a sink in a network. A binary relation is associated with any maximum flow in this network, and minimum cuts are identified with closures for this relation. As a consequence, finding all minimum cuts reduces to a straightforward enumeration. Applications of this results arise in sensitivity and parametric analyses of networks, the vertex packing and maximum closure problems, in unconstrained pseudo-boolean optimization and project selection, as well as in other areas of application of minimum cuts.

Selected Applications of Minimum Cuts in Networks

AbstractThis paper provides a review and a synthesis of applications of minimum cuts to a variety of combinatorial optimization problems, including linear and non-linear integer programming, problems in location theory, graph theory, sequencing and scheduling, and other areas. The central theme is a binary quadratic programming formulation of the minimum cut problem. After reviewing alternative formulations, extensions, and direct applications ofminimum cuts in network problems, the problem of finding a closure with maximum weight ina directed graph is considered. This problem can be solved as a minimum cut problem, which allows other related problems to be solved as a sequence of minimum cut problems. The paper concludes by exploring more difficult versions of the minimum cut problem involving negative capacities or additional constraints.

A Cut Approach to the Rectilinear Distance Facility Location Problem

This paper is concerned with the problem of locating n new facilities in the plane when there are m facilities already located. The objective is to minimize the weighted sum of rectilinear distances. Necessary and sufficient conditions for optimality are established. We show that the optimum locations of the new facilities are dependent on the relative orderings of old facilities along the two coordinate axes but not on the distances between them. Based on these results an algorithm is presented that requires the solution of at most m-1 minimum cut problems on networks with at most n + 2 vertices. All of these results are easily extended to the same location problem on a tree graph.

A network flow solution to some nonlinear 0‐1 programming problems, with applications to graph theory

A network flow technique is used to solve the unconstrained nonlinear 0-1 programming problem, which is maximizing the ratio of two polynomials, assuming that all the nonlinear coefficients in the numerator are non-negative and all the nonlinear coefficients in the denominator are nonpositive. Two examples are an investment selection problem to maximize the rate of return, and a decomposition approach to a scheduling problem studied by Sidney and Lawler. The proposed algorithm requires the solution of a sequence of minimum-cut problems in a related network, and can be extended to some more general problems of the same type. This approach is also applied to find the density of a graph (the maximum ratio, among its subgraphs, of the number of edges to the number of nodes) and its arboricity, for which polynomial algorithms are described. It is also useful in providing a bounding scheme for the maximum-clique and vertex packing problems.

An optimal algorithm for the mixed Chinese postman problem

The Chinese Postman Problem is well solved when the original graph contains only arcs or only edges. The mixed Chinese Postman Problem (MCPP) is, however, NP-complete, and very few papers have been devoted to this problem. In this paper, we present a new integer programming model and a new optimal algorithm for the MCPP. The simplex method is used iteratively to obtain sharp lower bounds by solving successive relaxations of this model. Optimality is achieved by using Gomory cuts, blossom inequalities and balanced set constraints. Detailed computational results are presented.

On the integer-valued variables in the linear vertex packing problem

Given a graph with weights on vertices, the vertex packing problem consists of finding a vertex packing (i.e. a set of vertices, no two of them being adjacent) of maximum weight. A linear relaxation of one binary programming formulation of this problem has these two well-known properties: (i) every basic solution is (0, 1/2, 1)-valued, (ii) in an optimum linear solution, an integer-valued variable keeps the same value in an optimum binary solution.As an answer to an open problem from Nemhauser and Trotter, it is shown that there is a unique maximal set of variables which are integral in optimal (VLP) solutions.

A Graph-Theoretic Equivalence for Integer Programs

This paper is concerned with the relation between 0-1 integer programs and graphs. An equivalence is established between solving 0-1 integer programs with quadratic or linear objective functions and solving a cut problem on a related graph.

On finding the K best cuts in a network

We show that the O(K . n^4) algorithm of Hamacher (1982) for finding the K best cut-sets fails because it may produce cuts rather than cut-sets. With the convention that two cuts (X, X) and (Y, Y) are different whenever X Y the K best cut problem can be solved in O(K . n^4).

An Efficient Implicit Enumeration Algorithm for the Maximum Clique Problem

We describe an implicit enumeration algorithm which can be used to find one or all maximum cliques in a graph. The procedure builds cliques one vertex at a time using depth-first search and a branching rule based on the number of triangles to which vertices belong. Computational results for moderately-sized graphs are reported and the algorithm is shown to be competitive for relatively sparse graphs.