Jennifer Ryan

Identifying Minimally Infeasible Subsystems of Inequalities

Given an infeasible system of linear inequalities, we show that the problem of identifying all minimally infeasible subsystems can be reduced to the problem of finding all vertices of a related polyhedron. This results in a shorter enumeration than that performed by previous method to solve this problem. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A column generation algorithm for bandwidth packing

We describe a column generation branch and bound procedure for optimally solving the bandwidth packing problem. The objective of this problem is to allocate bandwidth in a telecommunications network to maximize total revenue. The problem is formulated as an integer programming problem and the linear programming relaxation solved using column generation and the simplex algorithm. A branch and bound procedure which branches upon a particular path being used for a bandwidth routing is used to solve the IP. We present computational results.

Path assignment for call routing: an application of tabu search

We describe an implementation of tabu search that solves the path assignment problem, which is the problem of routing video data through an undercapacitated telecommunications network. Two versions of the tabu search were studied. Our results compare very favourably with those from other methods.

Tournament games and positive tournaments

Given a tournament T, the tournament game on T is as follows: Two players independently pick a node of T. If both pick the same node, the game is tied. Otherwise, the player whose node is at the tail of the arc connecting the two nodes wins. We show that the optimal mixed strategy for this game is unique and uses an odd number of nodes. A tournament is positive if the optimal strategy for its tournament game uses all of its nodes. The uniqueness of the optimal strategy then gives a new tournament decomposition: any tournament can be uniquely partitioned into positive subtournaments P1, P2, ,Pk, so Pi “beats” Pj for all 1 ≤ i > j ≤ k. We count the number of n node positive tournaments and list them for n ≤ 7.

LEAST-COST NETWORK TOPOLOGY DESIGN FOR A NEW SERVICE: An application of tabu search*

We describe an implementation of the tabu search metaheuristic that effectively finds a low-cost topology for a communications network to provide a centralized new service. Our results are compared to those of a greedy algorithm which applies corresponding decision rules, but without the guidance of the tabu search framework. These problems are difficult computationally, representing integer programs that can involve as many as 10,000 integer variables and 2000 constraints in practical applications. The tabu search results approach succeeded in obtaining significant improvements over the greedy approach, yielding optimal solutions to problems small enough to allow independent verification of optimality status and, more generally, yielding both absolute and percentage cost improvements that did not deteriorate with increasing problem size.

Finding the minimum weight IIS cover of an infeasible system of linear inequalities

Given an inconsistent set of inequalities Ax ⩽b, theirreducible inconsistent subsystems (IISs) designate subsets of the inequalities such that at least one member of each subset must be deleted in order to achieve a feasible system. By solving a set covering problem over the IISs, one can determine a minimum weight set of inequalities that must be deleted in order to achieve feasibility. Since the number of IISs is generally exponential in the size of the original subsystem, we generate the IISs only when they are violated by a trial solution. Computational results on the NETLIB infeasible LP library are given.

A Branch and Cut Algorithm for a Steiner Tree-Star Problem

This paper deals with a Steiner tree-star problem that is a special case of the degree constrained node-weighted Steiner tree problem. This problem arises in the context of designing telecommunications networks for digital data service, provided by regional telephone companies. In this paper, we develop an effective branch and cut procedure coupled with a suitable separation procedure that identifies violated facets for fractional solutions to the relaxed formulation. Computational results indicate that large problem instances with up to 200 nodes can be solved within acceptable time bounds.

The depth and width of local minima in discrete solution spaces

Abstract Heuristic search techniques such as simulated annealing and tabu search require “tuning” of parameters (i.e., the cooling schedule in simulated annealing, and the tabu list length in tabu search), to achieve optimum performance. In order for a user to anticipate the best choice of parameters, thus avoiding extensive experimentation, a better understanding of the solution space of the problem to be solved is needed. Two functions of the solution space, the maximum depth and the maximum width of local minima are discussed here, and sharp bounds on the value of these functions are given for the 0–1 knapsack problem and the cardinality set covering problem.

Matroid Applications and Algorithms

Matroid theory provides a set of modeling tools with which many combinatorial and algebraic problems may be treated. Generic algorithms for the resulting matroid problems can be used to solve problems from a variety of application areas including engineering, scheduling, mathematics, and mathematical programming. In this paper, we give an introduction to matroid theory and algorithms, and a survey of algorithmic applications. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Fractional domination of strong direct products

Let G be a graph with edges E(G). A subset of the nodes dominates G if each node of G is either in or is adjacent to a member of the subset. The domination number of G, y(G), is the minimum size of a dominating set. In 1963, Vizing [3] conjectured that for all graphs G and H, y(G 0 H) 3 y(G)?(H) w h ere G @ H is the Cartesian product of G and H. We prove an analogous result for the fractional domination number. We can redefine y(G) as the value of the integer programming problem. For n-vectors x and y, let x 2 y (X < y) mean Xi > yi (xi < yi) for all i. Let 1, and 0, be the n-vectors whose components are all one or all zero, respectively. If N(G) is the neighborhood matrix of G (the adjacency matrix plus the identity matrix), then