Yash P. Aneja

An integer linear programming approach to the steiner problem in graphs

Consider a connected undirected graph G[N; E] with N = S ∪ P, the set of nodes, where P is designated as the set of Steiner points. A weight is associated with each edge ei of the set E. The problem of obtaining a minimal weighted tree which spans the set S of nodes has been termed in literature as the Steiner problem in graphs. A specialized integer programming (set covering) formulation is presented for the problem. The number of constraints in this formulation grows exponentially with the size of the problem. A method called the row generation scheme is developed to solve the above problem. The method requires knowing the constraints only implicitly. Several other problems which can be put in a similar framework can also be handled by the above scheme. The generality of the scheme and its efficiency is discussed. Finally the computational result is demonstrated.

Shortest chain subject to side constraints

Certain properties of a shortest chain subject to several side constraints are established. Based on these an implicit enumeration algorithm, that is, a generalization of the one given by Dijkstra for the case without any side constraint, is presented. Validation of the algorithm and an illustrative example are included.

Minimal spanning tree subject to a side constraint

Abstract This paper addresses the problem of finding a minimal spanning tree in a network subject to a knapsack-type constraint. The problem is shown to belong to the class of NP -complete problems. Certain properties of an optimal solution to this problem are established considering a bicriteria spanning tree problem. Based on these, an algorithm is proposed which is mainly a branch and bound scheme branching from the test upper bound and generating efficient frontiers successively. It also uses certain effective heuristics and bounds that are obtained in polynomially bounded operations. The algorithm is validated and a numerical example is included.

On a class of quadratic programs

Abstract This paper considers a class of quadratic programs where the constraints ae linear and the objective is a product of two linear functions. Assuming the two linear factors to be non-negative, maximization and minimization cases are considered. Each case is analyzed with the help of a bicriteria linear program obtained by replacing the quadratic objective with the two linear functions. Global minimum (maximum) is attained at an efficient extreme point (efficient point) of the feasible set in the solution space and corresponds to an efficient extreme point (efficient point) of the feasible set in the bicriteria space. Utilizing this fact and certain other properties, two finite algorithms, including validations are given for solving the respective problems. Each of these, essentially, consists of solving a sequence of linear programs. Finally, a method is provided for relaxing the non-negativity assumption on the two linear factors of the objective function.

Maximal expected flow in a network subject to arc failures

In a network subject to arc failures, each chain has a probability of failure. Therefore the maximal flow in the network is a random variable. The problem considered here is that of maximizing the expected flow. An arc-chain formulation of the problem, and an algorithm for computing an optimal solution are provided. The algorithm involves a column generation technique, and a constrained chain in the network provides a desired column at each step of the simplex algorithm. The technique presented here is an extension of that of Ford and Fulkerson. The algorithm is validated, and a geometric interpretation is included.

Network flows with age dependent decay rates

Abstract This paper considers the problem of maximizing the output flow in a multicommodity network in which flow entering an arc experiences a decay rate which is a function of three factors: the arc, the commodity, and the age of the commodity as it enters the arc. An arc-chain linear programming formulation of the problem is given. The algorithm for solving the problem involves a novel column generation scheme for basis entry embedded in the revised simplex algorithm. An efficient algorithm for generating, at each iteration, such a column is provided and illustrated with a numerical example.

Maximization Of The Vector-Flow In Multigommodity Networks

AbstractThis paper considers the problem of maximizing the vector-flow in a multicommodity network. The known techniques of multiobjective linear programming are shown to be of little practical help in solving this problem. The non-dominated set in the commodity space is considered. Methods for determining the extreme points of this set in the cases of two commodity directed and undirected graphs are presented.