K. P. K. Nair

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.

Fuzzy models for single-period inventory problem

In this paper, we consider the single-period inventory problem in the presence of uncertainties. Two types of uncertainties, one arising from randomness which can be incorporated through a probability distribution and the other from fuzziness which can be characterized by fuzzy numbers, are considered. We develop two models, in one the demand is probabilistic while the cost components are fuzzy and in the other the costs are deterministic but the demand is fuzzy. In each, the objective is maximization of profit which is fuzzy and optimization is achieved through fuzzy ordering of fuzzy numbers with respect to their total integral values. We show that the first model reduces to the classical newsboy problem, and therefore an optimal solution is easily available. In second model, we show that the objective function is concave and hence present a characterization of the optimal solution, from which one can readily compute an optimal solution. Besides discussion of the models, a relevant extension is outlined.

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.

Improved complexity bound for the maximum cardinality bottleneck bipartite matching problem

Let G(V1, V2, E) be a bipartite graph and for each edge e ∈ E a weight We is prescribed. Then the bottleneck bipartite matching problem (BBMP) is to find a maximum cardinality matching M in G such that the largest edge weight associated with M is as small as possible. The best known algorithm to solve this problem has a worst-case complexity of O(m n log n), where m = |E| and n = |V1| + |V2|. In this note we present an O(m n log nm) algorithm to solve BBMP, improving the best available bound by a factor of O(m m log n)/n.

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.

A variation of the assignment problem

Abstract In this paper, a meaningful variant of the cost minimizing assignment problem with the objective of minimizing the total cost of assignment plus an additional ‘supervisory’ cost, which depends on the total time of completion of the project is formulated. An efficient method of finding an optimal solution to such a problem is presented with a numerical example to illustrate the same.

On stochastic spanning tree problem

This paper considers a generalized version of the stochastic spanning tree problem in which edge costs are random variables and the objective is to find a spectrum of optimal spanning trees satisfying a certain chance constraint whose right-hand side also is treated as a decision variable. A special case of this problem with fixed right-hand side has been solved polynomially using a parameteric approach. Also, the same parametric method without increasing the complexity order has been extended to include the right-hand side also as a decision variable. In this paper, two different methods are given for solving the generalized problem. First, a different parametric method better than the earlier one is given. Then, a method that makes use of the efficient extreme points of the convex hull of the mappings of all the spanning trees in a bicriteria spanning tree problem is presented. But it is shown that in the worst case the bicriteria method is superior.

Maximizing residual flow under an arc destruction

In this paper, we consider two problems related to single-commodity flows on a directed network. In the first problem, for a given s - t flow, if an arc is destroyed, all the flow that Is passing through that arc Is destroyed. What is left flowing from s to t is the residual flow. The objective Is to determine a flow pattern such that the residual flow is maximized. We provide a strongly polynomial algorithm for this problem, called the maximum residual flow problem, and consider various extensions of this basic model. In the second problem, known as the most vital arc problem, the objective is to remove an arc so that the maximal flow on the residual network is as small as possible. Results are also derived which help implement an efficient scheme for solving this problem.

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.

Flows over edge-disjoint mixed multipaths and applications

For improving reliability of communication in communication networks, where edges are subject to failure, Kishimoto [Reliable flow with failures in a network, IEEE Trans. Reliability, 46 (1997) 308-315] defined a @d-reliable flow, for a given source-sink pair of nodes, in a network for @d@?(0,1], where no edge carries a flow more than a fraction @d of the total flow in the network, and proved a max-flow min-cut theorem with cut-capacites defined suitably. Kishimoto and Takeuchi in [A method for obtaining @d-reliable flow in a network, IECCE Fundamentals E-81A (1998) 776-783] provided an efficient algorithm for finding such a flow. When (1/@d) is an integer, say q, Kishimoto and Takeuchi [On m-route flows in a network, IEICE Trans. J-76-A (1993) 1185-1200 (in Japanese)] introduced the notion of a q-path flow. Kishimoto [A method for obtaining the maximum multi-route flows in a network, Networks 27 (1996) 279-291] proved a max-flow min-cut theorem for q-path flow between a given source-sink pair (s,t) of nodes and provided a strongly polynomial algorithm for finding a q-path flow from s to t of maximum flow-value. In this paper, we extend the concept of q-path flow to any real number q>=1. When q(=1/@d) is fractional, we show that this general q-path flow can be viewed as a sum of some q-path flow and some q-path flow. We discuss several applications of this results, which include a simpler proof and generalization of a known result on wavelength division multiplexing problem. Finally we present a strongly polynomial, combinatorial algorithm for synthesizing an undirected network with minimum sum of edge capacities that satisfies (non-simultaneously) specified minimum requirements of q-path flow-values between all pairs of nodes, for a given real number q>=1.