Santosh N. Kabadi

Some NP-complete problems in quadratic and nonlinear programming

AbstractIn continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether(a)a given feasible solution is not a local minimum, and(b)the objective function is not bounded below on the set of feasible solutions. We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.

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.

TSP Heuristics: Domination Analysis and Complexity

We show that the 2-Opt and 3-Opt heuristics for the traveling salesman problem (TSP) on the complete graph Kn produce a solution no worse than the average cost of a tour in Kn in a polynomial number of iterations. As a consequence, we get that the domination numbers of the 2- Opt , 3- Opt , Carlier—Villon, Shortest Path Ejection Chain, and Lin—Kernighan heuristics are all at least (n-2)! / 2 . The domination number of the Christofides heuristic is shown to be no more than

Domination analysis of some heuristics for the traveling salesman problem

\lceil{n}/{2}\rceil !

Polynomially Solvable Cases of the TSP

, and for the Double Tree heuristic and a variation of the Christofides heuristic the domination numbers are shown to be one (even if the edge costs satisfy the triangle inequality). Further, unless P = NP, no polynomial time approximation algorithm exists for the TSP on the complete digraph

On totally dual integral systems

\vec{K}_n

Integer Solution for Linear Complementarity Problem

with domination number at least (n-1)!-k for any constant k or with domination number at least (n-1)! - (( k /(k+1))(n+r))!-1 for any non-negative constants r and k such that (n+r)

A linear time algorithm for the bottleneck traveling salesman problem on a Halin graph

\equiv

Flows over edge-disjoint mixed multipaths and applications

0 mod (k+1). The complexities of finding the median value of costs of all the tours in

The Bottleneck TSP

\vec{K}_n