M. Kaykobad

Solving the Multidimensional Multiple-choice Knapsack Problem by constructing convex hulls

This paper presents a heuristic to solve the Multidimensional Multiple-choice Knapsack Problem (MMKP), a variant of the classical 0-1 Knapsack Problem. We apply a transformation technique to map the multidimensional resource consumption to single dimension. Convex hulls are constructed to reduce the search space to find the near-optimal solution of the MMKP. We present the computational complexity of solving the MMKP using this approach. A comparative analysis of different heuristics for solving the MMKP has been presented based on the experimental results.

On Nonnegative Factorization of Matrices

Abstract It is shown that a sufficient condition for a nonnegative real symmetric matrix to be completely positive is that the matrix is diagonally dominant.

On Hamiltonian cycles and Hamiltonian paths

A Hamiltonian cycle is a spanning cycle in a graph, i.e., a cycle through every vertex, and a Hamiltonian path is a spanning path. In this paper we present two theorems stating sufficient conditions for a graph to possess Hamiltonian cycles and Hamiltonian paths. The significance of the theorems is discussed, and it is shown that the famous Ores theorem directly follows from our result.

Positive solutions of positive linear systems

Easily verifiable sufficient conditions are obtained for the existence of a positive solution (componentwise) of a linear nonhomogeneous system of equations with positive coefficients.

Solving the multi-objective Vehicle Routing Problem with Soft Time Windows with the help of bees

Abstract This paper presents a new model and solution for the multi-objective Vehicle Routing Problem with Soft Time Windows ( VRPSTW ) using a hybrid metaheuristic technique. The proposed methodology is developed on the basics of a new swarm based Artificial Bee Colony ( ABC ) algorithm combined with two-step constrained local search for neighborhood selection. VRPSTW involves computing the routes of a set of vehicles with fixed capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. Here, the time window constraints are relaxed into “soft”, that is penalty terms are added to the solution cost whenever a vehicle serves a customer outside of his time window. The solution of routing problems with soft time windows has valuable practical applications. This paper uses a direct interpretation of the VRPSTW as a multi-objective optimization problem where the total traveling distance, number of window violations and number of required vehicles are minimized while capacity and time window constraints are met. Our work aims at using ABC inspired foraging behavior of honey bees which balances exploration and exploitation to avoid local optima and reach the global optima. The algorithm is applied to solve the well known benchmark Solomon׳s problem instances. Experimental results show that our suggested approach is quite effective, as it provides solutions that are competitive with the best known results in the literature. Finally, we present an analysis of our proposed algorithm in terms of computational time.

An efficient decoding technique for Huffman codes

We present a new data structure for Huffman coding in which in addition to sending symbols in order of their appearance in the Huffman tree one needs to send codes of all circular leaf nodes (nodes with two adjacent external nodes), the number of which is always bounded above by half the number of symbols. We decode the text by using the memory efficient data structure proposed by Chen et al. [Inform. Process. Lett. 69 (1999) 119–122].  2002 Elsevier Science B.V. All rights reserved.

A comprehensive analysis of degree based condition for Hamiltonian cycles

Rahman and Kaykobad introduced a shortest distance based condition for finding the existence of Hamiltonian paths in graphs as follows: Let G be a connected graph with n vertices, and if d(u) + d(v) + delta(u, v) ges n + 1, for each pair of distinct non-adjacent vertices u and v in G, where delta(u, v) is the length of a shortest path between u and v , then G has Hamiltonian path. Rao Li proved that under the same condition, the graph is Hamiltonian or belongs to two different classes of graphs. Recently, Mehedy, Hasan and Kaykobad showed case by case that under the condition of Rahman and Kaykobad, the graph is Hamiltonian with exceptions for delta(u, v) = 2. Shengjia Li et. al. mentions a graph to be Hamiltonian whenever d(u) + d(v) ges n - 1, for all delta(u, v) = 2, otherwise n is odd and the graph falls into a special class. This paper relates the results of Mehedy, Hasan and Kaykobad with the two exceptional classes of graphs introduced by Rao Li and the graph class introduced by Shengjia Li et. al. The paper also provides a thorough analysis of the graph classes and shows the characteristics of a graph when it falls into one of those classes.

An improved degree based condition for Hamiltonian cycles

A Hamiltonian cycle is a closed path through all the vertices of a graph. Since discovering whether a graph has a Hamiltonian path or a Hamiltonian cycle are both NP-complete problems, researchers concentrated on formulating sufficient conditions that ensure Hamiltonicity of a graph. A recent paper [M.S. Rahman, M. Kaykobad, On Hamiltonian cycles and Hamiltonian paths, Information Processing Letters 94 (2005) 37-41] presents distance based sufficient conditions for the existence of a Hamiltonian path. In this paper we establish that the same condition forces Hamiltonian cycle to be present excepting for the case where end points of a Hamiltonian path is at a distance of 2.

A New String Matching Algorithm

In this paper a new exact string-matching algorithm with sub-linear average case complexity has been presented. Unlike other sub-linear string-matching algorithms it never performs more than n text character comparisons while working on a text of length n . It requires only O ( m +σ) extra pre-processing time and space, where m is the length of the pattern and σ is the size of the alphabet.

Block huffman coding

Abstract Dynamic or adaptive Huffman coding, proposed by Gallager [1] and extended by Knuth [21, can be used for compressing a continuous stream. Our proposal for accomplishing the same task is termed here as block Huffman coding. This is an easy and simple solution to compress continuous data by applying simple Huffman coding in blocks of data. For each block, a different header is stored. This header is shipped with each block of compressed data. However, to keep the header overhead low, we have used the proposed storage efficient header [3].