Table of Contents
ISRN Electronics
Volume 2012, Article ID 859820, 10 pages
http://dx.doi.org/10.5402/2012/859820
Research Article

Polynomial Time Instances for the IKHO Problem

1Department of Computer Science, University of Verona, 37134 Verona, Italy
2Department of Information Engineering and Computer Science, University of Trento, 38123 Povo, Italy

Received 20 January 2012; Accepted 7 February 2012

Academic Editors: C. W. Chiou and T. L. Kunii

Copyright © 2012 Romeo Rizzi and Luca Nardin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Interactive Knapsacks Heuristic Optimization (IKHO) problem is a particular knapsacks model in which, given an array of knapsacks, every insertion in a knapsack affects also the other knapsacks, in terms of weight and profit. The IKHO model was introduced by Isto Aho to model instances of the load clipping problem. The IKHO problem is known to be APX-hard and, motivated by this negative fact, Aho exhibited a few classes of polynomial instances for the IKHO problem. These instances were obtained by limiting the ranges of two structural parameters, c and u, which describe the extent to which an insertion in a knapsack in uences the nearby knapsacks. We identify a new and broad class of instances allowing for a polynomial time algorithm. More precisely, we show that the restriction of IKHO to instances where ( 𝑐 + 2 𝑒 ) / 𝑐 is bounded by a constant can be solved in polynomial time, using dynamic programming.