Polynomial Time Instances for the IKHO Problem

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 influences 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 + 2u)/c is bounded by a constant can be solved in polynomial time, using dynamic programming