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New Polynomial-time Instances to Various Knapsack-type Problems

Aho I.

Fundamenta Informaticae

2002

Vol. 53, nr 3,4

199-228

We describe a special case of the interactive knapsack optimization problem (motivated by the load clipping problem) solvable in polynomial time. Given an instance parameterized by k, the solution can be found in polynomial time, where the polynomial has degree k. In the interactive knapsack problem, k is connected to the length induced by an item. A similar construction solves a special case of the 0-1 multi-dimensional knapsack and the 0-1 linear integer programming problems in polynomial time. In these problems the parameter determines the width of the restriction matrix, which is a band matrix. We extend the 0-1 multi-dimensional knapsack solution to 0-n multi-dimensional knapsack problems (and to 0-n IP problems). Our algorithms are based on the (resource bounded) shortest path search: we represent restrictions efficiently in a form of a graph such that each feasible solution has a path between given source and target vertices.

Interactive knapsacks

Aho I.

Fundamenta Informaticae

2000

Vol. 44, Nr 1,2

1-23

The interactive knapsack problems are generalizations of the classical knapsack problem. Three different new NP-complete problems, interactive knapsack heuristic decision problem (IKHD), interactive knapsack decision problem (IKD) and multidimensional cloned knapsack decision problem (MDCS), are presented for the interactive knapsack models. IKD and MDCS are shown to be strongly NP-complete. By using interactive knapsacks we can model many planning and scheduling problems in an innovative way. Possible application areas include electricity management, single and multiprocessor scheduling, and packing and tiling problems. As a by-product we show that the longest weight-constrained path problem is NP-complete.