A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem

• Published in: SIAM journal on computing Vol. 35; no. 3; pp. 713 - 728
• Main Authors: Chekuri, Chandra, Khanna, Sanjeev
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2005
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Assignment problem; Computer science; control theory; systems; Exact sciences and technology; Knapsack problem; Operations research; Profits; Restrictions; Theoretical computing; 68Q25; Bin packing problem; 68W25; Approximation algorithm; Knapsack problem; Polynomial time; Gap problem; multiple knapsack problem; Generalized assignment problem; Packing; approximation algorithm 68Q17; polynomial time approximation scheme; Best approximation
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/S0097539700382820

The multiple knapsack problem (MKP) is a natural and well-known generalization of the single knapsack problem and is defined as follows. We are given a set of $n$ items and $m$ bins (knapsacks) such that each item $i$ has a profit $p(i)$ and a size $s(i)$, and each bin $j$ has a capacity $c(j)$. The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the generalized assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APX-hard and a 2-approximation, for it is implicit in the work of Shmoys and Tardos [Math. Program. A, 62 (1993), pp. 461-474], and thus far, this was also the best known approximation for MKP\@. The main result of this paper is a polynomial time approximation scheme (PTAS) for MKP\@. Apart from its inherent theoretical interest as a common generalization of the well-studied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APX-hard. We substantiate this by showing that slight generalizations of MKP are APX-hard. Thus our results help demarcate the boundary at which instances of GAP become APX-hard. An interesting aspect of our approach is a PTAS-preserving reduction from an arbitrary instance of MKP to an instance with $O(\log n)$ distinct sizes and profits.