A fast approximation scheme for the multiple knapsack problem
SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
Tight approximation algorithms for scheduling with fixed jobs and nonavailability
ACM Transactions on Algorithms (TALG)
Approximation algorithms for scheduling and packing problems
WAOA'11 Proceedings of the 9th international conference on Approximation and Online Algorithms
Faster approximation algorithms for scheduling with fixed jobs
CATS '11 Proceedings of the Seventeenth Computing: The Australasian Theory Symposium - Volume 119
Faster approximation algorithms for scheduling with fixed jobs
CATS 2011 Proceedings of the Seventeenth Computing on The Australasian Theory Symposium - Volume 119
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The multiple knapsack problem (MKP) is a well-known generalization of the classical knapsack problem. We are given a set $A$ of $n$ items and set $B$ of $m$ bins (knapsacks) such that each item $a \in A$ has a size $size(a)$ and a profit value $profit(a)$, and each bin $b \in B$ has a capacity $c(b)$. The goal is to find a subset $U \subset A$ of maximum total profit such that $U$ can be packed into $B$ without exceeding the capacities. The decision version of MKP is strongly NP-complete, since it is a generalization of the classical knapsack and bin packing problem. Furthermore, MKP does not admit a fully time polynomial time approximation scheme (FPTAS) even if the number $m$ of bins is two. Kellerer gave a polynomial time approximation scheme (PTAS) for MKP with identical capacities and Chekuri and Khanna presented a PTAS for MKP with general capacities with running time $n^{O(\log(1/\epsilon)/\epsilon^8)}$. In this paper we propose an efficient PTAS (EPTAS) with parameterized running time $2^{O(\log(1/\epsilon)/\epsilon^5)} \cdot poly(n) + O(m)$ for MKP. This also solves an open question by Chekuri and Khanna.