An Almost Linear-Time Algorithm for the Dense Subset-Sum Problem

This paper describes a new approach for solving the subset-sum problem. It is useful for solving other NP-hard problems. The limits and potential of this approach are discussed. The approach yields an algorithm for solving the dense version of the subset-sum problem. It runs in time $O(\ell \log \el...

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Bibliographic Details
Published in:SIAM journal on computing Vol. 20; no. 6; pp. 1157 - 1189
Main Authors: Galil, Zvi, Margalit, Oded
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.12.1991
Subjects:
Algorithms
Approximation
Classical combinatorial problems
Combinatorics
Combinatorics. Ordered structures
Computer science
Dynamic programming
Exact sciences and technology
Integer programming
Mathematics
Sciences and techniques of general use
Integer
Integer programming
Partition
Algorithm complexity
NP hard problem
Combinatory problem
Progression arithmétique
ISSN:0097-5397, 1095-7111
Online Access:Get full text
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Summary:This paper describes a new approach for solving the subset-sum problem. It is useful for solving other NP-hard problems. The limits and potential of this approach are discussed. The approach yields an algorithm for solving the dense version of the subset-sum problem. It runs in time $O(\ell \log \ell )$, where $\ell $ is the bound on the size of the elements. But for dense enough inputs and target numbers near the middle sum, it runs in time $O(m)$, where $m$ is the number of elements. Consequently, it improves the previously best algorithms by at least one order of magnitude and sometimes by two. The algorithm yields a characterization of the set of subset sums as a collection of arithmetic progressions with the same difference. This characterization is derived by elementary number-theoretic and algorithmic techniques. Such a characterization was first obtained by using analytic number theory and yielded inferior algorithms.
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ISSN:0097-5397
1095-7111
DOI:10.1137/0220072