Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach

  • Authors:

  • Changlin Wan;Zhongzhi Shi

  • Affiliations:

  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China 100080 and Graduate University of Chinese Academy of Sciences, Beijing, China 100080;Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China 100080

  • Venue:
  • FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
  • Year:
  • 2008

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Abstract

The subset sum problem (SSP) can be briefly stated as: given a target integer Eand a set Acontaining npositive integer aj, find a subset of Asumming to E. The densitydof an SSP instance is defined by the ratio of nto m, where mis the logarithm of the largest integer within A. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme for SSP, and implement it as a complete and exact algorithm (EnumPlus). The algorithm always equivalently reduces an instance to be low-density, and then solve it by enumeration. Through this approach, we show the possibility to design a sole algorithm that can efficiently solve arbitrary density instance in a uniform way. Furthermore, our algorithm has considerable performance advantage over previous algorithms. Firstly, it extends the density scope, in which SSP can be solved in expected polynomial time. Specifically, It solves SSP in expected O(nlogn) time when density $d \geq c\cdot \sqrt{n}/\log{n}$, while the previously best density scope is d驴 c·n/(logn)2. In addition, the overall expected time and space requirement in the average case are proven to be O(n5logn) and O(n5) respectively. Secondly, in the worst case, it slightly improves the previously best time complexity of exact algorithms for SSP. The worst-case time complexity of our algorithm is proved to be O(n·2n/2驴 c·2n/2+ n), while the previously best result is O(n·2n/2).