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Faster algorithms for k-subset sum and variations

Author

Listed:
  • Antonis Antonopoulos

    (National Technical University of Athens)

  • Aris Pagourtzis

    (National Technical University of Athens)

  • Stavros Petsalakis

    (National Technical University of Athens)

  • Manolis Vasilakis

    (National Technical University of Athens)

Abstract

We present new, faster pseudopolynomial time algorithms for the k-Subset Sum problem, defined as follows: given a set Z of n positive integers and k targets $$t_1, \ldots , t_k$$ t 1 , … , t k , determine whether there exist k disjoint subsets $$Z_1,\dots ,Z_k \subseteq Z$$ Z 1 , ⋯ , Z k ⊆ Z , such that $$\Sigma (Z_i) = t_i$$ Σ ( Z i ) = t i , for $$i = 1, \ldots , k$$ i = 1 , … , k . Assuming $$t = \max \{ t_1, \ldots , t_k \}$$ t = max { t 1 , … , t k } is the maximum among the given targets, a standard dynamic programming approach based on Bellman’s algorithm can solve the problem in $$O(n t^k)$$ O ( n t k ) time. We build upon recent advances on Subset Sum due to Koiliaris and Xu, as well as Bringmann, in order to provide faster algorithms for k-Subset Sum. We devise two algorithms: a deterministic one of time complexity $${\tilde{O}}(n^{k / (k+1)} t^k)$$ O ~ ( n k / ( k + 1 ) t k ) and a randomised one of $${\tilde{O}}(n + t^k)$$ O ~ ( n + t k ) complexity. Additionally, we show how these algorithms can be modified in order to incorporate cardinality constraints enforced on the solution subsets. We further demonstrate how these algorithms can be used in order to cope with variations of k-Subset Sum, namely Subset Sum Ratio, k-Subset Sum Ratio and Multiple Subset Sum.

Suggested Citation

  • Antonis Antonopoulos & Aris Pagourtzis & Stavros Petsalakis & Manolis Vasilakis, 2023. "Faster algorithms for k-subset sum and variations," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-21, January.
  • Handle: RePEc:spr:jcomop:v:45:y:2023:i:1:d:10.1007_s10878-022-00928-0
    DOI: 10.1007/s10878-022-00928-0
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    References listed on IDEAS

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    1. Diego Recalde & Daniel Severín & Ramiro Torres & Polo Vaca, 2018. "An exact approach for the balanced k-way partitioning problem with weight constraints and its application to sports team realignment," Journal of Combinatorial Optimization, Springer, vol. 36(3), pages 916-936, October.
    2. Dell’Amico, Mauro & Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2019. "Mathematical models and decomposition methods for the multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 274(3), pages 886-899.
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