Faster Algorithms for $$k\text {-}\textsc {Subset}\,\textsc {Sum}$$ and Variations

Antonopoulos, Antonis; Pagourtzis, Aris; Petsalakis, Stavros; Vasilakis, Manolis

doi:10.1007/978-3-030-97099-4_3

Cited by 3 publications

(1 citation statement)

References 21 publications

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“…These problems are tightly connected to Subset Sum, which has seen impressive advances recently, due to Koiliaris and Xu [20] who gave a deterministic Õ( √ nt) algorithm, where n is the number of input elements and t is the target, and by Bringmann [7] who gave a Õ(n+t) randomized algorithm, which is essentially optimal under SETH [1]. See also [2] for an extension of these algorithms to a more general setting. Jin and Wu subsequently proposed a simpler randomized algorithm [16] achieving the same bounds as [7], which however seems to only solve the decision version of the problem.…”

Section: Related Workmentioning

confidence: 99%

Approximating Subset Sum Ratio via Subset Sum Computations

Alonistiotis¹,

Antonopoulos²,

Melissinos³

et al. 2022

Preprint

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We present a new FPTAS for the Subset Sum Ratio problem, which, given a set of integers, asks for two disjoint subsets such that the ratio of their sums is as close to 1 as possible. Our scheme makes use of exact and approximate algorithms for the closely related Subset Sum problem, hence any progress over those-such as the recent improvement due to Bringmann and Nakos [SODA 2021]-carries over to our FPTAS. Depending on the relationship between the size of the input set n and the error margin ε, we improve upon the best currently known algorithm of Melissinos and Pagourtzis [COCOON 2018] of complexity O(n 4 /ε). In particular, the exponent of n in our proposed scheme may decrease down to 2, depending on the Subset Sum algorithm used. Furthermore, while the aforementioned state of the art complexity, expressed in the form O((n + 1/ε) c ), has constant c = 5, our results establish that c < 5.

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Section: Related Workmentioning

confidence: 99%

Approximating Subset Sum Ratio via Subset Sum Computations

Alonistiotis¹,

Antonopoulos²,

Melissinos³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Approximating Subset Sum Ratio via Subset Sum Computations

Alonistiotis

Antonopoulos

Melissinos

et al. 2022

Lecture Notes in Computer Science

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Faster algorithms for k-subset sum and variations

et al. 2022

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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.

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Faster Algorithms for $$k\text {-}\textsc {Subset}\,\textsc {Sum}$$ and Variations

Cited by 3 publications

References 21 publications

Approximating Subset Sum Ratio via Subset Sum Computations

Approximating Subset Sum Ratio via Subset Sum Computations

Approximating Subset Sum Ratio via Subset Sum Computations

Faster algorithms for k-subset sum and variations

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