Approximating Subset Sum Ratio via Subset Sum Computations

Alonistiotis, Giannis; Antonopoulos, Antonis; Melissinos, Nikolaos; Pagourtzis, Aris; Petsalakis, Stavros; Vasilakis, Manolis

doi:10.1007/978-3-031-06678-8_6

Cited by 3 publications

(6 citation statements)

References 28 publications

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“…6. Prior work In the current paper we improve upon the results of the preliminary version [2], by using approximate and exact Partition algorithms instead of Subset Sum computations.…”

Section: Our Contributionmentioning

confidence: 99%

Approximating subset sum ratio via partition computations

Alonistiotis,

Antonopoulos,

Melissinos

et al. 2024

Acta Informatica

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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 Partition, and clearly showcases the close relationship between the two algorithmic problems. Depending on the relationship between the size of the input set n and the error margin $$\varepsilon $$ ε , we improve upon the best currently known algorithm of Melissinos and Pagourtzis [COCOON 2018] of complexity $$\mathcal {O} (n^4 / \varepsilon )$$ O ( n 4 / ε ) . In particular, the exponent of n in our proposed scheme may decrease down to 2, depending on the Partition algorithm used.

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“…6. Prior work In the current paper we improve upon the results of the preliminary version [2], by using approximate and exact Partition algorithms instead of Subset Sum computations.…”

Section: Our Contributionmentioning

confidence: 99%

Approximating subset sum ratio via partition computations

Alonistiotis,

Antonopoulos,

Melissinos

et al. 2024

Acta Informatica

Self Cite

View full text Add to dashboard Cite

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“…Equal Subset Sum has been proven NP-hard by Woeginger and Yu (1992) and several variations have been proven NP-hard by Cieliebak et al (2003aCieliebak et al ( , 2008. A 1.324-approximation algorithm has been proposed for Subset Sum Ratio in Woeginger and Yu (1992), while several FPTASs have appeared in bibliography (Bazgan et al 2002;Nanongkai 2013;Melissinos and Pagourtzis 2018;Alonistiotis et al 2022).…”

Section: Related Workmentioning

confidence: 99%

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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“…Here we interpret z{0 " 8 for any z ě 0 to exclude that X or Y are the empty set. Both Equal Subset Sum and Subset Sum Ratio are well-studied variants of the classic Subset Sum problem, see, e.g., [2,3,18,19,21,22].…”

Section: Introductionmentioning

confidence: 99%

“…Melissinos and Pagourtzis [18] improved the running time to Opn 4 {εq. Recently, Alonistiotis et al [2] designed approximation schemes running in time r Opn 2.3 {ε 2.6 q and r Opn 2 {ε 3 q, thus achieving different tradeoffs compared to the algorithm by Melissinos and Pagourtzis.…”

Section: Introductionmentioning

confidence: 99%

“…Recent conditonal lower bounds show that we cannot expect much faster algorithms: (1) An approximation scheme for Subset Sum with running time Oppn `1{εq 1.999 q would contradict the Min-Plus-Convolution Hypothesis [5]. Moreover, (2) if Subset Sum has an approximation scheme with running time 2 opnq {ε 0.999 then by setting ε ă 1{t we would obtain an exact algorithm running in pseudopolynomial time 2 opnq t 0.999 , which would contradict the Strong Exponential Time Hypothesis [1]. Note that the running time r Opn{ε 0.9386 q that we obtain for approximating Subset Sum Ratio is of the form (1) Oppn `1{εq 1.999 q and of the form (2) 2 opnq {ε 0.999 .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Fine-Grained Perspective on Approximating Subset Sum and Partition

Bringmann

Nakos

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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Subset Sum Ratio is the following optimization problem: Given a set of n positive numbers I, find disjoint subsets X, Y Ď I minimizing the ratio maxtΣpXq{ΣpY q, ΣpY q{ΣpXqu, where ΣpZq denotes the sum of all elements of Z. Subset Sum Ratio is an optimization variant of the Equal Subset Sum problem. It was introduced by Woeginger and Yu in '92 and is known to admit an FPTAS [Bazgan, Santha, Tuza '98]. The best approximation schemes before this work had running time Opn 4 {εq [Melissinos, Pagourtzis '18], r Opn 2.3 {ε 2.6 q and r Opn 2 {ε 3 q [Alonistiotis et al. '22].In this work, we present an improved approximation scheme for Subset Sum Ratio running in time Opn{ε 0.9386 q. Here we assume that the items are given in sorted order, otherwise we need an additional running time of Opn log nq for sorting. Our improved running time simultaneously improves the dependence on n to linear and the dependence on 1{ε to sublinear.For comparison, the related Subset Sum problem admits an approximation scheme running in time Opn{εq [Gens, Levner '79]. If one would achieve an approximation scheme with running time r Opn{ε 0.99 q for Subset Sum, then one would falsify the Strong Exponential Time Hypothesis [Abboud, Bringmann, Hermelin, Shabtay '19] as well as the Min-Plus-Convolution Hypothesis [Bringmann, Nakos '21]. We thus establish that Subset Sum Ratio admits faster approximation schemes than Subset Sum. This comes as a surprise, since at any point in time before this work the best known approximation scheme for Subset Sum Ratio had a worse running time than the best known approximation scheme for Subset Sum.

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Approximating Subset Sum Ratio via Subset Sum Computations

Cited by 3 publications

References 28 publications

Approximating subset sum ratio via partition computations

Approximating subset sum ratio via partition computations

Faster algorithms for k-subset sum and variations

A Fine-Grained Perspective on Approximating Subset Sum and Partition

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