Approximation Schemes for Subset Sum Ratio Problems

Melissinos, Nikolaos; Pagourtzis, Aris; Triommatis, Theofilos

doi:10.1007/978-3-030-59901-0_9

Cited by 2 publications

(1 citation statement)

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“…Equal Subset Sum has been proven NP-hard by Woeginger and Yu [30] (see also the full version of [24] for an alternative proof) and several variations have been proven NP-hard by Cieliebak et al in [11,12]. A 1.324-approximation algorithm has been proposed for Subset Sum Ratio in [30] and several FP-TASs appeared in [5,26,22], the fastest so far being the one in [22] of complexity O(n 4 /ε), the complexity of which seems to also apply to various meaningful special cases, as shown in [23].…”

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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