Approximation schemes for subset-sums ratio problems

Melissinos, Nikolaos; Pagourtzis, Aris; Triommatis, Theofilos

doi:10.1016/j.tcs.2022.07.027

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2024

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“…Equal Subset Sum has been proven NP-hard by Woeginger and Yu [33] (see also the full version of [27] for an alternative proof) and several variations have been proven NPhard by Cieliebak et al [11,14]. A 1.324-approximation algorithm has been proposed for Subset Sum Ratio in [33] and several FPTASes appeared in [6,25,29], the fastest so far being the one in [25] of complexity O(n 4 /ε), the complexity of which also applies to various meaningful special cases, as shown in [26].…”

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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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