A Faster FPTAS for the Subset-Sums Ratio Problem

Abstract

The Subset-Sums Ratio problem (SSR) is an optimization problem in which, given a set of integers, the goal is to find two subsets such that the ratio of their sums is as close to 1 as possible. In this paper we develop a new FPTAS for the SSR problem which builds on techniques proposed in [D. Nanongkai, Simple FPTAS for the subset-sums ratio problem, Inf. Proc. Lett. 113 (2013)]. One of the key improvements of our scheme is the use of a dynamic programming table in which one dimension represents the difference of the sums of the two subsets. This idea, together with a careful choice of a scaling parameter, yields an FP-TAS that is several orders of magnitude faster than the best currently known scheme of [C. Bazgan, M. Santha, Z. Tuza, Efficient approximation algorithms for the Subset-Sums Equality problem, J. Comp. System Sci. 64 (2) (2002)].

Approximate #Knapsack Computations to Count Semi-Fair Allocations

“…An interesting future work would be to improve the complexity of the FPTAS by exploring dimension reduction techniques (see e.g. [8]).…”

Approximating Subset Sum Ratio via Subset Sum Computations

“…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].…”

“…Our algorithm makes use of exact and approximation algorithms for Subset Sum, thus, any improvement over those carries over to our proposed scheme. Additionally, depending on the relationship between n and ε, our algorithm improves upon the best existing approximation scheme of [22].…”

“…We present a new approximation scheme for this problem, highlighting its close relationship with the classical Subset Sum problem. Our proposed algorithm is the first to associate these closely related problems and, depending on the relationship of the cardinality of the input set n and the value of the error margin ε, achieves better asymptotic bounds than the current state of the art [22]. Moreover, while the complexity of the current state of the art approximation scheme expressed in the form O((n + 1/ε) c ) has an exponent c = 5, we present an FPTAS with constant c < 5.…”

Faster Algorithms for $k$-Subset Sum and Variations

“…Equal Subset Sum has been proven NP-hard by Woeginger and Yu [27] and several variations have been proven NP-hard by Cieliebak et al in [9,10]. A 1.324-approximation algorithm has been proposed for Subset Sum Ratio in [27] and several FPTASs appeared in [2,23,21], the fastest so far being the one in [21] of complexity O(n 4 /ε).…”