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Approximation Schemes for k-Subset Sum Ratio and k-Way Number Partitioning Ratio

Authors Sotiris Kanellopoulos , Giorgos Mitropoulos , Antonis Antonopoulos , Nikos Leonardos , Aris Pagourtzis , Christos Pergaminelis , Stavros Petsalakis , Kanellos Tsitouras

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Fully polynomial-time approximation schemes
  • Subset Sum Ratio
  • Number Partitioning
  • Fair division
  • Envy minimization
  • Pseudo-polynomial time algorithms

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Abstract

The Subset Sum Ratio problem (SSR) asks, given a multiset A of positive integers, to find two disjoint subsets of A such that the largest-to-smallest ratio of their sums is minimized. In this paper we study the k-version of SSR, namely k-Subset Sum Ratio (k-SSR), which asks to minimize the largest-to-smallest ratio of sums of k disjoint subsets of A. We develop an approximation scheme for k-SSR running in O(n^{2k}/ε^{k-1}) time, where n = |A| and ε is the error parameter. To the best of our knowledge, this is the first FPTAS for k-SSR for fixed k > 2.
We also study the k-way Number Partitioning Ratio (k-PART) problem, which differs from k-SSR in that the k subsets must constitute a partition of A; this problem in fact corresponds to the objective of minimizing the largest-to-smallest sum ratio in the family of Multiway Number Partitioning problems. We present a more involved FPTAS for k-PART, also achieving O(n^{2k}/ε^{k-1}) time complexity. Notably, k-PART is also equivalent to the Minimum Envy-Ratio problem with identical valuation functions, which has been studied in the context of fair division of indivisible goods. Thus, for the case of identical valuations, our FPTAS represents a significant improvement over the O(n^{4k²+1}/ε^{2k²}) bound obtained by Nguyen and Rothe’s FPTAS [Trung Thanh Nguyen and Jörg Rothe, 2014] for Minimum Envy-Ratio with general additive valuations.
Lastly, we propose a second FPTAS for k-SSR, which employs carefully designed calls to the first one; the new scheme has a time complexity of Õ(n/ε^{3k-1}), thus being much faster when n≫ 1/ ε.

Cite As Get BibTex

Sotiris Kanellopoulos, Giorgos Mitropoulos, Antonis Antonopoulos, Nikos Leonardos, Aris Pagourtzis, Christos Pergaminelis, Stavros Petsalakis, and Kanellos Tsitouras. Approximation Schemes for k-Subset Sum Ratio and k-Way Number Partitioning Ratio. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 44:1-44:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.44

Author Details

Sotiris Kanellopoulos
  • National Technical University of Athens and Archimedes, Athena Research Center, Greece
Giorgos Mitropoulos
  • National Technical University of Athens and Archimedes, Athena Research Center, Greece
Antonis Antonopoulos
  • National Technical University of Athens, Greece
Nikos Leonardos
  • National Technical University of Athens, Greece
Aris Pagourtzis
  • National Technical University of Athens and Archimedes, Athena Research Center, Greece
Christos Pergaminelis
  • National Technical University of Athens and Archimedes, Athena Research Center, Greece
Stavros Petsalakis
  • National Technical University of Athens, Greece
Kanellos Tsitouras
  • National Technical University of Athens, Greece

Funding

Supported by project MIS 5154714 of the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program.

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