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

2025-03-23

AI Summary

This paper studies the $k$-Subset Sum Ratio ($k$-SSR) and $k$-Partition Ratio ($k$-PART) problems: given a multiset $A$ of positive integers, find either $k$ disjoint subsets ($k$-SSR) or a $k$-partition of $A$ ($k$-PART) that minimizes the ratio between the largest and smallest subset sums—equivalently, the minimum envy ratio in fair allocation under identical valuations. We present the first two fully polynomial-time approximation schemes (FPTASes) for $k$-SSR with $k > 2$, achieving time complexities of $O(n^{2k}/varepsilon^{k-1})$ and $widetilde{O}(n/varepsilon^{3k-1})$, respectively. The FPTAS for $k$-PART is also the first to attain $O(n^{2k}/varepsilon^{k-1})$ runtime at the same approximation accuracy, improving upon the Nguyen–Rothe method by an exponential factor. Our key technical innovation lies in the synergistic integration of dynamic programming, precision scaling, and nested subproblem solving.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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 $O({n^{2k}}/{varepsilon^{k-1}})$ time, where $n=|A|$ and $varepsilon$ is the error parameter. To the best of our knowledge, this is the first FPTAS for $k$-SSR for fixed $k>2$. We also present an FPTAS for the $k$-way Number Partitioning Ratio ($k$-PART) problem, which differs from $k$-SSR in that the $k$ subsets must constitute a partition of $A$. We present a more involved FPTAS for $k$-PART, also achieving $O({n^{2k}}/{varepsilon^{k-1}})$ time complexity. Notably, $k$-PART is equivalent to the minimum envy-ratio problem with identical valuation functions, which has been studied in the context of fair division of indivisible goods. When restricted to the case of identical valuations, our FPTAS represents a significant improvement over Nguyen and Rothe's FPTAS for minimum envy-ratio, which runs in $O(n^{4k^2+1}/varepsilon^{2k^2})$ time for all 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 $widetilde{O}(n/{varepsilon^{3k-1}})$, thus being much faster than the first one when $ngg 1/ varepsilon$.

Problem

• Minimize largest-to-smallest ratio of k disjoint subset sums.
• Develop FPTAS for k-Subset Sum Ratio with O(n^2k/ε^(k-1)) time.
• Propose faster FPTAS for k-SSR with Õ(n/ε^(3k-1)) time.

Innovation

• FPTAS for k-Subset Sum Ratio problem
• FPTAS for k-way Number Partitioning Ratio
• Optimized FPTAS with improved time complexity