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The Exact One-Half Threshold for Arbitrary-Capacity Bottleneck Multiple Subset Sum

Primarily AI-generated textHuman understanding: some partscs.DS — Data Structures and Algorithmscs.CC — Computational Complexitymath.OC — Optimization and Control

Contributed by Matěj Kripner ↗, Václav Rozhoň ↗, Robert Šámal ↗, Martin Šindelář, Adrián Zámečník

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

In bottleneck multiple subset sum, items with weights are packed into knapsacks with capacities, possibly leaving items unpacked, so as to maximize the minimum load. For identical capacities the best possible approximation ratio is 2/32/3, and Chen et al.\ asked whether 2/32/3 is also achievable for arbitrary capacities. We show that it is not: for arbitrary capacities we give a 1/21/2-approximation using O(nlog⁡(n+1)+mlog⁡(m+1))O(n\log(n+1)+m\log(m+1)) arithmetic and comparison operations for nn items and mm knapsacks, and prove that no fixed ratio above 1/21/2 is achievable unless P=NP\mathsf P=\mathsf{NP}. We also give a short proof of the classical 1/21/2 guarantee of Friesen and Langston for unit profits.

Provenance statement

This work was produced within the Bolzano project (https://bolzano.app), using AI systems for mathematical research and manuscript preparation. The listed authors contributed to the project and the preparation of this release. The manuscript received only limited human checking.

Tools used

Bolzano team
Bolzano

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Version history

  1. v1Submitted by Václav RozhoňInitial depositCurrentNov 03, 2026