Minimizing Symmetric Convex Functions over Hybrid of Continuous and Discrete Convex Sets

Authors Yasushi Kawase , Koichi Nishimura, Hanna Sumita



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Author Details

Yasushi Kawase
  • University of Tokyo, Japan
Koichi Nishimura
  • CRESCO LTD., Japan
Hanna Sumita
  • Tokyo Institute of Technology, Japan

Acknowledgements

We are grateful to Kazuo Murota, Warut Suksompong, and the anonymous reviewers for their helpful comments.

Cite AsGet BibTex

Yasushi Kawase, Koichi Nishimura, and Hanna Sumita. Minimizing Symmetric Convex Functions over Hybrid of Continuous and Discrete Convex Sets. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 96:1-96:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.96

Abstract

We study the problem of minimizing a given symmetric strictly convex function over the Minkowski sum of an integral base-polyhedron and an M-convex set. This problem has a hybrid of continuous and discrete structures. This emerges from the problem of allocating mixed goods, consisting of both divisible and indivisible goods, to agents with binary valuations so that the fairness measure, such as the Nash welfare, is maximized. It is known that both an integral base-polyhedron and an M-convex set have similar and nice properties, and the non-hybrid case can be solved in polynomial time. While the hybrid case lacks some of these properties, we show the structure of an optimal solution. Moreover, we exploit a proximity inherent in the problem. Through our findings, we demonstrate that our problem is NP-hard even in the fair allocation setting where all indivisible goods are identical. Moreover, we provide a polynomial-time algorithm for the fair allocation problem when all divisible goods are identical.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Algorithmic game theory
Keywords
  • Integral base-polyhedron
  • Fair allocation
  • Matroid

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