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Cycle Cancellation for Submodular Fractional Allocations and Applications

Authors Chandra Chekuri , Pooja Kulkarni , Ruta Mehta , Jan Vondràk

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Submodular Optimization
  • Allocation Problems
  • Fair Division

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Abstract

We consider discrete allocation problems where m indivisible goods need to be allocated among n agents. When agents' valuation functions are additive, the well-known cycle canceling lemma [Lenstra et al., 1990; Shmoys and Tardos, 1993; Plotkin et al., 1995] plays a key role in the design and analysis of rounding algorithms. In this paper, we prove an analogous lemma for the case of submodular valuations. Our algorithm removes cycles in the support graph of a fractional allocation while guaranteeing that each agent’s value, measured using the multilinear extension, does not decrease. 
We demonstrate applications of the cycle-canceling algorithm, along with other ideas, to obtain new algorithms and results for three well-studied allocation objectives: max-min (Santa Claus problem), Nash social welfare (NSW), and maximin-share (MMS). For the submodular NSW problem, we obtain a 1/5-approximation; for the MMS problem, we obtain a 1/2(1-1/e)-approximation through new simple algorithms. For various special cases where the goods are "small" valued or the number of agents is constant, we obtain tight/best-known approximation algorithms. All our results are in the value-oracle model.

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Chandra Chekuri, Pooja Kulkarni, Ruta Mehta, and Jan Vondràk. Cycle Cancellation for Submodular Fractional Allocations and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.21

Author Details

Chandra Chekuri
  • University of Illinois at Urbana Champaign, IL, USA
Pooja Kulkarni
  • University of Chicago, IL, USA
Ruta Mehta
  • University of Illinois at Urbana Champaign, IL, USA
Jan Vondràk
  • Stanford University, CA, USA

Funding

  • Chekuri, Chandra: Supported in part by NSF grant CCF-2402667.
  • Kulkarni, Pooja: Supported in part by NSF grant CCF-2334461 and in part by a gift from Adobe to S.Khuller.
  • Mehta, Ruta: Supported in part by NSF grant CCF-2334461.

Acknowledgements

We would like to thank László Végh for useful discussions, in particular posing a question about using continuous local search for solving the multilinear relaxation of the Nash social welfare problem.

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