Cycle Cancellation for Submodular Fractional Allocations and Applications

2025-11-26

AI Summary

This paper studies fair allocation of indivisible items under submodular valuations. Addressing the challenge that classical rounding algorithms fail to preserve non-decreasing multilinear extension (MLE) values, we generalize the cycle-elimination lemma to the submodular setting and propose a framework based on detecting and eliminating cycles in the support graph of a fractional allocation—ensuring discrete allocations while maintaining each agent’s MLE value. Our approach unifies treatment of three central objectives: Santa Claus (max-min fairness), Nash social welfare (NSW), and maximin share (MMS). We achieve a 1/5-approximation for general submodular NSW and a (1−1/e)/2-approximation for MMS. Moreover, we obtain optimal or state-of-the-art guarantees in special cases—including small-item instances and constant numbers of agents. The framework provides a clean, general, and analytically tractable paradigm for submodular fair allocation.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We consider discrete allocation problem where $m$ indivisible goods are to be divided among $n$ agents. When agents' valuations are additive, the well-known cycle cancelling lemma by Lenstra, Shmoys, and Tardos plays a key role in 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 $frac{1}{5}$-approximation; for the MMS problem, we obtain a $frac{1}{2}(1-1/e)$-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.

Problem

• Extends cycle cancellation to submodular valuations in allocation problems
• Develops algorithms for max-min, Nash welfare, and maximin-share objectives
• Provides approximation guarantees for submodular fair division problems

Innovation

• Cycle cancellation for submodular valuations
• Multilinear extension preserves agent values
• New approximation algorithms for allocation objectives