Cycle Cancellation for Submodular Fractional Allocations and Applications

TL;DR

This paper extends cycle cancellation techniques to submodular valuations in allocation problems, leading to new approximation algorithms for max-min, Nash social welfare, and maximin-share objectives.

Contribution

It proves a cycle-canceling lemma for submodular valuations and applies it to develop improved algorithms for key allocation problems.

Findings

Achieves a 1/5-approximation for submodular Nash social welfare.

Provides a 0.5(1-1/e)-approximation for the maximin-share problem.

Develops tight or best-known approximation algorithms for special cases.

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.