Cycle Cancellation for Submodular Fractional Allocations and Applications
Chandra Chekuri, Pooja Kulkarni, Ruta Mehta, Jan Vondrak
TL;DR
The paper develops a submodular analogue of the cycle-cancellation lemma using the multilinear extension, enabling a fractional allocation to be transformed into a forest-structured, acyclic-support solution without reducing agents’ multilinear-extension values. This cycle-cancellation tool is combined with a NonUniformPipageRounding scheme to derive new approximation algorithms for three core objectives: Santa Claus (max-min), NSW, and MMS, all within the value-oracle model. The main results include a $1/5$-approximation for submodular NSW and a $(1-1/e- ext{poly}(1/n))$-approximation for NSW with a constant number of agents, as well as a $(1/2)(1-1/e-o(1))$-approximation for MMS and a $(1-1/e- ext{epsilon})$-approximation in the small-items regime. The work provides a unified rounding framework that leverages submodularity via the multilinear extension to obtain tight or best-known guarantees in several special cases, advancing practical fair division with submodular valuations in the value-oracle setting.
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.