The Submodular Santa Claus Problem

TL;DR

The paper addresses max-min fairness in the Santa Claus problem with indivisible resources and monotone submodular valuations. It develops a layered augmentation framework that reduces the original problem to an augmentation problem, solves a strong recursive linear programming relaxation via Dantzig–Wolfe decomposition and multilinear extensions, and then rounds the LP to obtain an integral allocation. The main contributions are a direct generalization from additive to monotone submodular valuations, a polylogarithmic approximation running in time $n^{O(\log n/\log\log n)}$ and an $n^{\varepsilon}$-approximation in time $n^{O(1/\varepsilon)}$, all built on a novel recursive LP with a block structure and a robust rounding scheme. These results advance the understanding of fair allocation and connect to related scheduling problems by surpassing the previous $O(\sqrt{n})$ barrier for the submodular Santa Claus problem in this model.

Abstract

We consider the problem of allocating indivisible resources to players so as to maximize the minimum total value any player receives. This problem is sometimes dubbed the Santa Claus problem and its different variants have been subject to extensive research towards approximation algorithms over the past two decades. In the case where each player has a potentially different additive valuation function, Chakrabarty, Chuzhoy, and Khanna [FOCS'09] gave an $O(n^ε)$-approximation algorithm with polynomial running time for any constant $ε> 0$ and a polylogarithmic approximation algorithm in quasi-polynomial time. We show that the same can be achieved for monotone submodular valuation functions, improving over the previously best algorithm due to Goemans, Harvey, Iwata, and Mirrokni [SODA'09], which has an approximation ratio of more than $\sqrt{n}$. Our result builds up on a sophisticated LP relaxation, which has a recursive block structure that allows us to solve it despite having exponentially many variables and constraints.

Figures (3)

• Figure 1: Block structure of non-zero entries in constraint matrix of linear programming relaxation
• Figure 2: An instance of the one-level augmentation problem. The top sink on the left has valuation function equal to the total flow received, and the sink at the bottom has valuation function equal to the total flow received divided by $2$. The set of bold orange edges forms a feasible solution which covers all the sinks in $T$.
• Figure 3: An instance of $2$-level augmentation problem. The two levels are copies of the one level of Figure \ref{['fig:augmentation_one_level']}, with the same valuation functions for the sinks in each level. The second and fourth source in $S_1$ have a linking edge with the first and second sinks in $T_2$. The set of bold orange edges forms a feasible solution which covers all of $T_1$.

Theorems & Definitions (32)

• theorem 1
• theorem 2
• theorem 3
• proof : Proof of \ref{['thm:main']}
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7: Augmentation