Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.05229

Santa Claus and Weighted Nash Social Welfare with Subadditive Valuations

Yuhang Chen·Yuda Feng·Weijiang Hu·Shi Li·Ruilong Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study two fairness objectives for allocating $m$ indivisible items among $n$ agents with subadditive valuations: the max-min objective, also known as the Santa Claus problem, which maximizes the smallest value of an agent, and the weighted Nash social welfare, which maximizes the weighted geometric mean of the agents' values. For the max-min allocation problem, we give a randomized polynomial-time $O(\sqrt n)$-approximation algorithm for subadditive valuations with demand queries, improving the $2n-1$ bound of Khot and Ponnuswami. For additive valuations and for submodular valuations with value queries, we give simpler $O(\sqrt n)$-approximation algorithms. For additive valuations, this improves the $O(\sqrt{n\log n})$ bound of Saha and Srinivasan. For these two classes, $O(m^\varepsilon)$-approximation algorithms are also known for every constant $\varepsilon>0$. Our ratios depend only on $n$ and match, up to constants, the $Ω(\sqrt n)$ integrality gap of the configuration LP. For the weighted Nash social welfare problem with subadditive valuations and demand queries, we give an $O(Γ(w))$-approximation algorithm, where $Γ(w)=\exp\bigl((\ln n-H(w))/2\bigr)$ and $H(w)$ is the entropy of the weight vector $w$. The factor $Γ(w)$ equals $1$ for equal weights and is at most $\sqrt n$ for all weights. We show that this ratio is tight for algorithms analyzed against the configuration LP: for every weight vector, the configuration LP has an integrality gap of $Θ(Γ(w))$, even for XOS valuations, and this gap is $Ω(\sqrt n)$ when one agent has almost all the weight. For submodular valuations, the configuration LP has a constant integrality gap for all weights.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuhang Chen, Yuda Feng, Weijiang Hu, Shi Li, Ruilong Zhang. 2026-11-03. Santa Claus and Weighted Nash Social Welfare with Subadditive Valuations. https://arxiv.org/abs/2611.05229

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Loop unrolling of UCA models: distance labeling

A proper circular-arc (PCA) model is a pair $M = (C, A)$ where $C$ is a circle and $A$ is a family of inclusion-free arcs on $C$ whose extremes are pairwise different. The model $M$ represents a digraph $D$ that has one vertex $v(a)$ for each $a \in A$ and one edge $v(a) \to v(b)$ for each pair of arcs $a,b \in A(M)$ such that the beginning point of $b$ belongs to $a$. For $k \geq 0$, the $k$-th power $D^k$ of $D$ has the same vertices as $D$ and $v(a) \to v(b)$ is an edge of $D^k$ when $a\neq b$ and the distance from $v(a)$ to $v(b)$ in $D$ is at most $k$. A unit circular-arc (UCA) model is a PCA model $U = (C,A)$ in which all the arcs have the same length $\ell+1$. If $\ell$, the length $c$ of $C$, and the extremes of the arcs of $A$ are integer, then $U$ is a $(c,\ell)$-CA model. For $i \geq 0$, the model $i \times U$ of $U$ is obtained by replacing each arc $(s,s+\ell+1)$ with the arc $(s,s+i\ell+1)$. If $U$ represents a digraph $D$, then $U$ is $k$-multiplicative when $i \times U$ represents $D^i$ for every $0 \leq i \leq k$. In this article we design a linear time algorithm to decide if a PCA model $M$ is equivalent to a $k$-multiplicative UCA model when $k$ is given as input. The algorithm either outputs a $k$-multiplicative UCA model $U$ equivalent to $M$ or a negative certificate that can be authenticated in linear time. Our main technical tool is a new characterization of those PCA models that are equivalent to $k$-multiplicative UCA models. For $k=1$, this characterization yields a new algorithm for the classical representation problem that is simpler than the previously known algorithms.

cs.DS↗
arXiv2026

Improved Algorithms for Unrelated Crowd Worker Scheduling in Mobile Social Networks

This paper studies task scheduling with unrelated crowd workers in mobile social networks, where the service time of a task depends on the crowd worker that performs it, and the objective is to minimize the total weighted completion time. For identical crowd workers, we tighten the approximation ratio of the Largest-Ratio-First (LRF) algorithm and update the competitive ratio of its online version. For unrelated crowd workers, we show that the time-indexed rounding framework of Im and Li remains valid when worker-dependent contact delays are added, which yields a randomized algorithm with an expected approximation ratio of 1.45; this improves upon the expected ratio of $1.5+ε$ obtained in our previous work for identical crowd workers. We also derandomize this algorithm. To improve computational efficiency, we further propose an algorithm that exploits the optimality of the Smith-ratio order on a single crowd worker, and we prove that it is a deterministic 1.5-approximation algorithm and that its polynomial-time version achieves an approximation ratio arbitrarily close to 1.5. Experiments on synthetic instances and on instances derived from real contact traces show that this algorithm outperforms three variants of LRF in all tested settings, with the largest advantage when the crowd workers are highly heterogeneous, performs on par with a local-search heuristic for unrelated machines, and is within 0.4\% of the optimum on average on small instances that can be solved exactly.

cs.DS↗
arXiv2026

Fully Scalable MPC Algorithms for Euclidean Clustering: Constant Approximation in Constant Rounds

We study Euclidean $k$-Means and $k$-Median under the Massively Parallel Computation (MPC) model, focusing on the \emph{fully scalable} setting. Previously, fully scalable algorithms for $k$-Means and/or $k$-Median either required $O(\log\log n\cdot\log\log\log n)$ rounds [Cohen-Addad, Kuhn, and Parsaeian; SODA'26] or guaranteed only an $O(\log n)$-approximation [Goranci, Jiang, Kiss, Kong, Qian, and Szilagyi; SODA'26]. In this paper, we close this gap by giving the first fully scalable algorithms for both problems, which work even when $k$ is polynomially larger than the local memory, and \emph{simultaneously} achieve an $O(1)$-approximation and $O(1)$ round complexity. Technically, we devise a new variant of the MP algorithm [Mettu and Plaxton, SICOMP'03] that computes a \emph{fractional solution satisfying the Lagrangian Multiplier Preserving (LMP) property}. Importantly, this LMP guarantee holds even under arbitrary distance distortions, a feature that is crucial for efficient MPC implementations. We then give a novel MPC rounding algorithm that first rounds to a bicriteria solution with $k+O(n^{1-ε})$ centers and then uses an \emph{iterative kernelization} idea to convert it into one with exactly $k$ centers.

cs.DS↗