Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.27098

Improved Hardness Results for Nash Social Welfare, Budgeted Allocation and GAP via the Unique Games Conjecture

Vignesh Viswanathan
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the problem of dividing a set of indivisible goods among agents with additive valuations. This problem has been studied under various objectives in both the computer science and the operations research literature. Our main contribution is a novel dictator test using this problem, which can separate a dictator from any function sufficiently far from a dictator. We use this test to prove the following hardness results (assuming the unique games conjecture is true): (1) We show that it is NP-hard to approximate the max Nash welfare by a factor better than $\sqrt[3]{\frac{81}{65}} - \varepsilon \approx 1.0761$. This improves on the previous best known inapproximability factor of $\sqrt{\frac87} - \varepsilon \approx 1.069$. (2) We show that it is NP-hard to approximate the maximum budgeted allocation by a factor better than $\frac{243}{227} - \varepsilon \approx 1.07$. This improves on the previous best known inapproximability factor of $\frac{16}{15} - \varepsilon \approx 1.067$. (3) We show that it is NP-hard to approximate the max generalized assignment problem (GAP) by a factor better than $\frac{145}{129} - \varepsilon \approx 1.124$. This improves on the previous best known inapproximability factor of $\frac{11}{10} - \varepsilon \approx 1.10$.

Explore related subjects

cs.GTcs.DS

Keep this discovery

BibTeXRIS

Vignesh Viswanathan. 2026-05-26. Improved Hardness Results for Nash Social Welfare, Budgeted Allocation and GAP via the Unique Games Conjecture. https://arxiv.org/abs/2605.27098

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

KEEP EXPLORING

Related papers

arXiv2026

MMS Allocation for Chores with Online Agent Arrivals

We study the fair allocation of $m$ indivisible chores to $n$ agents with subadditive cost functions arriving online in an arbitrary order. Upon an agent's arrival, we are informed of her cost function and must irrevocably assign her a set of chores. We focus on the Maximin Share (MMS) fairness notion and aim to compute an allocation in which all items are assigned, and no agent incurs a cost more than $\alpha$ times her MMS. Without any prior information about the instance (other than $n$ and $m$), we design an algorithm with a competitive ratio of $O(\min\{n, k\log^{1+\epsilon}k, \log m\})$ for any constant $\epsilon > 0$, where $k$ denotes the number of cost function types. Our bound matches the best known offline approximation guarantees for MMS under subadditive costs and is nearly optimal with respect to all three parameters: we show that even for binary additive cost functions, no online algorithm can achieve a competitive ratio of $o(\min\{n, k\log k, \log m\})$. We then consider the setting in which the $k$ cost function types are known in advance (though the realized types of arriving agents are not). For additive cost functions, we provide an algorithm with a competitive ratio of $O(\min\{\log k, \log(kn)/\log\log(kn)\})$, and show that constant-competitive algorithms do not exist for general $k$, even for the binary additive setting. For binary additive functions when $k \le n$, we propose a $3$-competitive algorithm and establish a lower bound of $2$.

cs.GT↗
arXiv2026

Truncated Noisy Best-Response Algorithms: Toward Game Theoretic Learning with Safety Guarantees

We consider a game theoretic approach to solve multi-agent coordination problems with submodular maximization objectives. It is known for such problems that the Nash equilibria for the corresponding game are always within 50% of the optimal, but that the equilibria which achieve this worst-case bound are not stable. To exploit this instability, we propose a family of algorithms which we call Truncated Noisy Best-Response (TNBR) Algorithms. These algorithms are flexibly characterized by agents asynchronously and stochastically selecting actions from a neighbourhood of their best response payoffs. We compute bounds on the recurrent classes of TNBR algorithms' associated Markov chains. Our bounds fall into two categories: first, "Performance" bounds ensure that TNBR algorithms always have a high-value recurrent state; second, "Safety" bounds ensure that TNBR algorithms never have arbitrarily-bad recurrent states. Furthermore, these two types of bounds are linked by a waterbed-like effect: every game with a poor Safety guarantee necessarily has a favorable Performance guarantee.

cs.GT↗
arXiv2026

Existence of the Core in Approval-Based Committee Elections

We settle the main open question in the theory of approval-based multi-winner elections: we show that there always exists a committee in the core. The core is a stability and group fairness concept. The proof introduces a new voting rule that optimizes an entropy-like objective function over committees and payment systems. All local optima of this objective function lie in the core, which implies that a core committee can be found in polynomial time.

cs.GT↗