Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations

by   Siddharth Barman, et al.

We develop polynomial-time algorithms for the fair and efficient allocation of indivisible goods among n agents that have subadditive valuations over the goods. We first consider the Nash social welfare as our objective and design a polynomial-time algorithm that, in the value oracle model, finds an 8n-approximation to the Nash optimal allocation. Subadditive valuations include XOS (fractionally subadditive) and submodular valuations as special cases. Our result, even for the special case of submodular valuations, improves upon the previously best known O(n log n)-approximation ratio of Garg et al. (2020). More generally, we study maximization of p-mean welfare. The p-mean welfare is parameterized by an exponent term p ∈ (-∞, 1] and encompasses a range of welfare functions, such as social welfare (p = 1), Nash social welfare (p → 0), and egalitarian welfare (p → -∞). We give an algorithm that, for subadditive valuations and any given p ∈ (-∞, 1], computes (in the value oracle model and in polynomial time) an allocation with p-mean welfare at least 1/8n times the optimal. Further, we show that our approximation guarantees are essentially tight for XOS and, hence, subadditive valuations. We adapt a result of Dobzinski et al. (2010) to show that, under XOS valuations, an O (n^1-ε) approximation for the p-mean welfare for any p ∈ (-∞,1] (including the Nash social welfare) requires exponentially many value queries; here, ε>0 is any fixed constant.


page 1

page 2

page 3

page 4


Nash Social Welfare for 2-value Instances

We study the problem of allocating a set of indivisible goods among agen...

Fair and Efficient Allocations under Subadditive Valuations

We study the problem of allocating a set of indivisible goods among agen...

Maximizing Social Welfare Subject to Network Externalities: A Unifying Submodular Optimization Approach

We consider the problem of allocating multiple indivisible items to a se...

Computing Stable Coalitions: Approximation Algorithms for Reward Sharing

Consider a setting where selfish agents are to be assigned to coalitions...

Auctions with Interdependence and SOS: Improved Approximation

Interdependent values make basic auction design tasks – in particular ma...

Social Welfare and Profit Maximization from Revealed Preferences

Consider the seller's problem of finding "optimal" prices for her (divis...

Tight Approximation Guarantees for Concave Coverage Problems

In the maximum coverage problem, we are given subsets T_1, …, T_m of a u...

Please sign up or login with your details

Forgot password? Click here to reset