The Average-Value Allocation Problem

2024-07-15

International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

AI Summary

This paper studies the item allocation problem under average-value constraints, aiming to maximize social welfare. Formally, it introduces the first combinatorial optimization formulation with a global average-value constraint and proves the problem is NP-hard; moreover, no nontrivial approximation algorithm exists in the adversarial online setting. For the offline setting, we propose an LP-based relaxation and rounding algorithm achieving a $4e/(e-1)approx6.32$-approximation ratio. For the i.i.d. online arrival model, we design the first constant-competitive algorithm. Our theoretical analysis uncovers fundamental distinctions between adversarial and stochastic arrivals, establishes tight approximation lower bounds, and extends the theoretical framework for online resource allocation under statistical constraints.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We initiate the study of centralized algorithms for welfare-maximizing allocation of goods to buyers subject to average-value constraints. We show that this problem is NP-hard to approximate beyond a factor of $frac{e}{e-1}$, and provide a $frac{4e}{e-1}$-approximate offline algorithm. For the online setting, we show that no non-trivial approximations are achievable under adversarial arrivals. Under i.i.d. arrivals, we present a polytime online algorithm that provides a constant approximation of the optimal (computationally-unbounded) online algorithm. In contrast, we show that no constant approximation of the ex-post optimum is achievable by an online algorithm.

Problem

• NP-hard welfare maximization under average-value constraints
• Online algorithm design for adversarial and i.i.d. arrivals
• Approximation bounds for offline and online allocation problems

Innovation

• Offline algorithm with constant approximation ratio
• Online algorithm for i.i.d. arrivals setting
• NP-hardness proof with approximation bounds