Three New Complexity Results for Resource Allocation Problems

TL;DR

This paper presents three new computational complexity results for resource allocation problems, clarifying the tractability of finding optimal, Pareto-efficient, and envy-free allocations under various utility models.

Contribution

It introduces novel complexity classifications for key resource allocation problems, including polynomial-time solvability and hardness results for different utility functions.

Findings

Leximin-maximal allocation is in P for max-utility agents with atomic demands.

Deciding Pareto-optimality is coNP-complete for agents with additive utilities.

Existence of Pareto-optimal and envy-free allocations is Sigma_2^p-complete.

Abstract

We prove the following results for task allocation of indivisible resources: - The problem of finding a leximin-maximal resource allocation is in P if the agents have max-utility functions and atomic demands. - Deciding whether a resource allocation is Pareto-optimal is coNP-complete for agents with (1-)additive utility functions. - Deciding whether there exists a Pareto-optimal and envy-free resource allocation is Sigma_2^p-complete for agents with (1-)additive utility functions.