A Reduction from Chores Allocation to Job Scheduling

TL;DR

This paper links chores allocation to job scheduling, showing that existing algorithms can be improved and providing new practical algorithms that achieve near-optimal fairness guarantees.

Contribution

It reduces the chores allocation problem to a well-studied scheduling problem, establishing new approximation bounds and developing practical algorithms for fair division.

Findings

HFFD's approximation ratio equals that of MultiFit, known to be 13/11.

New polynomial-time algorithms achieve near-optimal maximin-share fairness.

Improved algorithms for n=3 agents find 15/13-maximin-share allocations.

Abstract

We consider allocating indivisible chores among agents with different cost functions, such that all agents receive a cost of at most a constant factor times their maximin share. The state-of-the-art was presented in In EC 2021 by Huang and Lu. They presented a non-polynomial-time algorithm, called HFFD, that attains an 11/9 approximation, and a polynomial-time algorithm that attains a 5/4 approximation. In this paper, we show that HFFD can be reduced to an algorithm called MultiFit, developed by Coffman, Garey and Johnson in 1978 for makespan minimization in job scheduling. Using this reduction, we prove that the approximation ratio of HFFD is in fact equal to that of MultiFit, which is known to be 13/11 in general, 20/17 for n at most 7, and 15/13 for n=3. Moreover, we develop an algorithm for (13/11+epsilon)-maximin-share allocation for any epsilon>0, with run-time polynomial in the problem size and 1/epsilon. For n=3, we can improve the algorithm to find a 15/13-maximin-share allocation with run-time polynomial in the problem size. Thus, we have practical algorithms that attain the best known approximation to maximin-share chore allocation.