Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Reduction from Chores Allocation to Job Scheduling

Xin Huang, Erel Segal-Halevi

TL;DR

This work studies fair chore allocation under maximin-share (MMS) guarantees for $n$ agents with heterogeneous costs, and establishes a formal reduction from HFFD to the scheduling framework via MultiFit, yielding a tight MMS-approximation link. The results include a polynomial-time $(\frac{13}{11}+\epsilon)$-MMS algorithm and a $15/13$-MMS algorithm for $n=3$, aligning chore allocation with classic makespan bounds. It also provides a FPTAS for nearly optimal MMS allocations when costs are integral and discusses monotonicity properties for small $n$, highlighting practical MMS guarantees. By connecting chore allocation and scheduling through a two-stage reduction (FFV then FFD), the paper tightens the theoretical bridge between these domains and offers practical, scalable methods for MMS-based chore 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.

A Reduction from Chores Allocation to Job Scheduling

TL;DR

This work studies fair chore allocation under maximin-share (MMS) guarantees for agents with heterogeneous costs, and establishes a formal reduction from HFFD to the scheduling framework via MultiFit, yielding a tight MMS-approximation link. The results include a polynomial-time -MMS algorithm and a -MMS algorithm for , aligning chore allocation with classic makespan bounds. It also provides a FPTAS for nearly optimal MMS allocations when costs are integral and discusses monotonicity properties for small , highlighting practical MMS guarantees. By connecting chore allocation and scheduling through a two-stage reduction (FFV then FFD), the paper tightens the theoretical bridge between these domains and offers practical, scalable methods for MMS-based chore 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.
Paper Structure (34 sections, 27 theorems, 8 equations, 3 figures, 1 table, 5 algorithms)

This paper contains 34 sections, 27 theorems, 8 equations, 3 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related Work
  4. Overview of the Technique
  5. Preliminaries
  6. Agents and Chores
  7. First Fit Decreasing and MultiFit
  8. From identical to different cost functions
  9. Proof Overview
  10. First Fit Abstraction
  11. Reduction from First Fit Valid to FFD
  12. Tidy Up Tuple
  13. 1. Removing unallocated chores.
  14. 2. Removing chores smaller than an unallocated chore.
  15. 3. Removing redundant chores.
  16. ...and 19 more sections

Key Result

Theorem 1

For any $n\geq 2$, HFFD among $n$ agents guarantees to each agent at most $\alpha^{(n)}$ of their MMS.

Figures (3)

  • Figure 1: A bird's eye view of the whole proof.
  • Figure 2: The sequence of reallocation after replacing the excessive chore $c_k^*$ with a suitable reduced chore $c^{\circ}$, in the proof of Claim \ref{['claim:fs(k)>v(c*)']} (for a medium fit-in space). Here we have $t=2$.
  • Figure 3: The shift of fallback chores after replacing the excessive chore, in the proof of Claim \ref{['claim:shift']} (for a large fit-in space).

Theorems & Definitions (85)

  • Theorem 1: Informal
  • Theorem 2: Informal
  • Theorem 3: Informal
  • Definition 1: Maximin share
  • Definition 2: Maximin share allocation
  • Definition 3: Identical-order preference (IDO) DBLP:conf/sigecom/HuangL21
  • Lemma 1: DBLP:conf/sigecom/HuangL21
  • Definition 4
  • Example 1
  • Definition 5: Lexicographic order
  • ...and 75 more