[Paper Review] Tight Lower Bounds on Envy-Free Makespan Approximation

Amos Fiat, Ariel Levavi|arXiv (Cornell University)|May 8, 2012

Optimization and Search Problems7 references4 citations

TL;DR

This paper establishes a tight lower bound of Ω(log m) on the approximation ratio for envy-free makespan minimization in scheduling unrelated machines with indivisible jobs. By constructing a cost matrix where any envy-free allocation must have a makespan of at least log m times the optimal, the authors prove that no mechanism can guarantee a better approximation ratio than O(log m), closing the gap between known upper and lower bounds.

ABSTRACT

In this work we give a tight lower bound on makespan approximations for envy-free allocation mechanism dedicated to scheduling tasks on unrelated machines. Specifically, we show that no mechanism exists that can guarantee an envy-free allocation of jobs to $m$ machines with a makespan of less than a factor of $O(\log m)$ of the minimal makespan. Combined with previous results, this paper definitively proves that the optimal algorithm for obtaining a minimal makespan for any envy-free division can at best approximate the makespan to a factor of $O(\log m)$.

Motivation & Objective

• To close the gap between known upper and lower bounds on envy-free makespan approximation in unrelated machine scheduling.
• To prove that no mechanism can achieve an envy-free allocation with a makespan better than O(log m) times the optimal makespan.
• To establish that the O(log m) upper bound from prior work is asymptotically tight.
• To demonstrate that even locally efficient envy-free allocations must incur a makespan of at least Ω(log m) for certain cost matrices.
• To formalize a construction of a cost matrix where optimal makespan is 1, but any envy-free allocation requires at least log m makespan.

Proposed method

• Construct a cost matrix with n jobs and m = n + l machines, where l = log n, such that c_{i,j} = 1 for i ≤ n and j = i, and c_{i,j} = 2^{i−n} for n < i ≤ n+l and all j.
• Define a makespan lower bound via contradiction: assume a locally efficient assignment with makespan < 2^l = log n, then show a permutation of job bundles to higher-indexed machines reduces total cost.
• Use harmonic number approximations to bound the total cost reduction from moving bundles from machines 1 to n−1 to machines 2 to n.
• Leverage bounds on job counts per machine: fewer than 2^{l+1} jobs per machine i ≤ n, and fewer than 2^l total jobs on machines n+1 to n+l.
• Show that cost reduction from moving jobs from early machines (low cost) to later machines (higher cost) is outweighed by the cost increase, leading to contradiction unless makespan ≥ log n.
• Conclude that every envy-free assignment must have makespan Ω(log m), matching the known O(log m) upper bound.

Experimental results

Research questions

• RQ1 Can any mechanism guarantee an envy-free allocation with a makespan better than O(log m) times the optimal makespan for unrelated machine scheduling with indivisible jobs?
• RQ2 Is the O(log m) upper bound on envy-free makespan approximation tight, or can a better approximation ratio be achieved?
• RQ3 What structural properties of cost matrices force any envy-free allocation to incur a makespan of at least Ω(log m)?
• RQ4 Does the existence of locally efficient envy-free allocations imply a lower bound on the makespan in certain hard instances?
• RQ5 Can a cost matrix be constructed such that the optimal makespan is 1, but every envy-free allocation has a makespan of at least Ω(log m)?

Key findings

• The paper establishes a tight lower bound of Ω(log m) on the approximation ratio for envy-free makespan minimization in unrelated machine scheduling with indivisible jobs.
• No mechanism can guarantee an envy-free allocation with a makespan less than O(log m) times the optimal makespan, proving the O(log m) upper bound is asymptotically tight.
• For a constructed cost matrix with optimal makespan 1, any envy-free allocation must have a makespan of at least log n = log m, establishing the lower bound.
• The lower bound holds even for locally efficient envy-free assignments, meaning the result is robust to refinement of the allocation process.
• The proof uses a cost matrix where machine costs increase exponentially beyond n machines, and shows that moving bundles to higher-cost machines cannot be avoided without violating envy-freeness.
• The result definitively closes the gap between the known O(log m) upper bound and the best possible lower bound, proving that O(log m) is the optimal approximation ratio for envy-free makespan.