New bounds for truthful scheduling on two unrelated selfish machines

TL;DR

This paper establishes new bounds for the best possible approximation ratio of truthful scheduling algorithms on two unrelated selfish machines, improving understanding of their efficiency and near-optimality.

Contribution

It introduces a novel Min-Max formulation for the approximation ratio and derives tighter bounds using distribution approximations, especially for small task numbers.

Findings

Almost tight bounds for n=2 with ratio approximately 1.506

Improved bounds on R_n for small n

New analytical methods for bounding approximation ratios

Abstract

We consider the minimum makespan problem for $n$ tasks and two unrelated parallel selfish machines. Let $R_n$ be the best approximation ratio of randomized monotone scale-free algorithms. This class contains the most efficient algorithms known for truthful scheduling on two machines. We propose a new $Min-Max$ formulation for $R_n$, as well as upper and lower bounds on $R_n$ based on this formulation. For the lower bound, we exploit pointwise approximations of cumulative distribution functions (CDFs). For the upper bound, we construct randomized algorithms using distributions with piecewise rational CDFs. Our method improves upon the existing bounds on $R_n$ for small $n$. In particular, we obtain almost tight bounds for $n=2$ showing that $|R_2-1.505996|<10^{-6}$.