Smoothed Performance Guarantees for Local Search

TL;DR

This paper analyzes the robustness of local search and greedy algorithms for scheduling under random noise, revealing that some performance guarantees are much more resilient in practice than worst-case bounds suggest.

Contribution

It introduces a smoothed analysis framework for scheduling algorithms, showing that certain performance bounds are independent of the number of machines under noise.

Findings

Smoothed performance guarantees for jump and lex-jump algorithms are Theta(phi) and Theta(log(phi)).

Bounds are robust for restricted machines but fragile for unrestricted machines.

The results extend to routing games, showing similar bounds for the price of anarchy.

Abstract

We study popular local search and greedy algorithms for scheduling. The performance guarantee of these algorithms is well understood, but the worst-case lower bounds seem somewhat contrived and it is questionable if they arise in practical applications. To find out how robust these bounds are, we study the algorithms in the framework of smoothed analysis, in which instances are subject to some degree of random noise. While the lower bounds for all scheduling variants with restricted machines are rather robust, we find out that the bounds are fragile for unrestricted machines. In particular, we show that the smoothed performance guarantee of the jump and the lex-jump algorithm are (in contrast to the worst case) independent of the number of machines. They are Theta(phi) and Theta(log(phi)), respectively, where 1/phi is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is Theta(log(phi)) for routing games on parallel links. Additionally we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of Theta(log(phi)).