Smoothed Performance Guarantees for Local Search

We study popular local search and greedy algorithms for standard machine scheduling problems. The performance guarantee of these algorithms is well understood, but the worst-case lower bounds seem somewhat contrived and it is questionable whether 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 Θ(φ) and Θ(log φ), respectively, where 1/φ is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is Θ(log φ) for routing games on parallel links. Additionally, we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of Θ(log φ).