Smoothed performance guarantees for local search

• Published in: Mathematical programming Vol. 146; no. 1-2; pp. 185 - 218
• Main Authors: Brunsch, Tobias, Röglin, Heiko, Rutten, Cyriel, Vredeveld, Tjark
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014; Springer Nature B.V
• Subjects: Algorithms; Analysis; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Data smoothing; Full Length Paper; Games; Guarantees; Job shops; Lower bounds; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Noise; Numerical Analysis; Schedules; Scheduling; Searching; Studies; Texts; Theoretical; 68Q25; 68W40; 68W25; 68Q87; 90B35
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-013-0683-7

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  and  , respectively, where  is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is  for routing games on parallel links. Additionally, we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of   .