Exponential-time approximation of weighted set cover

• Published in: Information processing letters Vol. 109; no. 16; pp. 957 - 961
• Main Authors: Cygan, Marek, Kowalik, Łukasz, Wykurz, Mateusz
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 31.07.2009; Elsevier; Elsevier Sequoia S.A
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Approximation algorithms; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Designs and configurations; Exact sciences and technology; Exponential-time algorithms; Information processing; Instance reduction; Mathematics; Miscellaneous; Sciences and techniques of general use; Set-cover; Studies; Theoretical computing; Trade-off; Exponential-time algorithms; Approximation algorithms; Instance reduction; Set-cover; Trade-off; Polynomial; Approximation; Computer theory; Research; Approximation algorithm; Polynomial time; Trade; NP hard problem; Information processing; Time complexity; Algorithm analysis
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2009.05.003

The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponential-time algorithms for problems of that kind. The goal is getting the time complexity still of order O ( c n ) , but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation. In fact, we describe a kind of approximation schemes—trade-offs between approximation factor and the time complexity. We use general transformations from exponential-time exact algorithms to approximations that are faster but still exponential-time. For example, we show that for any reduction rate r, one can transform any O ∗ ( c n ) -time 1 algorithm for Set Cover into a ( 1 + ln r ) -approximation algorithm running in time O ∗ ( c n / r ) . We believe that results of that kind extend the applicability of exact algorithms for NP-hard problems.