Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

Giannopoulou, Archontia C.; Mertzios, George B.; Niedermeier, Rolf

Abstract

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path, that is, given an undirected graph, find a maximum-length path in G. Longest Path is NP-hard in general but known to be solvable in O(n4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial-time solvable problems similarly to what classical parameterized complexity analysis does for NP-hard problems.

Citation

Giannopoulou, A. C., Mertzios, G. B., & Niedermeier, R. (2017). Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs. Theoretical Computer Science, 689, 67-95. https://doi.org/10.1016/j.tcs.2017.05.017

• Journal Article Type: Article
• Acceptance Date: May 21, 2017
• Online Publication Date: May 25, 2017
• Publication Date: May 25, 2017
• Deposit Date: Jun 2, 2017
• Publicly Available Date: May 25, 2018
• Journal: Theoretical Computer Science
• Print ISSN: 0304-3975
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 689
• Pages: 67-95
• DOI: https://doi.org/10.1016/j.tcs.2017.05.017
• Public URL: https://durham-repository.worktribe.com/output/1356106