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

Giannopoulou, A.C.; Mertzios, G.B.; Niedermeier, R.

doi:10.4230/lipics.ipec.2015.102

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

Giannopoulou, A.C.; Mertzios, G.B.; Niedermeier, R.

Authors

A.C. Giannopoulou

Dr George Mertzios george.mertzios@durham.ac.uk
Associate Professor

R. Niedermeier

Contributors

Thore Husfeldt
Editor

Iyad Kanj
Editor

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; it is NP-hard in general but known to be solvable in O(n^4) 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(k^9n) 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 for polynomial-time solvable problems offers a very fertile ground for future studies for all sorts of algorithmic problems. It 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., Mertzios, G., & Niedermeier, R. (2015). Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs. In T. Husfeldt, & I. Kanj (Eds.), 10th International Symposium on Parameterized and Exact Computation (IPEC 2015) (102-113). https://doi.org/10.4230/lipics.ipec.2015.102

Conference Name	10th International Symposium on Parameterized and Exact Computation (IPEC)
Conference Location	Patras, Greece
Start Date	Sep 16, 2015
End Date	Sep 18, 2015
Acceptance Date	Sep 30, 2015
Publication Date	Nov 9, 2015
Deposit Date	Sep 30, 2015
Publicly Available Date	Oct 12, 2015
Pages	102-113
Series Title	LIPIcs : Leibniz international proceedings in informatics
Series Number	43
Series ISSN	1868-8969
Book Title	10th International Symposium on Parameterized and Exact Computation (IPEC 2015).
DOI	https://doi.org/10.4230/lipics.ipec.2015.102
Keywords	Fixed-parameter algorithm, Preprocessing, Data reduction, Polynomialtime algorithm, Longest path problem, Interval graphs, Proper interval vertex deletion set.
Public URL	https://durham-repository.worktribe.com/output/1152540