2026-07-24T17:55:54Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:5575 2024-03-06T10:36:35Z ddc:004 open_access

Polynomial Fixed-parameter Algorithms: A Case Study for Longest Path on Interval Graphs Giannopoulou, Archontia C. Mertzios, George B. Niedermeier, Rolf fixed-parameter algorithm preprocessing data reduction polynomial-time algorithm longest path problem interval graphs proper interval vertex del 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Archontia C. Giannopoulou and George B. Mertzios and Rolf Niedermeier 2015 Is Part Of LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2015.102 urn:nbn:de:0030-drops-55750 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.102 eng https://creativecommons.org/licenses/by/3.0/legalcode