Polynomial Fixed-Parameter Algorithms: A Case Study for Longest Path on Interval Graphs

TL;DR

This paper develops a fixed-parameter algorithm for the Longest Path problem on interval graphs, improving efficiency by leveraging parameterization related to proper interval graphs and demonstrating practical approximation methods.

Contribution

It introduces a fixed-parameter algorithm for Longest Path on interval graphs based on vertex deletion to proper interval graphs, enhancing computational efficiency.

Findings

Fixed-parameter algorithm runs in O(k^9 n) time.

Longest Path solvable in linear time on proper interval graphs.

Vertex deletion parameter k can be approximated within factor 4.

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 $O(n^{4})$ 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 $O(k^{9}n)$ 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.