Results

**1 - 2**of**2**### Polynomial Fixed-Parameter Algorithms: A Case Study for Longest Path on Interval Graphs

, 2015

"... 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 un-attractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path; it is NP-hard ..."

Abstract
- Add to MetaCart

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 un-attractive 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(n4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by ver-tex 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 the idea of 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.