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.
Highlights
Parameterized complexity analysis [16, 18, 30] is a flourishing field dealing with the exact solvability of NP-hard problems
For illustrating the potential algorithmic challenges posed by the “fixed-parameter tractability (FPT) inside P” framework, we focus on Longest Path on Interval Graphs, which is known to be solvable in O(n4) time [23], and we derive an PL-FPT-algorithm that satisfies all three desirable algorithmic properties described above
We provide a polynomial fixed-parameter algorithm that runs in O(k9n) time, proving that Longest Path on Interval Graphs is in the class PL-FPT when parameterized by the size of a minimum proper interval deletion set
Summary
Parameterized complexity analysis [16, 18, 30] is a flourishing field dealing with the exact solvability of NP-hard problems. The key idea is to lift classical complexity analysis, rooted in the P versus NP phenomenon, from a one-dimensional to a two- (or even multi-)dimensional perspective, the key concept being “fixed-parameter tractability (FPT)”. Why should this natural and successful approach be limited to intractable (i.e., NP-hard) problems? We refine the class P by introducing, for every polynomial p(n), the class P-FPT (p(n)) (Polynomial Fixed-Parameter Tractable), containing the problems solvable in O(kt · p(n)) time for some constant t ≥ 1, i.e., the dependency of the complexity on the parameter k is at most polynomial. As this research direction is still only little explored, we suggest and follow a focus first on problems for which the best known upper bounds of the time complexity are polynomials of high degree, e.g., O(n4) or higher
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