Faster deterministic parameterized algorithm for k-Path

TL;DR

This paper presents a faster deterministic algorithm for the k-Path problem in directed graphs, reducing the exponential time complexity and improving upon previous methods, with potential applications to related problems.

Contribution

Introduces a new deterministic algorithm for k-Path with improved exponential time complexity, applicable to multiple related graph problems.

Findings

Deterministic algorithm for k-Path with time $O^*(2.554^k)$

Improves previous deterministic time complexity of $O^*(2.597^k)$

Technique applicable to other problems like k-Tree and Graph Motif

Abstract

In the k-Path problem, the input is a directed graph $G$ and an integer $k\geq 1$, and the goal is to decide whether there is a simple directed path in $G$ with exactly $k$ vertices. We give a deterministic algorithm for k-Path with time complexity $O^*(2.554^k)$. This improves the previously best deterministic algorithm for this problem of Zehavi [ESA 2015] whose time complexity is $O^*(2.597^k)$. The technique used by our algorithm can also be used to obtain faster deterministic algorithms for k-Tree, r-Dimensional k-Matching, Graph Motif, and Partial Cover.