arXiv subjects

Radovan Červený

Publications and source records attributed to Radovan Červený.

3 recordsLinked to original sources

arXiv2023

Generating faster algorithms for d-Path Vertex Cover

Many algorithms which exactly solve hard problems require branching on more or less complex structures in order to do their job. Those who design such algorithms often find themselves doing a meticulous analysis of numerous different cases in order to identify these structures and design suitable branching rules, all done by hand. This process tends to be error prone and often the resulting algorithm may be difficult to implement in practice. In this work, we aim to automate a part of this process and focus on simplicity of the resulting implementation. We showcase our approach on the following problem. For a constant $d$, the $d$-Path Vertex Cover problem ($d$-PVC) is as follows: Given an undirected graph and an integer $k$, find a subset of at most $k$ vertices of the graph, such that their deletion results in a graph not containing a path on $d$ vertices as a subgraph. We develop a fully automated framework to generate parameterized branching algorithms for the problem and obtain algorithms outperforming those previously known for $3 \le d \le 8$. E.g., we show that $5$-PVC can be solved in $O(2.7^k\cdot n^{O(1)})$ time.

cs.DS↗

arXiv2022

On Kernels for d-Path Vertex Cover

In this paper we study the kernelization of the $d$-Path Vertex Cover ($d$-PVC) problem. Given a graph $G$, the problem requires finding whether there exists a set of at most $k$ vertices whose removal from $G$ results in a graph that does not contain a path (not necessarily induced) with $d$ vertices. It is known that $d$-PVC is NP-complete for $d\geq 2$. Since the problem generalizes to $d$-Hitting Set, it is known to admit a kernel with $\mathcal{O}(dk^d)$ edges. We improve on this by giving better kernels. Specifically, we give kernels with $\mathcal{O}(k^2)$ vertices and edges for the cases when $d=4$ and $d=5$. Further, we give a kernel with $\mathcal{O}(k^4d^{2d+9})$ vertices and edges for general $d$.

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arXiv2022

Faster FPT Algorithm for 5-Path Vertex Cover

The problem of $d$-Path Vertex Cover, $d$-PVC lies in determining a subset $F$ of vertices of a given graph $G=(V,E)$ such that $G \setminus F$ does not contain a path on $d$ vertices. The paths we aim to cover need not to be induced. It is known that the $d$-PVC problem is NP-complete for any $d \ge 2$. When parameterized by the size of the solution $k$, 5-PVC has direct trivial algorithm with $\mathcal{O}(5^kn^{\mathcal{O}(1)})$ running time and, since $d$-PVC is a special case of $d$-Hitting Set, an algorithm running in $\mathcal{O}(4.0755^kn^{\mathcal{O}(1)})$ time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in $\mathcal{O}(4^kn^{\mathcal{O}(1)})$ time.

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