Parameterized Complexity of Graph Burning
Year: 2020
Type: preprint
Abstract: Graph Burning asks, given a graph $G = (V,E)$ and an integer $k$, whether there exists $(b_{0},\dots,b_{k-1}) \in V^{k}$ such that every vertex in $G$ has distance at most $i$ from some $b_{i}$. This problem is known to be NP-complete even on connected caterpillars of maximum degree $3$. We study the parameterized complexity of this problem and answer all questions arose by Kare and Reddy [IWOCA 2019] about parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by $k$ and that it does no admit a polynomial kernel parameterized by vertex cover number unless $\mathrm{NP} \subseteq \mathrm{coNP/poly}$. We also show that the problem is fixed-parameter tractable parameterized by clique-width plus the maximum diameter among all connected components. This implies the fixed-parameter tractability parameterized by modular-width, by treedepth, and by distance to cographs. Although the parameterization by distance to split graphs cannot be handled with the clique-width argument, we show that this is also tractable by a reduction to a generalized problem with a smaller solution size.
Source: arXiv (Cornell University)
Authors: Kobayashi, Yasuaki, Otachi, Yota
Institutions: Kyoto University, Nagoya University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Speeding up FPT algorithms with special tree decompositions, Research on algorithms and data structures for solving theoretically hard problems in practical time, Research on fast algorithmic meta-theorems for tree decomposable graphs, General algorithms for fixed-parameter intractable problems
OpenAlex ID: W4297780261