Parameterized Complexity of Graph Burning

Graph Burning asks, given a graph G = ( V , E ) and an integer k , whether there exists ( b 0 , ⋯ , b k - 1 ) ∈ 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 parameterize...

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Published in:	Algorithmica Vol. 84; no. 8; pp. 2379 - 2393
Main Authors:	Kobayashi, Yasuaki, Otachi, Yota
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2022 Springer Nature B.V
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Parameterization Parameters Polynomials Special Issue: Parameterized and Exact Computation (IPEC 2020) Theory of Computation Fixed-parameter tractability Parameterized complexity Graph burning
ISSN:	0178-4617, 1432-0541
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Abstract	Graph Burning asks, given a graph G = ( V , E ) and an integer k , whether there exists ( b 0 , ⋯ , b k - 1 ) ∈ 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 by Kare and Reddy [IWOCA 2019] about the parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by k and that it does not admit a polynomial kernel parameterized by vertex cover number unless NP ⊆ 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. Using a different technique, we show that parameterization by distance to split graphs is also tractable. We finally show that the problem parameterized by max leaf number is XP.
AbstractList	Graph Burning asks, given a graph $$G = (V,E)$$ G = ( V , E ) and an integer k , whether there exists $$(b_{0},\dots ,b_{k-1}) \in V^{k}$$ ( b 0 , ⋯ , b k - 1 ) ∈ V k such that every vertex in G has distance at most i from some $$b_{i}$$ 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 by Kare and Reddy [IWOCA 2019] about the parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by k and that it does not admit a polynomial kernel parameterized by vertex cover number unless $$\mathrm {NP} \subseteq \mathrm {coNP/poly}$$ NP ⊆ 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. Using a different technique, we show that parameterization by distance to split graphs is also tractable. We finally show that the problem parameterized by max leaf number is XP. Graph Burning asks, given a graph G=(V,E) and an integer k, whether there exists (b0,⋯,bk-1)∈Vk such that every vertex in G has distance at most i from some bi. 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 by Kare and Reddy [IWOCA 2019] about the parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by k and that it does not admit a polynomial kernel parameterized by vertex cover number unless NP⊆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. Using a different technique, we show that parameterization by distance to split graphs is also tractable. We finally show that the problem parameterized by max leaf number is XP. Graph Burning asks, given a graph G = ( V , E ) and an integer k , whether there exists ( b 0 , ⋯ , b k - 1 ) ∈ 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 by Kare and Reddy [IWOCA 2019] about the parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by k and that it does not admit a polynomial kernel parameterized by vertex cover number unless NP ⊆ 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. Using a different technique, we show that parameterization by distance to split graphs is also tractable. We finally show that the problem parameterized by max leaf number is XP.
Author	Kobayashi, Yasuaki Otachi, Yota
Author_xml	– sequence: 1 givenname: Yasuaki surname: Kobayashi fullname: Kobayashi, Yasuaki organization: Hokkaido University – sequence: 2 givenname: Yota orcidid: 0000-0002-0087-853X surname: Otachi fullname: Otachi, Yota email: otachi@nagoya-u.jp organization: Nagoya University
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Snippet	Graph Burning asks, given a graph G = ( V , E ) and an integer k , whether there exists ( b 0 , ⋯ , b k - 1 ) ∈ V k such that every vertex in G has distance at... Graph Burning asks, given a graph $$G = (V,E)$$ G = ( V , E ) and an integer k , whether there exists $$(b_{0},\dots ,b_{k-1}) \in V^{k}$$ ( b 0 , ⋯ , b k - 1... Graph Burning asks, given a graph G=(V,E) and an integer k, whether there exists (b0,⋯,bk-1)∈Vk such that every vertex in G has distance at most i from some...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Parameterization Parameters Polynomials Special Issue: Parameterized and Exact Computation (IPEC 2020) Theory of Computation
Title	Parameterized Complexity of Graph Burning
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