Grundy Coloring and Friends, Half-Graphs, Bicliques

The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ , the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ , i.e., the maximum number of colors that the first-fit coloring r...

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Vydané v:	Algorithmica Ročník 85; číslo 1; s. 1 - 28
Hlavní autori:	Aboulker, Pierre, Bonnet, Édouard, Kim, Eun Jung, Sikora, Florian
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.01.2023 Springer Nature B.V Springer Verlag
Predmet:	Algorithm Analysis and Problem Complexity Algorithms Color Coloring Combinatorial analysis Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graphs Mathematics of Computing Parameterization Questions Theory of Computation ETH lower bounds free graphs Grundy coloring Half-graphs Theory of computation → Fixed parameter tractability Theory of computation → Graph algorithms analysis Parameterized complexity
ISSN:	0178-4617, 1432-0541
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Abstract	The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ , the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ , i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f ( k ) n 2 k - 1 -time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b - Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS ’17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K t , t -free graphs for b - Chromatic Core and Partial Grundy Coloring , making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.
AbstractList	The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f(k)n2k-1-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS ’17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on Kt,t-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest. The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ , the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ , i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f ( k ) n 2 k - 1 -time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b - Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS ’17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K t , t -free graphs for b - Chromatic Core and Partial Grundy Coloring , making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.
Author	Sikora, Florian Kim, Eun Jung Aboulker, Pierre Bonnet, Édouard
Author_xml	– sequence: 1 givenname: Pierre surname: Aboulker fullname: Aboulker, Pierre organization: DI/ENS, PSL University – sequence: 2 givenname: Édouard orcidid: 0000-0002-1653-5822 surname: Bonnet fullname: Bonnet, Édouard email: edouard.bonnet@ens-lyon.fr organization: CNRS, ENS de Lyon, LIP UMR5668, Univ Lyon, Université Claude Bernard Lyon 1 – sequence: 3 givenname: Eun Jung orcidid: 0000-0002-6824-0516 surname: Kim fullname: Kim, Eun Jung organization: CNRS UMR7243, LAMSADE, Université Paris-Dauphine, PSL University – sequence: 4 givenname: Florian orcidid: 0000-0003-2670-6258 surname: Sikora fullname: Sikora, Florian organization: CNRS UMR7243, LAMSADE, Université Paris-Dauphine, PSL University
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Keywords	ETH lower bounds free graphs Grundy coloring Half-graphs Theory of computation → Fixed parameter tractability Theory of computation → Graph algorithms analysis Parameterized complexity
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Snippet	The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ , the smallest available color. The problem Grundy Coloring... The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ, the smallest available color. The problem Grundy Coloring...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Color Coloring Combinatorial analysis Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graphs Mathematics of Computing Parameterization Questions Theory of Computation
Title	Grundy Coloring and Friends, Half-Graphs, Bicliques
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