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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Vydáno v:Algorithmica Ročník 85; číslo 1; s. 1 - 28
Hlavní autoři: Aboulker, Pierre, Bonnet, Édouard, Kim, Eun Jung, Sikora, Florian
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.01.2023
Springer Nature B.V
Springer Verlag
Témata:
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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Shrnutí: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.
Bibliografie:ObjectType-Article-1
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-022-01001-2