Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q - Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. I...

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Vydané v:Algorithmica Ročník 81; číslo 10; s. 3865 - 3889
Hlavní autori: Jansen, Bart M. P., Pieterse, Astrid
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.10.2019
Springer Nature B.V
Predmet:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Computer Science
Computer Systems Organization and Communication Networks
Data reduction
Data Structures and Information Theory
Graph coloring
Homomorphisms
Kernels
Lower bounds
Mathematics of Computing
Parameterization
Parameters
Polynomials
Special Issue: Parameterized and Exact Computation
Theory of Computation
Graph homomorphism
Graph coloring
05C85
Kernelization
05C15
ISSN:0178-4617, 1432-0541
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Shrnutí:The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q - Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q - Coloring . First, we obtain a kernel of bitsize O ( k q - 1 log k ) for q - Coloring parameterized by Vertex Cover for any q ≥ 3 . This size bound is optimal up to  k o ( 1 ) factors assuming  NP ⊈ coNP / poly , and improves on the previous-best kernel of size  O ( k q ) . We generalize this result for deciding q -colorability of a graph G , to deciding the existence of a homomorphism from G to an arbitrary fixed graph H . Furthermore, we can replace the parameter vertex cover by the less restrictive parameter twin-cover. We prove that H - Coloring parameterized by Twin-Cover has a kernel of size O ( k Δ ( H ) log k ) . Our second result shows that 3- Coloring does not admit non-trivial sparsification: assuming NP ⊈ coNP / poly , the parameterization by the number of vertices  n admits no (generalized) kernel of size  O ( n 2 - ε ) for any ε > 0 . Previously, such a lower bound was only known for coloring with q ≥ 4 colors.
Bibliografia:ObjectType-Article-1
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content type line 14
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00578-5