Finer Tight Bounds for Coloring on Clique-Width
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| Title: | Finer Tight Bounds for Coloring on Clique-Width |
|---|---|
| Authors: | Michael Lampis |
| Contributors: | Michael Lampis |
| Source: | SIAM Journal on Discrete Mathematics. 34:1538-1558 |
| Publication Status: | Preprint |
| Publisher Information: | Society for Industrial & Applied Mathematics (SIAM), 2020. |
| Publication Year: | 2020 |
| Subject Terms: | FOS: Computer and information sciences, Programmation, Clique-width, 0211 other engineering and technologies, 0102 computer and information sciences, 02 engineering and technology, Computational Complexity (cs.CC), 01 natural sciences, organisation des données, Computer Science - Computational Complexity, Computer Science - Data Structures and Algorithms, SETH, Coloring, Data Structures and Algorithms (cs.DS), ddc:004, logiciels |
| Description: | We revisit the complexity of the classical $k$-Coloring problem parameterized by clique-width. This is a very well-studied problem that becomes highly intractable when the number of colors $k$ is large. However, much less is known on its complexity for small, concrete values of $k$. In this paper, we completely determine the complexity of $k$-Coloring parameterized by clique-width for any fixed $k$, under the SETH. Specifically, we show that for all $k\ge 3,��>0$, $k$-Coloring cannot be solved in time $O^*((2^k-2-��)^{cw})$, and give an algorithm running in time $O^*((2^k-2)^{cw})$. Thus, if the SETH is true, $2^k-2$ is the "correct" base of the exponent for every $k$. Along the way, we also consider the complexity of $k$-Coloring parameterized by the related parameter modular treewidth ($mtw$). In this case we show that the "correct" running time, under the SETH, is $O^*({k\choose \lfloor k/2\rfloor}^{mtw})$. If we base our results on a weaker assumption (the ETH), they imply that $k$-Coloring cannot be solved in time $n^{o(cw)}$, even on instances with $O(\log n)$ colors. To appear in ICALP 2018 |
| Document Type: | Article Conference object |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 1095-7146 0895-4801 |
| DOI: | 10.1137/19m1280326 |
| DOI: | 10.48550/arxiv.1804.07975 |
| DOI: | 10.4230/lipics.icalp.2018.86 |
| Access URL: | https://drops.dagstuhl.de/opus/volltexte/2018/9090/pdf/LIPIcs-ICALP-2018-86.pdf http://arxiv.org/abs/1804.07975 https://dblp.uni-trier.de/db/journals/siamdm/siamdm34.html#Lampis20 https://epubs.siam.org/doi/pdf/10.1137/19M1280326 https://doi.org/10.1137/19M1280326 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.86 |
| Rights: | CC BY arXiv Non-Exclusive Distribution |
| Accession Number: | edsair.doi.dedup.....9810db4144baebfed6c46fe1a03e0dce |
| Database: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstract: | We revisit the complexity of the classical $k$-Coloring problem parameterized by clique-width. This is a very well-studied problem that becomes highly intractable when the number of colors $k$ is large. However, much less is known on its complexity for small, concrete values of $k$. In this paper, we completely determine the complexity of $k$-Coloring parameterized by clique-width for any fixed $k$, under the SETH. Specifically, we show that for all $k\ge 3,��>0$, $k$-Coloring cannot be solved in time $O^*((2^k-2-��)^{cw})$, and give an algorithm running in time $O^*((2^k-2)^{cw})$. Thus, if the SETH is true, $2^k-2$ is the "correct" base of the exponent for every $k$. Along the way, we also consider the complexity of $k$-Coloring parameterized by the related parameter modular treewidth ($mtw$). In this case we show that the "correct" running time, under the SETH, is $O^*({k\choose \lfloor k/2\rfloor}^{mtw})$. If we base our results on a weaker assumption (the ETH), they imply that $k$-Coloring cannot be solved in time $n^{o(cw)}$, even on instances with $O(\log n)$ colors.<br />To appear in ICALP 2018 |
|---|---|
| ISSN: | 10957146 08954801 |
| DOI: | 10.1137/19m1280326 |