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# Tight Lower Bounds for List Edge Coloring

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LIPIcs.SWAT.2018.28.pdf
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## Cite As

Lukasz Kowalik and Arkadiusz Socala. Tight Lower Bounds for List Edge Coloring. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 28:1-28:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SWAT.2018.28

## Abstract

The fastest algorithms for edge coloring run in time 2^m n^{O(1)}, where m and n are the number of edges and vertices of the input graph, respectively. For dense graphs, this bound becomes 2^{Theta(n^2)}. This is a somewhat unique situation, since most of the studied graph problems admit algorithms running in time 2^{O(n log n)}. It is a notorious open problem to either show an algorithm for edge coloring running in time 2^{o(n^2)} or to refute it, assuming the Exponential Time Hypothesis (ETH) or other well established assumptions. We notice that the same question can be asked for list edge coloring, a well-studied generalization of edge coloring where every edge comes with a set (often called a list) of allowed colors. Our main result states that list edge coloring for simple graphs does not admit an algorithm running in time 2^{o(n^2)}, unless ETH fails. Interestingly, the algorithm for edge coloring running in time 2^m n^{O(1)} generalizes to the list version without any asymptotic slow-down. Thus, our lower bound is essentially tight. This also means that in order to design an algorithm running in time 2^{o(n^2)} for edge coloring, one has to exploit its special features compared to the list version.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Problems, reductions and completeness
• Theory of computation → Design and analysis of algorithms
• Mathematics of computing → Graph coloring
##### Keywords
• list edge coloring
• complexity
• ETH lower bound

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## References

1. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. J. Comput. Syst. Sci., 87:119-139, 2017. URL: http://dx.doi.org/10.1016/j.jcss.2017.03.003.
2. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546-563, 2009.
3. O. V. Borodin, A. V. Kostochka, and D. R. Woodall. List edge and list total colourings of multigraphs. J. of Comb. Theory, Ser. B, 71:184-204, 1997.
4. Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socala. Tight lower bounds on graph embedding problems. J. ACM, 64(3):18:1-18:22, 2017. URL: http://dx.doi.org/10.1145/3051094.
5. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
6. Marek Cygan, Marcin Pilipczuk, and Michal Pilipczuk. Known algorithms for edge clique cover are probably optimal. SIAM J. Comput., 45(1):67-83, 2016. URL: http://dx.doi.org/10.1137/130947076.
7. Fedor V. Fomin, Kazuo Iwama, Dieter Kratsch, Petteri Kaski, Mikko Koivisto, Lukasz Kowalik, Yoshio Okamoto, Johan van Rooij, and Ryan Williams. 08431 open problems - moderately exponential time algorithms. In Fedor V. Fomin, Kazuo Iwama, and Dieter Kratsch, editors, Moderately Exponential Time Algorithms, number 08431 in Dagstuhl Seminar Proceedings, Dagstuhl, Germany, 2008. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany. URL: http://drops.dagstuhl.de/opus/volltexte/2008/1798.
8. Fred Galvin. The list chromatic index of a bipartite multigraph. J. Comb. Theory, Ser. B, 63(1):153-158, 1995. URL: http://dx.doi.org/10.1006/jctb.1995.1011.
9. Ian Holyer. The np-completeness of some edge-partition problems. SIAM J. Comput., 10(4):713-717, 1981. URL: http://dx.doi.org/10.1137/0210054.
10. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
11. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
12. Moshe Lewenstein, Seth Pettie, and Virginia Vassilevska Williams. Structure and hardness in P (dagstuhl seminar 16451). Dagstuhl Reports, 6(11):1-34, 2016. URL: http://dx.doi.org/10.4230/DagRep.6.11.1.
13. Dániel Marx. Can you beat treewidth? Theory of Computing, 6(1):85-112, 2010. URL: http://dx.doi.org/10.4086/toc.2010.v006a005.
14. Craig A. Tovey. A simplified NP-complete satisfiability problem. Discrete Appl. Math., 8(1):85-89, 1984.
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