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# Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget

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LIPIcs.ESA.2022.25.pdf
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## Acknowledgements

The authors want to thank an anonymous reviewer for very useful comments.

## Cite As

Bartłomiej Bosek and Anna Zych-Pawlewicz. Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.25

## Abstract

In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals are allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains k-colorable at all times, the updates consist of insertions only, and the final instance consists of n intervals, then we can achieve an amortized recourse budget of 𝒪({k⁷ log n}) while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [Bartłomiej Bosek et al., 2020] in terms of both k and n. We complement this result by showing the lower bound of Ω(n) on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented. As an additional application of our techniques we include a new combinatorial result on coloring unit circular arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs 𝒜. We show that if there is a set 𝒜' of non-intersecting unit arcs of size L²-1 such that 𝒜 ∪ 𝒜' does not contain L+1 arcs intersecting in one point, then it is possible to color 𝒜 with L colors. This complements the work on circular arc coloring [Belkale and Chandran, 2009; Tucker, 1975; Valencia-Pabon, 2003], which specifies sufficient conditions needed to color 𝒜 with L+1 colors or more.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• dynamic algorithms
• unit interval graphs
• coloring
• recourse budget
• parametrized dynamic algorithms

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## References

1. Josh Alman, Matthias Mnich, and Virginia Vassilevska Williams. Dynamic parameterized problems and algorithms. In Proceedings of the 44^th International Colloquium on Automata, Languages, and Programming, ICALP 2017, volume 80 of LIPIcs, pages 41:1-41:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.41.
2. Luis Barba, Jean Cardinal, Matias Korman, Stefan Langerman, André van Renssen, Marcel Roeloffzen, and Sander Verdonschot. Dynamic graph coloring. In Algorithms and data structures, volume 10389 of Lecture Notes in Comput. Sci., pages 97-108. Springer, Cham, 2017. URL: https://doi.org/10.1007/978-3-319-62127-2_9.
3. Dwight R. Bean. Effective coloration. The Journal of Symbolic Logic, 41(2):469-480, 1976. URL: http://www.jstor.org/stable/2272247.
4. Naveen Belkale and L. Sunil Chandran. Hadwiger’s conjecture for proper circular arc graphs. European J. Combin., 30(4):946-956, 2009. URL: https://doi.org/10.1016/j.ejc.2008.07.024.
5. Aaron Bernstein, Jacob Holm, and Eva Rotenberg. Online bipartite matching with amortized O(log² n) replacements. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 947-959. SIAM, Philadelphia, PA, 2018. URL: https://doi.org/10.1137/1.9781611975031.61.
6. Bartłomiej Bosek, Stefan Felsner, Kamil Kloch, Tomasz Krawczyk, Grzegorz Matecki, and Piotr Micek. On-line chain partitions of orders: a survey. Order, 29(1):49-73, 2012. URL: https://doi.org/10.1007/s11083-011-9197-1.
7. Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych. Online bipartite matching in offline time. In 55th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2014, pages 384-393. IEEE Computer Soc., Los Alamitos, CA, 2014. URL: https://doi.org/10.1109/FOCS.2014.48.
8. Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych-Pawlewicz. Shortest augmenting paths for online matchings on trees. Theory Comput. Syst., 62(2):337-348, 2018. URL: https://doi.org/10.1007/s00224-017-9838-x.
9. Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych-Pawlewicz. A tight bound for shortest augmenting paths on trees. In LATIN 2018: Theoretical informatics, volume 10807 of Lecture Notes in Comput. Sci., pages 201-216. Springer, Cham, 2018. URL: https://doi.org/10.1007/978-3-319-77404-6_1.
10. Bartłomiej Bosek, Yann Disser, Andreas Emil Feldmann, Jakub Pawlewicz, and Anna Zych-Pawlewicz. Recoloring Interval Graphs with Limited Recourse Budget. In Susanne Albers, editor, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020), volume 162 of Leibniz International Proceedings in Informatics (LIPIcs), pages 17:1-17:23, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SWAT.2020.17.
11. Bartłomiej Bosek and Anna Zych-Pawlewicz. Recoloring unit interval graphs with logarithmic recourse budget, 2022. URL: https://doi.org/10.48550/ARXIV.2202.08006.
12. Jiehua Chen, Wojciech Czerwiński, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk, Manuel Sorge, Bartłomiej Wróblewski, and Anna Zych-Pawlewicz. Efficient fully dynamic elimination forests with applications to detecting long paths and cycles, pages 796-809. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.50.
13. Mitre Costa Dourado, Van Bang Le, Fábio Protti, Dieter Rautenbach, and Jayme Luiz Szwarcfiter. Mixed unit interval graphs. Discret. Math., 312:3357-3363, 2012.
14. Zdeněk Dvořák, Martin Kupec, and Vojtěch Tůma. A dynamic data structure for MSO properties in graphs with bounded tree-depth. In Proceedings of the 22^nd Annual European Symposium on Algorithms, ESA 2014, volume 8737 of Lecture Notes in Computer Science, pages 334-345. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_28.
15. Leah Epstein and Meital Levy. Online interval coloring and variants. In Proceedings of the 32nd International Conference on Automata, Languages and Programming, ICALP'05, pages 602-613, Berlin, Heidelberg, 2005. Springer-Verlag. URL: https://doi.org/10.1007/11523468_49.
16. M. R. Garey, D. S. Johnson, G. L. Miller, and C. H. Papadimitriou. The complexity of coloring circular arcs and chords. SIAM Journal on Algebraic Discrete Methods, 1(2):216-227, 1980. URL: https://doi.org/10.1137/0601025.
17. Martin Charles Golumbic. Chapter 8 - interval graphs. In Martin Charles Golumbic, editor, Algorithmic Graph Theory and Perfect Graphs, volume 57 of Annals of Discrete Mathematics, pages 171-202. Elsevier, 2004. URL: https://doi.org/10.1016/S0167-5060(04)80056-6.
18. Magnús M. Halldórsson and Mario Szegedy. Lower bounds for on-line graph coloring, 1994.
19. Makoto Imase and Bernard M. Waxman. Dynamic Steiner tree problem. SIAM J. Discrete Math., 4(3):369-384, 1991. URL: https://doi.org/10.1137/0404033.
20. Henry A. Kierstead and William T. Trotter, Jr. An extremal problem in recursive combinatorics. Congr. Numer., 33:143-153, 1981.
21. Jakub Łącki, Jakub Oćwieja, Marcin Pilipczuk, Piotr Sankowski, and Anna Zych. The power of dynamic distance oracles: efficient dynamic algorithms for the Steiner tree. In STOC'15 - Proceedings of the 2015 ACM Symposium on Theory of Computing, pages 11-20. ACM, New York, 2015.
22. Dániel Marx. Precoloring extension on unit interval graphs. Discrete Applied Mathematics, 154(6):995-1002, 2006. URL: https://doi.org/10.1016/j.dam.2005.10.008.
23. Nicole Megow, Martin Skutella, José Verschae, and Andreas Wiese. The power of recourse for online MST and TSP. In Automata, languages, and programming. Part I, volume 7391 of Lecture Notes in Comput. Sci., pages 689-700. Springer, Heidelberg, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_58.
24. Stephan Olariu. An optimal greedy heuristic to color interval graphs. Inform. Process. Lett., 37(1):21-25, 1991. URL: https://doi.org/10.1016/0020-0190(91)90245-D.
25. James B. Orlin, Maurizio A. Bonuccelli, and Daniel P. Bovet. An O(n²) algorithm for coloring proper circular arc graphs. SIAM Journal on Algebraic Discrete Methods, 2(2):88-93, 1981. URL: https://doi.org/10.1137/0602012.
26. H. Maehara P. Frankl. Open-interval graphs versus closed-interval graphs. Discret. Math., 63:97-100, 1987.
27. Rasmus V. Rasmussen and Michael A. Trick. Round robin scheduling – a survey. European Journal of Operational Research, 188(3):617-636, 2008. URL: https://doi.org/10.1016/j.ejor.2007.05.046.
28. Wei-Kuan Shih and Wen-Lian Hsu. An o(n1.5) algorithm to color proper circular arcs. Discrete Applied Mathematics, 25(3):321-323, 1989. URL: https://doi.org/10.1016/0166-218X(89)90011-5.
29. Shay Solomon and Nicole Wein. Improved dynamic graph coloring. In 26th European Symposium on Algorithms, volume 112 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 72, 16. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018.
30. Alan Tucker. Coloring a family of circular arcs. SIAM Journal on Applied Mathematics, 29(3):493-502, 1975. URL: https://doi.org/10.1137/0129040.
31. Mario Valencia-Pabon. Revisiting Tucker’s algorithm to color circular arc graphs. SIAM Journal on Computing, 32(4):1067-1072, 2003. URL: https://doi.org/10.1137/S0097539700382157.
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