2 Search Results for "Bosek, Bartłomiej"

Document
DOI: 10.4230/LIPIcs.ESA.2022.25
Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget

Authors: Bartłomiej Bosek and Anna Zych-Pawlewicz

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

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.
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)

Copy BibTex To Clipboard

@InProceedings{bosek_et_al:LIPIcs.ESA.2022.25,
  author =	{Bosek, Bart{\l}omiej and Zych-Pawlewicz, Anna},
  title =	{{Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.25},
  URN =		{urn:nbn:de:0030-drops-169637},
  doi =		{10.4230/LIPIcs.ESA.2022.25},
  annote =	{Keywords: dynamic algorithms, unit interval graphs, coloring, recourse budget, parametrized dynamic algorithms}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2020.17
Recoloring Interval Graphs with Limited Recourse Budget

Authors: Bartłomiej Bosek, Yann Disser, Andreas Emil Feldmann, Jakub Pawlewicz, and Anna Zych-Pawlewicz

Published in: LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Abstract
We consider the problem of coloring an interval graph dynamically. Intervals arrive one after the other and have to be colored immediately such that no two intervals of the same color overlap. In each step only a limited number of intervals may be recolored to maintain a proper coloring (thus interpolating between the well-studied online and offline settings). The number of allowed recolorings per step is the so-called recourse budget. Our main aim is to prove both upper and lower bounds on the required recourse budget for interval graphs, given a bound on the allowed number of colors. For general interval graphs with n vertices and chromatic number k it is known that some recoloring is needed even if we have 2k colors available. We give an algorithm that maintains a 2k-coloring with an amortized recourse budget of 𝒪(log n). For maintaining a k-coloring with k ≤ n, we give an amortized upper bound of 𝒪(k⋅ k! ⋅ √n), and a lower bound of Ω(k) for k ∈ 𝒪(√n), which can be as large as Ω(√n). For unit interval graphs it is known that some recoloring is needed even if we have k+1 colors available. We give an algorithm that maintains a (k+1)-coloring with at most 𝒪(k²) recolorings per step in the worst case. We also give a lower bound of Ω(log n) on the amortized recourse budget needed to maintain a k-coloring. Additionally, for general interval graphs we show that if one does not insist on maintaining an explicit coloring, one can have a k-coloring algorithm which does not incur a factor of 𝒪(k ⋅ k! ⋅ √n) in the running time. For this we provide a data structure, which allows for adding intervals in 𝒪(k² log³ n) amortized time per update and querying for the color of a particular interval in 𝒪(log n) time. Between any two updates, the data structure answers consistently with some optimal coloring. The data structure maintains the coloring implicitly, so the notion of recourse budget does not apply to it.
Cite as

Bartłomiej Bosek, Yann Disser, Andreas Emil Feldmann, Jakub Pawlewicz, and Anna Zych-Pawlewicz. Recoloring Interval Graphs with Limited Recourse Budget. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 17:1-17:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{bosek_et_al:LIPIcs.SWAT.2020.17,
  author =	{Bosek, Bart{\l}omiej and Disser, Yann and Feldmann, Andreas Emil and Pawlewicz, Jakub and Zych-Pawlewicz, Anna},
  title =	{{Recoloring Interval Graphs with Limited Recourse Budget}},
  booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
  pages =	{17:1--17:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-150-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{162},
  editor =	{Albers, Susanne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.17},
  URN =		{urn:nbn:de:0030-drops-122649},
  doi =		{10.4230/LIPIcs.SWAT.2020.17},
  annote =	{Keywords: Colouring, Dynamic Algorithms, Recourse Budget, Interval Graphs}
}
  • Refine by Author
  • 2 Bosek, Bartłomiej
  • 2 Zych-Pawlewicz, Anna
  • 1 Disser, Yann
  • 1 Feldmann, Andreas Emil
  • 1 Pawlewicz, Jakub

  • Refine by Classification
  • 1 Theory of computation → Data structures design and analysis
  • 1 Theory of computation → Design and analysis of algorithms
  • 1 Theory of computation → Dynamic graph algorithms
  • 1 Theory of computation → Online algorithms

  • Refine by Keyword
  • 1 Colouring
  • 1 Dynamic Algorithms
  • 1 Interval Graphs
  • 1 Recourse Budget
  • 1 coloring
  • Show More...

  • Refine by Type
  • 2 document

  • Refine by Publication Year
  • 1 2020
  • 1 2022

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact