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Recoloring Interval Graphs with Limited Recourse Budget

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

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Author Details

Bartłomiej Bosek
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Poland
Yann Disser
  • Department of Mathematics, TU Darmstadt, Germany
Andreas Emil Feldmann
  • Department of Applied Mathematics, Charles University in Prague, Czech Republic
Jakub Pawlewicz
  • University of Warsaw, Poland
Anna Zych-Pawlewicz
  • University of Warsaw, Poland

Funding

  • Bosek, Bartłomiej: National Science Centre of Poland, project number 2017/26/D/ST6/00264.
  • Disser, Yann: `Excellence Initiative' of the German Federal and State Governments, and Graduate School CE at TU Darmstadt.
  • Feldmann, Andreas Emil: Czech Science Foundation GAČR (grant #17-10090Y), and Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004).
  • Zych-Pawlewicz, Anna: National Science Centre of Poland, project number 2017/26/D/ST6/00264.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.SWAT.2020.17

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Theory of computation → Online algorithms
  • Theory of computation → Data structures design and analysis
Keywords
  • Colouring
  • Dynamic Algorithms
  • Recourse Budget
  • Interval Graphs

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