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Parameterized Reunion with Achromatic Number

Authors Satyabrata Jana , Souvik Saha , Saket Saurabh , Anannya Upasana

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Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Achromatic number
  • Coloring
  • Fixed-parameter tractability
  • Kernelization
  • Lower bound
  • W-hardness

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Abstract

In this paper, we study the Achromatic Number problem. Given a graph G and an integer k, the task is to determine whether there exists a proper coloring of G, using at least k colors, in which every pair of distinct colors appears on the endpoints of some edge. It was established early on that the problem is fixed-parameter tractable (FPT)- even before the formal development of parameterized complexity. In fact, Farber, Hahn, Hell, and Miller [JCTB, 1986] devised an algorithm with a running time of 𝒪(f(k) ⋅ |E(G)|). Although the exact form of f(k) was not specified, it appears to be at least doubly exponential in k. In our work, we first present an algorithm with an explicit dependence on k, and then introduce another algorithm that is parameterized by the vertex cover number of the graph. More formally, we show the following.
- Achromatic Number is solvable in time 2^𝒪(k⁵)+𝒪(|E(G)|). 
- Achromatic Number admits a polynomial kernel when the input is restricted to a d-degenerate graph and a more efficient kernel on trees. 
- We also study the parameterized complexity of the problem with respect to Vertex Cover and show that it admits an FPT algorithm running in time 2^𝒪(𝓁²) ⋅ n^𝒪(1), where 𝓁 is the size of a vertex cover.

Cite As Get BibTex

Satyabrata Jana, Souvik Saha, Saket Saurabh, and Anannya Upasana. Parameterized Reunion with Achromatic Number. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.42

Author Details

Satyabrata Jana
  • University of Warwick, UK
Souvik Saha
  • The Institute of Mathematical Sciences, HBNI, Chenai, India
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chenai, India
  • University of Bergen, Norway
Anannya Upasana
  • The Institute of Mathematical Sciences, HBNI, Chenai, India

Funding

  • Jana, Satyabrata: Supported by the Engineering and Physical Sciences Research Council (EPSRC) via the project MULTIPROCESS (grant no. EP/V044621/1).
  • Saurabh, Saket: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416).

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