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Minimum Consistent Subset in Trees and Interval Graphs

Authors Aritra Banik, Sayani Das, Anil Maheshwari , Bubai Manna, Subhas C. Nandy, Krishna Priya K. M., Bodhayan Roy, Sasanka Roy, Abhishek Sahu

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Author Details

Aritra Banik
  • National Institute of Science, Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, India
Sayani Das
  • Theoretical Computer Science, The Institute of Mathematical Sciences, Chennai, India
Anil Maheshwari
  • School of Computer Science, Carleton University, Ottawa, Canada
Bubai Manna
  • Department of Mathematics, Indian Institute of Technology Kharagpur, India
Subhas C. Nandy
  • Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India
Krishna Priya K. M.
  • National Institute of Science, Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, India
Bodhayan Roy
  • Department of Mathematics, Indian Institute of Technology Kharagpur, India
Sasanka Roy
  • Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India
Abhishek Sahu
  • National Institute of Science, Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, India

Funding

  • Banik, Aritra: Supported by the Science and Engineering Research Board (SERB) via the project MTR/2022/000253.
  • Maheshwari, Anil: Supported by Natural Sciences and Engineering Research Council of Canada (NSERC)
  • Roy, Bodhayan: Supported by the Science and Engineering Research Board (SERB) via the project MTR/2021/000474.

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Aritra Banik, Sayani Das, Anil Maheshwari, Bubai Manna, Subhas C. Nandy, Krishna Priya K. M., Bodhayan Roy, Sasanka Roy, and Abhishek Sahu. Minimum Consistent Subset in Trees and Interval Graphs. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.7

Abstract

In the Minimum Consistent Subset (MCS) problem, we are presented with a connected simple undirected graph G, consisting of a vertex set V(G) of size n and an edge set E(G). Each vertex in V(G) is assigned a color from the set {1,2,…, c}. The objective is to determine a subset V' ⊆ V(G) with minimum possible cardinality, such that for every vertex v ∈ V(G), at least one of its nearest neighbors in V' (measured in terms of the hop distance) shares the same color as v. The decision problem, indicating whether there exists a subset V' of cardinality at most l for some positive integer l, is known to be NP-complete even for planar graphs.
In this paper, we establish that the MCS problem is NP-complete on trees. We also provide a fixed-parameter tractable (FPT) algorithm for MCS on trees parameterized by the number of colors (c) running in O(2^{6c} n^6) time, significantly improving the currently best-known algorithm whose running time is O(2^{4c} n^{2c+3}). In an effort to comprehensively understand the computational complexity of the MCS problem across different graph classes, we extend our investigation to interval graphs. We show that it remains NP-complete for interval graphs, thus enriching graph classes where MCS remains intractable.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Nearest-Neighbor Classification
  • Minimum Consistent Subset
  • Trees
  • Interval Graphs
  • Parameterized complexity
  • NP-complete

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References

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