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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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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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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.

Cite As Get BibTex

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

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.

References

  1. Hiroki Arimura, Tatsuya Gima, Yasuaki Kobayashi, Hiroomi Nochide, and Yota Otachi. Minimum consistent subset for trees revisited. CoRR, abs/2305.07259, 2023. URL: https://doi.org/10.48550/arXiv.2305.07259.
  2. Sandip Banerjee, Sujoy Bhore, and Rajesh Chitnis. Algorithms and hardness results for nearest neighbor problems in bicolored point sets. In Michael A. Bender, Martín Farach-Colton, and Miguel A. Mosteiro, editors, LATIN 2018: Theoretical Informatics, pages 80-93, 2018. URL: https://doi.org/10.1007/978-3-319-77404-6_7.
  3. Ahmad Biniaz, Sergio Cabello, Paz Carmi, Jean-Lou De Carufel, Anil Maheshwari, Saeed Mehrabi, and Michiel Smid. On the minimum consistent subset problem. Algorithmica, 83(7):2273-2302, 2021. URL: https://doi.org/10.1007/S00453-021-00825-8.
  4. Ahmad Biniaz and Parham Khamsepour. The minimum consistent spanning subset problem on trees. Journal of Graph Algorithms and Applications, 28(1):81-93, 2024. URL: https://doi.org/10.7155/JGAA.V28I1.2929.
  5. Rajesh Chitnis. Refined lower bounds for nearest neighbor condensation. In Sanjoy Dasgupta and Nika Haghtalab, editors, International Conference on Algorithmic Learning Theory, 29 March - 1 April 2022, Paris, France, volume 167 of Proceedings of Machine Learning Research, pages 262-281. PMLR, 2022. URL: https://proceedings.mlr.press/v167/chitnis22a.html.
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer Publishing Company, Incorporated, 1st edition, 2015. Google Scholar
  7. Sanjana Dey, Anil Maheshwari, and Subhas C. Nandy. Minimum consistent subset problem for trees. In Evripidis Bampis and Aris Pagourtzis, editors, Fundamentals of Computation Theory - 23rd International Symposium, FCT 2021, Athens, Greece, September 12-15, 2021, Proceedings, volume 12867 of Lecture Notes in Computer Science, pages 204-216. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86593-1_14.
  8. Sanjana Dey, Anil Maheshwari, and Subhas C. Nandy. Minimum consistent subset of simple graph classes. Discret. Appl. Math., 338:255-277, 2023. URL: https://doi.org/10.1016/J.DAM.2023.05.024.
  9. Reinhard Diestel. Graph theory, volume 173 of. Graduate texts in mathematics, page 7, 2012. Google Scholar
  10. Michael R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976. URL: https://doi.org/10.1016/0304-3975(76)90059-1.
  11. Peter E. Hart. The condensed nearest neighbor rule (corresp.). IEEE Transactions on Information Theory, 14(3):515-516, 1968. URL: https://doi.org/10.1109/TIT.1968.1054155.
  12. Kamyar Khodamoradi, Ramesh Krishnamurti, and Bodhayan Roy. Consistent subset problem with two labels. In B.S. Panda and Partha P. Goswami, editors, Algorithms and Discrete Applied Mathematics, pages 131-142, 2018. URL: https://doi.org/10.1007/978-3-319-74180-2_11.
  13. Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability pcp characterization of np. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, STOC '97, pages 475-484, New York, NY, USA, 1997. Association for Computing Machinery. URL: https://doi.org/10.1145/258533.258641.
  14. Vijay V. Vazirani. Approximation Algorithms. Springer Publishing Company, Incorporated, 2010. Google Scholar
  15. Gordon Wilfong. Nearest neighbor problems. In Proceedings of the Seventh Annual Symposium on Computational Geometry, SCG ’91, pages 224-233, 1991. URL: https://doi.org/10.1145/109648.109673.
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