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International Conference on Combinatorial Optimization and Applications

COCOA 2021: Combinatorial Optimization and Applications pp 152–166Cite as

The Complexity of Finding Optimal Subgraphs to Represent Spatial Correlation

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 13135)

Abstract

Understanding spatial correlation is vital in many fields including epidemiology and social science. Lee, Meeks and Pettersson (Stat. Comput. 2021) recently demonstrated that improved inference for areal unit count data can be achieved by carrying out modifications to a graph representing spatial correlations; specifically, they delete edges of the planar graph derived from border-sharing between geographic regions in order to maximise a specific objective function. In this paper we address the computational complexity of the associated graph optimisation problem. We demonstrate that this problem cannot be solved in polynomial time unless P = NP; we further show intractability for two simpler variants of the problem. We follow these results with two parameterised algorithms that exactly solve the problem in polynomial time in restricted settings. The first of these utilises dynamic programming on a tree decomposition, and runs in polynomial time if both the treewidth and maximum degree are bounded. The second algorithm is restricted to problem instances with maximum degree three, as may arise from triangulations of planar surfaces, but is an FPT algorithm when the maximum number of edges that can be removed is taken as the parameter.

Keywords

  • Parameterised complexity
  • Treewidth
  • Colour coding
  • Spatial statistics

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References

  1. Bradley, J.R., Wikle, C.K., Holan, S.H.: Bayesian spatial change of support for count-valued survey data with application to the American community survey. J. Am. Statist. Assoc. 111(514), 472–487 (2016)

    Google Scholar 

  2. Cygan, M., et al.: Parameterized Algorithms, vol. 5. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

  3. Enright, J., Lee, D., Meeks, K., Pettersson, W., Sylvester, J.: The complexity of finding optimal subgraphs to represent spatial correlation (2021). arXiv:2010.10314

  4. Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33(4), 892–922 (2004)

    CrossRef  MathSciNet  Google Scholar 

  5. He, Q., Zeng, C., Xie, P., Liu, Y., Zhang, M.: An assessment of forest biomass carbon storage and ecological compensation based on surface area: a case study of Hubei Province. China Ecol. Indicat. 90, 392–400 (2018)

    CrossRef  Google Scholar 

  6. bibitemch13Karp72 Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Proceedings of a Symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pp. 85–103. Plenum Press, New York (1972)

    Google Scholar 

  7. Lee, D., Meeks, K., Pettersson, W.: Improved inference for areal unit count data using graph-based optimisation. Stat. Comput. 31(4), 1–17 (2021). https://doi.org/10.1007/s11222-021-10025-7

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Mindell, J.S., et al.: Using triangulation to assess a suite of tools to measure community severance. J. Transp. Geogr. 60, 119–129 (2017)

    Google Scholar 

  9. Mitchell, R., Lee, D.: Is there really a “wrong side of the tracks” in urban areas and does it matter for spatial analysis? Ann. Assoc. Am. Geogr. 104(3), 432–443 (2014)

    Google Scholar 

  10. Cristopher Moore and John Michael Robson: Hard tiling problems with simple tiles. Discrete Comput. Geom. 26(4), 573–590 (2001)

    CrossRef  MathSciNet  Google Scholar 

  11. Stoner, O., Economou, T., Marques da Silva, G.D.: A hierarchical framework for correcting under-reporting in count data. J. Am. Stat. Assoc. 114(528), 1481–1492 (2019)

    Google Scholar 

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Acknowledgements

All authors gratefully acknowledge funding from the Engineering and Physical Sciences Research Council (ESPRC) grant number EP/T004878/1 for this work, while Meeks was also supported by a Royal Society of Edinburgh Personal Research Fellowship (funded by the Scottish Government).

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Authors and Affiliations

  1. School of Computing Science, University of Glasgow, Glasgow, UK

    Jessica Enright, Kitty Meeks, William Pettersson & John Sylvester

  2. School of Mathematics and Statistics, University of Glasgow, Glasgow, UK

    Duncan Lee

Authors
  1. Jessica Enright
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  2. Duncan Lee
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  3. Kitty Meeks
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  4. William Pettersson
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  5. John Sylvester
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Corresponding author

Correspondence to William Pettersson .

Editor information

Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Dr. Ding-Zhu Du

  2. University of New Brunswick, Fredericton, NB, Canada

    Donglei Du

  3. Tianjin University of Technology, Tianjin, China

    Chenchen Wu

  4. Beijing University of Technology, Beijing, China

    Dachuan Xu

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Enright, J., Lee, D., Meeks, K., Pettersson, W., Sylvester, J. (2021). The Complexity of Finding Optimal Subgraphs to Represent Spatial Correlation. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Combinatorial Optimization and Applications. COCOA 2021. Lecture Notes in Computer Science(), vol 13135. Springer, Cham. https://doi.org/10.1007/978-3-030-92681-6_13

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  • DOI: https://doi.org/10.1007/978-3-030-92681-6_13

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  • Online ISBN: 978-3-030-92681-6

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