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The Parameterized Complexity of Finding Point Sets with Hereditary Properties

Authors David Eppstein, Daniel Lokshtanov

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • parameterized complexity
  • fixed-parameter tractability
  • point set pattern matching
  • largest pattern-avoiding subset
  • order type

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Abstract

We consider problems where the input is a set of points in the plane and an integer k, and the task is to find a subset S of the input points of size k such that S satisfies some property. We focus on properties that depend only on the order type of the points and are monotone under point removals. We exhibit a property defined by three forbidden patterns for which finding a k-point subset with the property is W[1]-complete and (assuming the exponential time hypothesis) cannot be solved in time n^{o(k/log k)}. However, we show that problems of this type are fixed-parameter tractable for all properties that include all collinear point sets, properties that exclude at least one convex polygon, and properties defined by a single forbidden pattern.

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David Eppstein and Daniel Lokshtanov. The Parameterized Complexity of Finding Point Sets with Hereditary Properties. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.IPEC.2018.11

Author Details

David Eppstein
  • Computer Science Department, University of California, Irvine, USA
Daniel Lokshtanov
  • Department of Informatics, University of Bergen, Norway

Funding

  • Eppstein, David: Supported in part by NSF grants CCF-1618301 and CCF-1616248.
  • Lokshtanov, Daniel: Supported by Pareto-Optimal Parameterized Algorithms, ERC Starting Grant 715744

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