Discretization of Planar Geometric Cover Problems

We consider discretization of the 'geometric cover problem' in the plane: Given a set $P$ of $n$ points in the plane and a compact planar object $T_0$, find a minimum cardinality collection of planar translates of $T_0$ such that the union of the translates in the collection contains all the points in $P$. We show that the geometric cover problem can be converted to a form of the geometric set cover, which has a given finite-size collection of translates rather than the infinite continuous solution space of the former. We propose a reduced finite solution space that consists of distinct canonical translates and present polynomial algorithms to find the reduce solution space for disks, convex/non-convex polygons (including holes), and planar objects consisting of finite Jordan curves.

[1]  Jon Louis Bentley,et al.  The Complexity of Finding Fixed-Radius Near Neighbors , 1977, Inf. Process. Lett..

[2]  David S. Johnson.  The NP-Completeness Column: An Ongoing Guide , 1986, J. Algorithms.

[3]  David S. Johnson,et al.  The NP-Completeness Column: An Ongoing Guide , 1982, J. Algorithms.

[4]  Raimund Seidel,et al.  Constructing arrangements of lines and hyperplanes with applications , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[5]  Alejandro López-Ortiz,et al.  On the discrete Unit Disk Cover Problem , 2011, Int. J. Comput. Geom. Appl..

[6]  Petr Vojtechovský,et al.  An Improved Approximation Factor For The Unit Disk Covering Problem , 2006, CCCG.

[7]  Gautam K. Das,et al.  Unit Disk Cover Problem in 2D , 2013, ICCSA.

[8]  Matthew J. Katz,et al.  Covering Points by Unit Disks of Fixed Location , 2007, ISAAC.

[9]  Sören Laue.  Geometric Set Cover and Hitting Sets for Polytopes in $R^3$ , 2008, STACS.

[10]  Alejandro López-Ortiz,et al.  On the discrete Unit Disk Cover Problem , 2012, Int. J. Comput. Geom. Appl..

[11]  Robert J. Fowler,et al.  Optimal Packing and Covering in the Plane are NP-Complete , 1981, Inf. Process. Lett..

[12]  Teofilo F. Gonzalez,et al.  Covering a Set of Points in Multidimensional Space , 1991, Inf. Process. Lett..

[13]  Michael T. Goodrich,et al.  Almost optimal set covers in finite VC-dimension , 1995, Discret. Comput. Geom..

[14]  Leonidas J. Guibas,et al.  Arrangements of Curves in the Plane - Topology, Combinatorics and Algorithms , 2018, Theor. Comput. Sci..

[15]  Alejandro López-Ortiz,et al.  The Within-Strip Discrete Unit Disk Cover Problem , 2017, CCCG.

[16]  Wolfgang Maass,et al.  Approximation schemes for covering and packing problems in image processing and VLSI , 1985, JACM.

[17]  Nabil H. Mustafa,et al.  PTAS for geometric hitting set problems via local search , 2009, SCG '09.

[18]  David G. Kirkpatrick,et al.  Fast Detection of Polyhedral Intersection , 1983, Theor. Comput. Sci..

[19]  Gill Barequet,et al.  Translating a Convex Polygon to Contain a Maximum Number of Points , 1997, Comput. Geom..

[20]  Kenneth L. Clarkson,et al.  Improved Approximation Algorithms for Geometric Set Cover , 2005, Discret. Comput. Geom..

[21]  Stephane Durocher,et al.  An Improved Line-Separable Algorithm for Discrete Unit Disk Cover , 2010, Discret. Math. Algorithms Appl..

[22]  Micha Sharir,et al.  On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles , 1986, Discret. Comput. Geom..

[23]  Shiyan Hu,et al.  Polynomial time approximation schemes for minimum disk cover problems , 2010, J. Comb. Optim..

[24]  Nimrod Megiddo,et al.  On the Complexity of Some Common Geometric Location Problems , 1984, SIAM J. Comput..

[25]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[26]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[27]  Bernard Chazelle,et al.  On a circle placement problem , 1986, Computing.

[28]  David Eppstein,et al.  Iterated nearest neighbors and finding minimal polytopes , 1993, SODA '93.

[29]  Micha Sharir,et al.  Computing the Smallest K-enclosing Circle and Related Problems , 1993, Comput. Geom..

[30]  Ion I. Mandoiu,et al.  Selecting Forwarding Neighbors in Wireless Ad Hoc Networks , 2001, DIALM '01.

[31]  Leonidas J. Guibas,et al.  Polyhedral Assembly Partitioning Using Maximally Covered Cells in Arrangements of Convex Polytopes , 1998, Int. J. Comput. Geom. Appl..