Maximizing Covered Area in the Euclidean Plane with Connectivity Constraint

Given a set D of n unit disks in the plane and an integer k ≤ n, the maximum area connected subset problem asks for a set D′ ⊆ D of size k that maximizes the area of the union of disks, under the constraint that this union is connected. This problem is motivated by wireless router deployment and is a special case of maximizing a submodular function under a connectivity constraint. We prove that the problem is NP-hard and analyze a greedy algorithm, proving that it is a 2 approximation. We then give a polynomial-time approximation scheme (PTAS) for this problem with resource augmentation, i.e., allowing an additional set of εk disks that are not drawn from the input. Additionally, for two special cases of the problem we design a PTAS without resource augmentation. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms

[1]  Nabil H. Mustafa,et al.  Improved Results on Geometric Hitting Set Problems , 2010, Discret. Comput. Geom..

[2]  Harry B. Hunt,et al.  NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs , 1998, J. Algorithms.

[3]  Kate Ching-Ju Lin,et al.  Maximizing Submodular Set Function With Connectivity Constraint: Theory and Application to Networks , 2013, IEEE/ACM Transactions on Networking.

[4]  Yuval Rabani,et al.  Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem , 2008, SWAT.

[5]  Giri Narasimhan,et al.  Geometric spanner networks , 2007 .

[6]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[7]  Steven Chaplick,et al.  Approximation Schemes for Geometric Coverage Problems , 2018, ESA.

[8]  Tomomi Matsui,et al.  Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs , 1998, JCDCG.

[9]  Yuval Filmus,et al.  Monotone Submodular Maximization over a Matroid via Non-Oblivious Local Search , 2012, SIAM J. Comput..

[10]  Hadas Shachnai,et al.  Maximizing submodular set functions subject to multiple linear constraints , 2009, SODA.

[11]  Csaba D. Tóth,et al.  Constant-Factor Approximation for TSP with Disks , 2015, ArXiv.

[12]  Joseph S. B. Mitchell,et al.  Connecting a Set of Circles with Minimum Sum of Radii , 2011, WADS.

[13]  Sariel Har-Peled,et al.  Being Fat and Friendly is Not Enough , 2009, ArXiv.

[14]  L. Wolsey.  Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems , 1982, Math. Oper. Res..

[15]  David S. Johnson,et al.  The Rectilinear Steiner Tree Problem is NP Complete , 1977, SIAM Journal of Applied Mathematics.

[16]  Jan Vondrák,et al.  Maximizing a Monotone Submodular Function Subject to a Matroid Constraint , 2011, SIAM J. Comput..

[17]  Himanshu Gupta,et al.  Connected sensor cover: self-organization of sensor networks for efficient query execution , 2003, IEEE/ACM Transactions on Networking.

[18]  Charles J. Colbourn,et al.  Unit disk graphs , 1991, Discret. Math..

[19]  Harry B. Hunt,et al.  Simple heuristics for unit disk graphs , 1995, Networks.

[20]  Lee-Ad Gottlieb,et al.  The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme , 2011, STOC '12.

[21]  Chandra Chekuri,et al.  Submodular function maximization via the multilinear relaxation and contention resolution schemes , 2011, STOC '11.

[22]  Michael Segal,et al.  Improved approximation algorithms for connected sensor cover , 2004, ADHOC-NOW.

[23]  T-H. Hubert Chan,et al.  Reducing Curse of Dimensionality: Improved PTAS for TSP (with Neighborhoods) in Doubling Metrics , 2016, SODA.

[24]  Suman Banerjee,et al.  Node Placement for Connected Coverage in Sensor Networks , 2003 .

[25]  Eli Upfal,et al.  Algorithms for Detecting Significantly Mutated Pathways in Cancer , 2010, RECOMB.

[26]  Alexander Grigoriev,et al.  Approximation schemes for the generalized geometric problems with geographic clustering , 2005, EuroCG.

[27]  Samir Khuller,et al.  Analyzing the Optimal Neighborhood: Algorithms for Budgeted and Partial Connected Dominating Set Problems , 2013, SODA.

[28]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[29]  Jan Vondrák,et al.  Submodular Maximization over Multiple Matroids via Generalized Exchange Properties , 2009, Math. Oper. Res..

[30]  Khaled M. Elbassioni,et al.  On Approximating the TSP with Intersecting Neighborhoods , 2006, ISAAC.