Multiobjective Disk Cover Admits a PTAS

We introduce multiobjective disk cover problems and study their approximability. We construct a polynomial-time approximation scheme (PTAS) for the multiobjective problem where k types of points (customers) in the plane have to be covered by disks (base stations) such that the number of disks is minimized and for each type of points, the number of covered points is maximized. Our approximation scheme can be extended so that it works with the following additional features: interferences, different services for different types of customers, different shapes of supply areas, weighted customers, individual costs for base stations, and payoff for the quality of the obtained service. Furthermore, we show that it is crucial to solve this problem in a multiobjective way, where all objectives are optimized at the same time. The constrained approach (i.e., the restriction of a multiobjective problem to a single objective) often used for such problems can significantly degrade their approximability. We can show non-approximability results for several single-objective restrictions of multiobjective disk cover problems. For example, if there are 2 types of customers, then maximizing the supplied customers of one type is not even approximable within a constant factor, unless P = NP.

[1]  Evripidis Bampis,et al.  On the approximate tradeoff for bicriteria batching and parallel machine scheduling problems , 2003, Theor. Comput. Sci..

[2]  Friedhelm Meyer auf der Heide,et al.  Algorithms — ESA 2001 , 2001, Lecture Notes in Computer Science.

[3]  Mihalis Yannakakis,et al.  Small Approximate Pareto Sets for Biobjective Shortest Paths and Other Problems , 2009, SIAM J. Comput..

[4]  Heribert Vollmer,et al.  The complexity of base station positioning in cellular networks , 2005, Discret. Appl. Math..

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

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

[7]  Mihalis Yannakakis,et al.  On the approximability of trade-offs and optimal access of Web sources , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[8]  Matthew J. Katz,et al.  Approximating the Visible Region of a Point on a Terrain , 2008, GeoInformatica.

[9]  Erik Jan van Leeuwen,et al.  Approximating geometric coverage problems , 2008, SODA '08.

[10]  Timothy M. Chan.  Polynomial-time approximation schemes for packing and piercing fat objects , 2003, J. Algorithms.

[11]  E. Polak,et al.  On Multicriteria Optimization , 1976 .

[12]  Sergei Vassilvitskii,et al.  Efficiently computing succinct trade-off curves , 2005, Theor. Comput. Sci..

[13]  Emmanuel Jeannot,et al.  Bi-objective scheduling algorithms for optimizing makespan and reliability on heterogeneous systems , 2007, SPAA '07.

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

[15]  Lenore Cowen,et al.  Approximation Algorithms for the Class Cover Problem , 2004, Annals of Mathematics and Artificial Intelligence.

[16]  J. Orlin,et al.  Fast Approximation Schemes for Multi-Criteria Flow, Knapsack, and Scheduling Problems , 1995 .

[17]  T. C. Edwin Cheng,et al.  Bicriterion Single Machine Scheduling with Resource Dependent Processing Times , 1998, SIAM J. Optim..

[18]  Evripidis Bampis,et al.  A FPTAS for Approximating the Unrelated Parallel Machines Scheduling Problem with Costs , 2001, ESA.

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

[20]  Klaus Jansen,et al.  Polynomial-Time Approximation Schemes for Geometric Intersection Graphs , 2005, SIAM J. Comput..

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

[22]  S. Khuller,et al.  Approximation algorithms for partial covering problems , 2001, J. Algorithms.

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

[24]  Esther M. Arkin,et al.  Minimum-cost coverage of point sets by disks , 2006, SCG '06.

[25]  H. Safer.  Fast approximation schemes for multi-criteria combinatorial optimization , 1992 .

[26]  J. van Leeuwen,et al.  Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[27]  Mihalis Yannakakis,et al.  Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other Problems , 2007, APPROX-RANDOM.