Article Outline
Keywords and Phrases
Introduction
Definitions
Applications
Models
Cliques
Independent Sets
Vertex Cover
Domination
Coloring and Clique Partitioning
Related Results
Conclusions
References
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alber J, Fiala J (2004) Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J Algorithm 52(2):134–151
Alzoubi KM, Wan PJ, Frieder O (2002) Distributed heuristics for connected dominating sets in wireless ad hoc networks. J Commun Netw 4:22–29
Ambühl C, Erlebach T, Mihal'ák M, Nunkesser M (2006) Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In: Diaz J, Jansen K, Rolim JDP, Zwick U (eds) Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques vol 4110 of Lecture Notes in Computer Science. Springer. pp 3–14
Baker BS (1994) Approximation algorithms for NP‑complete problems on planar graphs. J ACM 41(1):153–180
Balasundaram B, Butenko S (2006) Graph domination, coloring and cliques in telecommunications. In: Resende MGC, Pardalos PM (eds) Handbook of Optimization in Telecommunications. Spinger, New York, pp 865–890
Bar-Yehuda R, Even S (1982) On approximating a vertex cover for planar graphs. In: STOC '82: Proceeding of the Fourteenth Annual ACM Symposium on Theory of Computing, pp 303–309
Bar-Yehuda R, Even S (1985) A local-ratio theorem for approximating the weighted vertex cover problem. Ann Discret Math 25:27–46
Basagni S (1999) Distributed clustering for ad hoc networks. In Proceedings of the (1999) International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN '99) p 310
Breu H (1996) Algorithmic Aspects of Constrained Unit Disk Graphs, PhD thesis. University of British Columbia
Breu H, Kirkpatrick DG (1998) Unit disk graph recognition is NP-hard. Comput Geometr Theory Appl 9(1–2):3–24
Brooks RL (1941) On coloring the nodes of a network. Proc Cambridge Philos Soc 37:194–197
Butenko S, Cheng X, Du DZ, Pardalos P (2003) On the construction of virtual backbone for ad hoc wireless networks. In Butenko S, Murphey R, Pardalos PM (eds), Cooperative Control: Models, Applications and Algorithms. Kluwer Academic, pp 43–54
Callahan PB, Kosaraju SR (1995) Algorithms for dynamic closest pair and n-body potential fields. In: Proceedings of the sixth annual ACM-SIAM symposium on discrete algorithms (SODA '95). Society for Industrial and Applied Mathematics, Philadelphia, pp 263–272
Cardei M, Cheng X, Cheng X, Du DZ (2002) Connected domination in multihop ad hoc wireless networks. In: Caulfield HJ, Chen SH, Cheng HD, Duro RJ, Honavar V, Kerre EE, Lu M, Romay MG, Shih TK, Wang DVPP, Yang Y (eds) Proceedings of the 6th Joint Conference on Information Science JCIS/Association for Intelligent Machinery, Inc, pp 251–255
Cerioli MR, Faria L, Ferreira TO, Protti F (2004) On minimum clique partition and maximum independent set on unit disk graphs and penny graphs: complexity and approximation. In: Latin-American Conference on Combinatorics, Graphs and Applications, vol 18 of Electronic Notes in Discrete Mathematics. Elsevier, Amsterdam, pp 73–79 (electronic)
Chatterjee M, Das S, Turgut D (2002) WCA: A weighted clustering algorithm for mobile ad hoc networks. J Cluster Computing, Special Issue on Mobile Ad hoc Netw 5:193–204
Cheng X, Huang X, Li D, Wu W, Du DZ (2003) A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks. Netw 42(4):202–208
Clark BN, Colbourn CJ, Johnson DS (1990) Unit disk graphs. Discret Math 86:165–177
Das B, Bharghavan V (1997) Routing in ad-hoc networks using minimum connected dominating sets. In: Proceedings of IEEE International Conference on Communications, pp 376–380
Das B, Sivakumar R, Bharghavan V (1997) Routing in ad-hoc networks using a virtual backbone. In: Proceedings of the International Conference on Computers and Communication Networks (IC3N), pp 1–20
Diaz J, Kaminski M (2007) Max-cut and max-bisection are np-hard on unit disk graphs. Theoret Comput Sci 377(1–3):271–276
Du H, Jia X, Li D, Wu W (2004) Coloring of double disk graphs. J Global Optim 28(1):115–119
Erlebach T, Fiala J (2006) Independence and coloring problems on intersection graphs of disks. In: Bampis E, Jansen K, Kenyon C (eds) Efficient Approximation and Online Algorithms, vol 3484 of Lecture Notes in Computer Science. Springer, pp 135–155
Erlebach T, Jansen K, Seidel E (2001) Polynomial-time approximation schemes for geometric graphs. In: SODA '01: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, pp 671–679
Even G, Lotker Z, Ron D, Smorodinsky S (2004) Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM J Comput 33(1):94–136
Fiala J, Fishkin AV, Fomin F (2004) On distance constrained labeling of disk graphs. Theoret Comp Sci 326(1–3):261–292
Fishkin AV (2004) Disk graphs: A short survey. In: Jansen K, Solis-Oba R (eds) Approximation and Online Algorithms, vol 2909 of Lecture Notes in Computer Science. Springer, Berlin, pp 260–264
Funke S, Kesselman A, Meyer U, Segal M (2006) A simple improved distributed algorithm for minimum CDS in unit disk graphs. ACM Trans Sensor Netw 2(3):444–453
Gao J, Zhang L (2005) Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM J Comput 35(1):151–169
Garey MR, Johnson DS (1979) Computers and Intractability: A Guide to the Theory of NP-completeness. WH Freeman, New York
Gräf A, Stumpf M, Weißenfels G (1998) On coloring unit disk graphs. Algorithmica 20(3):277–293
Gupta R, Walrand J (2004) Approximating maximal cliques in ad-hoc networks. In: Proceedings of the 15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '04), pp 365–369
Hale WK (1980) Frequency assignment: Theory and applications. Proc IEEE 68(12):1497–1514
Havel TF (1982) The Combinatorial Distance Geometry Approach to the Calculation of Molecular Conformation. PhD thesis, University of California, Berkeley
Havet F, Kang RJ, Sereni J-S (2005) Improper colouring of unit disk graphs. Electron Notes Discret Math 22:123–128
Hliněný P, Kratochvíl J (2001) Representing graphs by disks and balls (a survey of recognition-complexity results). Discret Math 229(1–3):101–124
Hochbaum DS (1983) Efficient bounds for the stable set, vertex cover and set packing problems. Discret Appl Math 6(3):243–254
Hochbaum DS, Maass W (1985) Approximation schemes for covering and packing problems in image processing and vlsi. J ACM 32(1):130–136
Hunt HB 3rd, Marathe MV, Radhakrishnan V, Ravi SS, Rosenkrantz DJ, Stearns RE (1998) Nc-approximation schemes for np- and pspace-hard problems for geometric graphs. J Algorithms 26(2):238–274
Krishna P, Chatterjee M, Vaidya NH, Pradhan DK (1995) A cluster-based approach for routing in ad-hoc networks. In: Proceedings of the USENIX Symposium on Location Independent and Mobile Computing, pp 1–8
Kuhn F, Moscibroda T, Wattenhofer R (2004) Unit disk graph approximation. In: Proceedings of the 2004 joint workshop on foundations of mobile computing (DIALM-POMC '04) ACM Press, New York, pp 17–23
Kuhn F, Zollinger A (2003) Ad-hoc networks beyond unit disk graphs. In: Proceedings of the 2003 joint workshop on Foundations of mobile computing (DIALM-POMC '03). ACM Press, New York, pp 69–78
Li J, Liang X, Selveraj H, Muthukumar V, Gewali LP (2005) A novel data structure for unit disk graphs. J Comb Math Comb Comput 54:145–156
Malesińska E, Piskorz S, Weißenfels G (1998) On the chromatic number of disk graphs. Networks 32(1):13–22
Marathe MV, Breu H, Hunt HB 3rd, Ravi S, Rosenkrantz DJ (1995) Simple heuristics for unit disk graphs. Networks 25:59–68
Marathe MV, Radhakrishnan V, Hunt HB 3rd, Ravi SS (1997) Hierarchically specified unit disk graphs. Theor Comp Sci 174(1–2):23–65
Matsui T (2000) Approximation algorithms for maximum independent set problems and fractional coloring problems on unit disk graphs. In: Akiyama J, Kano M, Urabe M (eds) Discrete and Computational Geometry, vol 1763 of Lecture Notes in Computer Science. Springer, pp 194–200
Min M, Du H, Jia X, Huang CX, Huang SC-H, Wu W (2006) Improving construction for connected dominating set with steiner tree in wireless sensor networks. J Global Optim 35(1):444–453
Miyamoto Y, Matsui T (2005) Multicoloring unit disk graphs on triangular lattice points. In: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms (SODA '05). Society for Industrial and Applied Mathematics, Philadelphia, pp 895–896
Nemhauser GL, Trotter LE (1975) Vertex packing: structural properties and algorithms. Math Program 8:232–248
Nieberg T, Hurink J, Kern W (2004) A robust PTAS for maximum weight independent sets in unit disk graphs. In: Hromkovic J, Nagl M, Westfechtel B (eds) Graph-Theoretic Concepts in Computer Science, vol 3353 of Lecture Notes in Computer Science. Springer, pp 214–221
Nieberg T, Hurink JL (2006) A PTAS for the minimum dominating set problem in unit disk graphs. In: Erlebach T, Persiano G (eds) Approximation and Online Algorithms, vol 3879 of Lecture Notes in Computer Science. Springer, pp 296–306
Nolan J (2004) Bisectored unit disk graphs. Networks 43(3):141–152
Peeters R (1991) On coloring j-unit sphere graphs. Technical report, Department of Economics, Tilburg University
Raghavan V, Spinrad J (2003) Robust algorithms for restricted domains. J Algorithm 48(1):160–172
Ramaswami R, Parhi KK (1989) Distributed scheduling of broadcasts in a radio network. In: Proceedings of the Eighth Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM '89) vol 2, pp 497–504
Stojmenovic I (ed) (2002) Handbook of Wireless Networks and Mobile Computing. Wiley InterScience
Szekeres G, Wilf HS (1968) An inequality for the chromatic number of a graph. J Comb Theory 4:1–3
van Leeuwen EJ (2005) Approximation algorithms for unit disk graphs. In: Kratsch D (ed) Graph-theoretic concepts in computer science, vol 3787 of Lecture Notes in Computer Science. Springer, Berlin, pp 351–361
Wan P-J, Alzoubi KM, Frieder O (2004) Distributed construction of connected dominating set in wireless ad hoc networks. Mobile Netw Appl 9(2):141–149
Wang DW, Kuo Y-S (1988) A study on two geometric location problems. Informat Proc Let 28(6):281–286
Wu W, Du H, Jia X, Li Y, Huang SC-H (2006) Minimum connected dominating sets and maximal independent sets in unit disk graphs. Theoret Comp Sci 352(1):1–7
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag
About this entry
Cite this entry
Balasundaram, B., Butenko, S. (2008). Optimization Problems in Unit-Disk Graphs . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_486
Download citation
DOI: https://doi.org/10.1007/978-0-387-74759-0_486
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-74758-3
Online ISBN: 978-0-387-74759-0
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering