## Simple heuristics for unit disk graphs (1995)

Venue: | NETWORKS |

Citations: | 128 - 6 self |

### BibTeX

@ARTICLE{Marathe95simpleheuristics,

author = {M. V. Marathe and H. Breu and H. B. Hunt III and S. S. Ravi and D. J. Rosenkrantz},

title = {Simple heuristics for unit disk graphs},

journal = {NETWORKS},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.

### Citations

11403 | Computers and Intractability: A Guide to the Theory of NP-Completeness - Garey, Johnson - 1979 |

1175 | Algorithmic Graph Theory and Perfect Graphs - Golumbic - 2004 |

718 | Proof verification and hardness of approximation problems - Arora, Lund, et al. - 1998 |

700 |
Approximation algorithms for combinatorial problems
- Johnson
- 1974
(Show Context)
Citation Context ...he minimum total domination and the minimum connected domination problems. (For general graphs, the minimum dominating set problem can be approximated to within a factor O(log n) of the optimal value =-=[Jo74]-=-; it is also known that no polynomial time algorithm can provide a performance guarantee of o(log n), unless every problem in NP can be solved in deterministic time O(n poly log n ) [LY93]. For genera... |

426 |
Sphere Packings, Lattices and Groups
- H, Conway, et al.
- 1998
(Show Context)
Citation Context ...f of this property relies on a geometric observation concerning packing of unit disks in the plane. (The problems of Packing and Covering have been of interest to researchers for quite some time. See =-=[CS88]-=- for more on this subject.) Lemma 3.1 Let C be a circle of radius r and let S be a set of circles of radius r such that every circle in S intersects C and no two circles in S intersect each other. The... |

390 |
On the hardness of approximating minimization problems
- Lund, Yannakakis
- 1994
(Show Context)
Citation Context ... disk graphs. (For general graphs, unless P = NP, there is an ffl ? 0 such that no polynomial time algorithm for the off-line minimum coloring problem can provide a performance guarantee of O(n ffl ) =-=[LY93]-=-. On-line coloring algorithms with constant competitive ratios have been obtained previously for several other classes of graphs [HMR+94, Sl94, MHR93, Ki91, GL88, Sl87, KT81]. It is also known that no... |

311 |
Approximation algorithms for NPcomplete problems on planar graphs
- Baker
- 1983
(Show Context)
Citation Context ...nput, it is possible to obtain polynomial time approximation schemes for maximum independent set, minimum vertex cover and minimum dominating set problems. These results are obtained using ideas from =-=[Ba83]-=- and [HM85]. Our heuristics for the maximum independent set problem for unit disk graphs and circle intersection graphs also provide a slight generalization of a result in [Ho83]. In that paper, Hochb... |

257 |
Frequency assignment: theory and applications
- Hale
(Show Context)
Citation Context ...isk graphs. The minimum dominating set problem corresponds to selecting a minimum number of transmitters so that all the other stations are within the range of at least one of the chosen transmitters =-=[Ha80]-=-. As observed in [CCJ90], unit disk graphs are not perfect graphs. (An odd cycle of length five or more is a unit disk graph but not perfect.) Also, unit disk graphs are not planar. (A clique of size ... |

251 |
Unit disk graphs
- Clark, Colbourn, et al.
- 1990
(Show Context)
Citation Context ... dominating set problem corresponds to selecting a minimum number of transmitters so that all the other stations are within the range of at least one of the chosen transmitters [Ha80]. As observed in =-=[CCJ90]-=-, unit disk graphs are not perfect graphs. (An odd cycle of length five or more is a unit disk graph but not perfect.) Also, unit disk graphs are not planar. (A clique of size five or more is a unit d... |

162 |
A local-ratio theorem for approximating the weighted vertex cover problem
- Bar-Yehuda, Even
- 1985
(Show Context)
Citation Context ...or class S of maximum cardinality and return the set V 1 [P [ (Q \Gamma S) as the approximate vertex cover. To determine the performance guarantee of this heuristic, we need the following result from =-=[BE85]-=-. 6 1. V 1 = ;; V 0 = V . 2. while G(V 0 ) contains triangles do 3. begin (a) Let X ` V 0 be such that G(X) is a triangle. (b) V 1 = V 1 [ X. (c) V 0 = V 0 \Gamma X. 4. end 5. Obtain the sets P and Q ... |

122 | Vertex packings: structural properties and algorithms - Nemhauser, Trotter - 1975 |

88 | Unit disk graph recognition is NP-hard
- Breu, Kirkpatrick
- 1998
(Show Context)
Citation Context ...n of a given unit disk graph in the intersection model into a description in the proximity model and vice versa in linear time. The recognition problem for unit disk graphs was shown to be NP-hard in =-=[BK93]-=-. As already mentioned, none of our heuristics requires a geometric representation of a unit disk graph as part of the input. An approximation algorithm for an optimization problem \Pi provides a perf... |

81 |
Efficient bounds for the stable set, vertex cover and set packing problems
- Hochbaum
- 1983
(Show Context)
Citation Context ...y optimum vertex cover V C (G) of G, jV C (G)jsjP j + jQj=2. Moreover, the sets P and Q can be found in polynomial time. 2 The second result that is used in the heuristic is a theorem due to Hochbaum =-=[Ho83]-=-. This theorem points out how a near-optimal vertex cover can be obtained in some cases starting from an NT decomposition. We have included the proof from [Ho83] since our heuristic uses the method gi... |

45 | An extremal problem in recursive combinatorics - Kierstead, Trotter - 1981 |

43 | On-line and first fit colorings of graphs - Gyárfás, Lehel - 1988 |

43 | Coloring inductive graphs on-line
- Irani
- 1994
(Show Context)
Citation Context ...aphs. We note that unless P 6= NP, there is no PTAS for the minimum coloring problem for unit disk graphs since the 3-coloring problem for unit disk graphs is NP-hard [CCJ90]. We also give an on-line =-=[Ira90]-=- coloring heuristic with a competitive ratio of 6 for unit disk graphs. (For general graphs, unless P = NP, there is an ffl ? 0 such that no polynomial time algorithm for the off-line minimum coloring... |

22 | Polynomial Time Approximation Algorithm for Dynamic Storage Allocation - Kierstead - 1991 |

18 |
Approximation Schemes for Covering and Packing
- Hochbaum, Maass
- 1985
(Show Context)
Citation Context ... possible to obtain polynomial time approximation schemes for maximum independent set, minimum vertex cover and minimum dominating set problems. These results are obtained using ideas from [Ba83] and =-=[HM85]-=-. Our heuristics for the maximum independent set problem for unit disk graphs and circle intersection graphs also provide a slight generalization of a result in [Ho83]. In that paper, Hochbaum observe... |

16 | On approximating the minimum independent dominating set - Irving - 1991 |

14 | A unified approach to approximation schemes for NP- and PSPACE-hard problems for geometric graphs - Marathe, Radhakrishnan, et al. - 1994 |

13 |
Polynomially bounded minimization problems which are hard to approximate
- Kann
- 1993
(Show Context)
Citation Context ... log n ) [LY93]. For general graphs, it is also known that there is an ffl ? 0 such that the minimum independent domination problem cannot be approximated to within a factor of O(n ffl ) unless P = NP=-=[Kan93, Ir91]-=-.) For intersection graphs of circles of arbitrary radii in the plane, we present heuristics with performance guarantees of 5/3, 6 and 5 respectively for vertex cover, off-line coloring and independen... |

12 |
On approximating a vertex cover for planar graphs
- Bar-Yehuda, Even
- 1982
(Show Context)
Citation Context ...heoretic problems. We start with a heuristic for the minimum vertex cover problem. 4.1 Minimum Vertex Cover The heuristic given here is essentially the same as the one given in in Bar-Yehuda and Even =-=[BE82]-=- for planar graphs but the analysis which leads to a performance guarantee of 1.5 is different. It is interesting that an algorithm which provides a performance guarantee of 1.5 for planar graphs prov... |

10 | Efficient approximation algorithms for domatic partition and on-line coloring of circular arc graphs - Marathe, Hunt, et al. - 1996 |

9 |
Graph Theory and Its Applications to
- Roberts
- 1978
(Show Context)
Citation Context ...ported by NSF Grant CCR-90-06396. 1 Introduction, motivation and summary of results Intersection graphs of geometric objects have been both widely studied and used to model many problems in real life =-=[Ro78]-=-. In this paper we consider intersection graphs of regular polygons, emphasizing intersection graphs of unit disks. A graph is a unit disk graph if and only if its vertices can be put in one to one co... |

8 | A Coloring Algorithm for Interval Graphs - Slusarek - 1989 |

4 | C900- An advanced mobile radio telephone system with optimum frequency utilization - Kammerlander - 1984 |

1 | Proof Verification and Hardness of Approximation Problems,” Proc. 33rd Annual Symp. on Foundations of Computer Science (FOCS - Arora, Lund, et al. - 1992 |

1 | An Extremal Problem in Recursive - Kierstead, Trotter - 1981 |