Journal of Algorithms
Batch processing with interval graph compatibilities between tasks
Discrete Applied Mathematics
Approximation schemes for wireless networks
ACM Transactions on Algorithms (TALG)
Partition into cliques for cubic graphs: Planar case, complexity and approximation
Discrete Applied Mathematics
Approximation Algorithms for Domatic Partitions of Unit Disk Graphs
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Good quality virtual realization of unit ball graphs
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Approximation algorithms for intersection graphs
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Shifting strategy for geometric graphs without geometry
Journal of Combinatorial Optimization
| Hi-index | 0.00 |
We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the best known ratio of 3. Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths and ε0 that either (i) computes a clique partition, or (ii) produces a certificate proving that the graph is not a UDG; if the graph is a UDG, then our partition is guaranteed to be within (1+ε) ratio of the optimum; however, if the graph is not a UDG, it either computes a clique partition, or detects that the graph is not a UDG. Noting that recognition of UDG's is NP-hard even with edge lengths, this is a significant weakening of the input model. We consider a weighted version of the problem on vertex weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas crucial in obtaining a PTAS breakdown. Nevertheless, the weighted version admits a (2+ε)-approximation algorithm even when the graph is expressed, say, as an adjacency matrix. This is an improvement on the best known 8-approximation for the unweighted case for UDGs expressed in standard form.