Discrete Applied Mathematics
Resolvability in graphs and the metric dimension of a graph
Discrete Applied Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
On the Approximability of the Minimum Test Collection Problem
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
On Metric Generators of Graphs
Mathematics of Operations Research
Tight approximability results for test set problems in bioinformatics
Journal of Computer and System Sciences
On the Metric Dimension of Cartesian Products of Graphs
SIAM Journal on Discrete Mathematics
Approximation hardness of dominating set problems in bounded degree graphs
Information and Computation
Linear time approximation schemes for the Gale-Berlekamp game and related minimization problems
Proceedings of the forty-first annual ACM symposium on Theory of computing
Network Discovery and Verification
IEEE Journal on Selected Areas in Communications
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We study the approximation complexity of the Metric Dimension problem in bounded degree, dense as well as in general graphs. For the general case, we prove that the Metric Dimension problem is not approximable within (1-ε)lnn for any ε0, unless NP⊆DTIME(nloglogn), and we give an approximation algorithm which matches the lower bound. Even for bounded degree instances it is APX-hard to determine (compute) the exact value of the metric dimension which we prove by constructing an approximation preserving reduction from the bounded degree Vertex Cover problem. The special case, in which the underlying graph is superdense turns out to be APX-complete. In particular, we present a greedy constant factor approximation algorithm for these kind of instances and construct a approximation preserving reduction from the bounded degree Dominating Set problem. We also provide first explicit approximation lower bounds for the Metric Dimension problem restricted to dense and bounded degree graphs.