We show that, for any ∊ > 0, there is a deterministic embedding of edge-weighted planar graphs of diameter D into bounded-treewidth graphs. The embedding has additive error ∊D. We use this construction to obtain the first efficient bicriteria approximation schemes for weighted planar graphs addressing k-Center (equivalently d-Domination), and a metric generalization of independent set, d-independent SET. The approximation schemes employ a metric generalization of Baker's framework that is based on our embedding result.

We present an implementation of a linear-time approximation scheme for the traveling salesman problem on planar graphs with edge weights. We observe that the theoretical algorithm involves constants that are too large for practical use. Our implementation, which is not subject to the theoretical algorithm's guarantee, can quickly find good tours in very large planar graphs.

We provide an implementation of an algorithm that, given a triangulated planar graph with m edges, returns a simple cycle that is a 2/3-balanced separator consisting of at most edges. An efficient construction of a short and balanced separator that forms a simple cycle is essential in numerous planar graph algorithms, e.g., for computing shortest paths, minimum cuts, or maximum flows. To the best of our knowledge, this is the first implementation of such a cycle separator algorithm with a worst-case guarantee on the cycle length. We evaluate the performance of our algorithm and compare it to the planar separator algorithms recently studied by Holzer et al. [ESA 2005, ACM Journal of Experimental Algorithms 2009]. Out of these algorithms, only the Fundamental Cycle Separator (FCS) produces a simple cycle separator. However, FCS does not provide a worst-case size guarantee. We demonstrate that (i) our algorithm is competitive across all test cases in terms of running time, balance and cycle length, (ii) it provides worst-case guarantees on the cycle length, significantly outperforming FCS on some instances, and (iii) it scales to large graphs.