# Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs

## Abstract

We consider two optimization problems in planar graphs. In Maximum Weight Independent Set of Objects we are given a graph *G* and a family \({\mathcal {D}}\) of *objects*, each being a connected subgraph of *G* with a prescribed weight, and the task is to find a maximum-weight subfamily of \({\mathcal {D}}\) consisting of pairwise disjoint objects. In Minimum Weight Distance Set Cover we are given a graph *G* in which the edges might have different lengths, two sets \({\mathcal {D}},{\mathcal {C}}\) of vertices of *G*, where vertices of \({\mathcal {D}}\) have prescribed weights, and a nonnegative radius *r*.
The task is to find a minimum-weight subset of \({\mathcal {D}}\) such that every vertex of \({\mathcal {C}}\) is at distance at most *r* from some selected vertex. Via simple reductions, these two problems generalize a number of geometric optimization tasks, notably Maximum Weight Independent Set for polygons in the plane and Weighted Geometric Set Cover for unit disks and unit squares. We present *quasi-polynomial time approximation schemes* (QPTASs) for both of the above problems in planar graphs: given an accuracy parameter \(\epsilon >0\) we can compute a solution whose weight is within multiplicative factor of \((1+\epsilon )\) from the optimum in time \(2^{{\mathrm {poly}}(1/\epsilon ,\log |{\mathcal {D}}|)}\cdot n^{{\mathcal {O}}(1)}\), where *n* is the number of vertices of the input graph. We note that a QPTAS for Maximum Weight Independent Set of Objects would follow from existing work. However, our main contribution is to provide a unified framework that works for both problems in both a planar and geometric setting and to transfer the techniques used for recursive approximation schemes for geometric problems due to Adamaszek and Wiese (in Proceedings of the FOCS 2013, IEEE, 2013; in Proceedings of the SODA 2014, SIAM, 2014) and Har-Peled and Sariel (in Proceedings of the SOCG 2014, SIAM, 2014) to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods as a phenomenon in planar graphs.

## Keywords

Approximation schemes Planar graphs Independent set of objects Geometric set cover## Notes

### Acknowledgements

We thank Dániel Marx for insightful discussions on the approach to optimization problems in geometric and planar settings via Voronoi diagrams.

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