Quasi-polynomial time approximation schemes for packing and covering problems in planar graphs

Michał Pilipczuk, Erik Jan Van Leeuwen, Andreas Wiese

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


We consider two optimization problems in planar graphs. In MAXIMUM WEIGHT INDEPENDENT SET OF OBJECTS we are given a graph G and a family 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 D consisting of pairwise disjoint objects. In MINIMUM WEIGHT DISTANCE SET COVER we are given an edge-weighted graph G, two sets D, C of vertices of G, where vertices of D have prescribed weights, and a nonnegative radius r. The task is to find a minimum-weight subset of D such that every vertex of 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 ϵ > 0 we can compute a solution whose weight is within multiplicative factor of (1 + ϵ) from the optimum in time 2poly(1/ϵ,log|D|) · no(1), where n is the number of vertices of the input graph. Our main technical contribution is to transfer the techniques used for recursive approximation schemes for geometric problems due to Adamaszek, Har-Peled, and Wiese [1, 2, 4] to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods.

Original languageEnglish
Title of host publication26th European Symposium on Algorithms, ESA 2018
EditorsHannah Bast, Grzegorz Herman, Yossi Azar
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770811
StatePublished - 1 Aug 2018
Externally publishedYes
Event26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland
Duration: 20 Aug 201822 Aug 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference26th European Symposium on Algorithms, ESA 2018


  • Planar graphs
  • Voronoi diagram


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