2026-07-25T14:06:59Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9528 2024-03-06T10:43:58Z ddc:004 open_access

Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs Pilipczuk, Michal van Leeuwen, Erik Jan Wiese, Andreas QPTAS planar graphs Voronoi diagram 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 epsilon>0 we can compute a solution whose weight is within multiplicative factor of (1+epsilon) from the optimum in time 2^{poly(1/epsilon,log |D|)}* n^{O(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 [Adamaszek and Wiese, 2013; Adamaszek and Wiese, 2014; Sariel Har-Peled, 2014] to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Michal Pilipczuk and Erik Jan van Leeuwen and Andreas Wiese 2018 Is Part Of LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2018.65 urn:nbn:de:0030-drops-95282 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.65 eng https://creativecommons.org/licenses/by/3.0/legalcode