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

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 a graph G in which the edges might have different lengths, 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. 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.