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

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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.