Approximation Schemes for Minimum 2-Connected Spanning Subgraphs in Weighted Planar Graphs

Abstract

Abstract. We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edge-weighted undirected planar graphs that are resistant to edge or vertex removal. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs, which augments any connected subgraph A of a weighted planar graph G to a (1 + ε)-spanner of G with total weight bounded by weight(A)/ε. From this we derive quasi-polynomial time approximation schemes for the problems of finding the minimum-weight 2-edge-connected or biconnected spanning subgraph in planar graphs. We also design approximation schemes for the minimum-weight 1-2-connectivity problem, which is the variant of the survivable network design problem where vertices have non-uniform (1 or 2) connectivity constraints. Prior to our work, for all these problems no polynomial or quasi-polynomial time algorithms were known to achieve an approximation ratio better than 2.

Approximation algorithms via contraction decomposition

“…Our result can also be viewed as a generalization of these results. Furthermore, we obtain a PTAS for minimum-weight c-edge-connected submultigraph 1 in bounded-genus graphs, for any constant c ≥ 2, which generalizes and improves previous algorithms for c = 2 on planar graphs [BCGZ05,CGSZ04]. We also extend our results in Section 4 toward general H-minor-free graphs, where significant additional difficulties arise, and we show how to solve all but one.…”

A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

“…This paper proved a spanner result for minor-excluded graphs. Berger, Czumaj, Grigni, and Zhao ( [7], building on [16]) give a PTAS for the problem of finding a minimum-weight 2-edge-connected spanning multi-subgraph 1 of an edge-weighted planar graph, and a quasipolynomial approximation scheme for finding a 1 Duplicate edges of the input graph are allowed in the solution.…”

“…6 Finding a planar embedding for a planar graph is a well-studied problem, and linear-time algorithms are known. 7 , so we assume throughout this paper that every planar graph comes equipped with an embedding. It follows from Euler's formula that an n-node planar graph with no parallel edges has O(n) edges.…”

“…However, it is known (see, .e.g, [35]) more generally that for any graph embedded on a closed, orientable surface, the corresponding combinatorial embedding determines the geometric embedding up to homeomorphism. 7 The first was due to Hopcroft and Tarjan [29]. See [12] for a discussion of later work.…”

The Two-Edge Connectivity Survivable Network Problem in Planar Graphs

“…Berger et al [2] and Berger and Grigni [3] gave PTASes for the unweighted and weighted cases, respectively, in planar graphs. In both cases, the degrees of the polynomial depend on the desired precision .…”