An efficient polynomial-time approximation scheme for Steiner forest in planar graphs

In the Steiner forest problem, we are given an undirected graph G with edge-lengths and a set D of pairs (s i , t i ) of vertices. The pairs are called demands, and the vertices that appear in demands are called terminals. The goal is to find a minimum-length forest F that, for every demand (s i , t i ), contains a path in F from s i to t i . This problem generalizes the Steiner tree problem in networks.

There is a polynomial-time 2-approximation algorithm [1], but the problem doesn’t have an approximation scheme unless P=NP [3,16]. However, for restricted input classes, polynomial-time approximation schemes have been found. For the case where the vertices are the points on the plane and edgelengths are Euclidean distances, Borradaile, Klein, and Mathieu [5] give an approximation scheme that can be implemented in O(n log n) time where n is the number of terminals.

For planar graphs, Bateni, Hajiaghayi, and Marx [2] give a polynomial-time approximation scheme. The running time, however, for obtaining a (1 + )-approximate solution has the form n -c . The degree of the polynomial grows as gets smaller. An efficient polynomial-time approximation scheme is an approximation scheme whose running time has the form O(f ( )n c ) for some function f and some constant c independent of . Thus the approximation scheme of Bateni, Hajiaghayi, and Marx is not an efficient PTAS in this sense. Our main result is an efficient PTAS:

Theorem 1.1. For planar Steiner forest, there is an approximation scheme whose running time is O(n log 3 n). 2 Techniques

Tree-decomposition and branch-decomposition are ways to map pieces of the graph to nodes of a tree so that, loosely speaking, pieces have small overlap. We formally define branch-decomposition. The general paradigm is to reduce the problem to graphs of bounded tree-or branch-width.

A carving of a ground set is a maximal family C of mutually noncrossing subsets of the ground set. In this paper, we refer to the sets in C as clusters. The Hasse diagram of the inclusion partial order on the clusters is a tree in which each node has zero or two children.

A branch-decomposition of a graph G is a carving C of the edges of G. The boundary ∂C of a cluster C is the set of vertices v such that C contains a proper nonempty subset of the edges incident to v. The width of a branch-decomposition C of G is max{|∂C| : C ∈ C}. The branchwidth of G is the minimum width over all branch-decompositions of G.

Treewidth (not defined here) is within a constant factor of branchwidth. Graphs of bounded treewidth and branchwidth are tree-like, and many problems can be solved exactly in linear time on such graphs. As we will see, this is not the case for Steiner forest.

The approximation scheme of Bateni, Hajiaghayi, and Marx fits into the framework of Klein [13], which consists of the following steps:

spanner Find a subgraph G 1 (called the Steiner tree spanner) of the input graph G 0 such that, for constants c and d,

, where OPT(G 0 ) is the total length of the edges used by an optimum solution, and 2. OPT(G 1 ) ≤ (1 + d ) OPT(G 0 ).

thinning Partition the edges into p subsets such that the contraction of any subset yields a graph of branchwidth O(p). Let G 2 be the graph obtained from G 1 by contracting the subset S having the smallest total length.

dynamic programming Find an (approximately) optimal solution in G 2 .

lifting Lift the solution in G 2 to a solution in G 0 by uncontracting edges of S and adding them to the solution as needed.

This presentation of the framework differs from the original in [13] in that, in the original, the dynamicprogramming step finds an optimal solution. Suppose the solution in the dynamic-programming step has length at most (1 + c ) OPT(G 2 ). Since contraction preserves connectivity, OPT(G 2 ) ≤ OPT(G 1 ). By the spanner property, OPT(G 1 ) ≤ (1 + d ) OPT(G 0 ). The addition of some edges from S in the lifting step increases the length by at most length(G 1 )/p. We choose p = c/ so the additional length is at most OPT(G 0 ). Hence the length of the final solution is ((

The dynamic-programming step is straightforward and takes linear time; the construction is given in [13]. (See also [9,8]. (It has been generalized to bounded-genus graphs [9] and, more recently, to minor-excluded graphs [8].) The lifting step is problem-dependent but straightforward for the problems (such as TSP, Steiner tree, and Steiner forest) to which the framework has been successfully applied. The spanner and dynamic-programming steps are problem-dependent. It is in those steps that Bateni, Hajiaghayi, and Marx [2] introduced new techniques, and it is there that our improvements go.

The spanner step of Bateni et al. built on the same step in the Steiner tree PTAS of Borradaile, Klein, and Mathieu [4,6]. The proof of the latter can be adapted to show the following. For an instance of Steiner forest consisting of a graph G and a set D of terminal pairs, let OPT(G, D) denote the optim

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