Minimum Weight 2-Edge-Connected Spanning Subgraphs in Planar Graphs

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1 Minimum Weight 2-Edge-Connected Spanning Subgraphs in Planar Graphs André Berger 1 and Michelangelo Grigni 2 1 Technische Universität Berlin, Institut für Mathematik, Berlin, Germany. berger@math.tu-berlin.de 2 Department of Mathematics and Computer Science, Emory University, Atlanta GA 30322, USA. mic@mathcs.emory.edu Abstract. We present a linear time algorithm exactly solving the 2-edge connected spanning subgraph (2-ECSS) problem in a graph of bounded treewidth. Using this with Klein s diameter reduction technique [15], we find a linear time PTAS for the problem in unweighted planar graphs, and the first PTAS for the problem in weighted planar graphs. 1 Introduction A graph G = (V, E) is 2-edge-connected (2-EC) if G e is connected for every edge e E. Given G and an non-negative edge weight vector w R E +, the 2-edge-connected spanning subgraph (2-ECSS) problem is to find a 2-ECSS of G of minimum weight. We consider PTAS (polytime approximation scheme) results. For our purposes, a PTAS is an algorithm taking an instance (G, w) and ε > 0 as inputs, running in polynomial time for each fixed ε, and finding a solution with weight at most 1 + ε times optimal. Like metric TSP, the 2-ECSS problem is NP-hard even when restricted to unweighted planar graphs, and MAX-SNP-hard for bounded degree graphs [9]. In particular 2-ECSS is NP-hard to approximate within 1573/1572 on graphs of maximum degree 3 [7]. For unweighted graphs the best known polytime algorithm has approximation ratio 5/4 [12], and for weighted graphs the best known ratio is 2 [13, 14]. In Section 2 we present an algorithm solving the 2-ECSS problem exactly in linear time for weighted graphs of bounded treewidth. Graphs of bounded treewidth have particular algorithmic interest, since they adapt very well to dynamic programming techniques. Moreover, graphs of small treewidth are also related to fixed parameter algorithms and the bidimensionality theory [10].

2 A PTAS is known for 2-ECSS in complete geometric graphs of low dimension [9], and for unweighted planar graphs [8]. The latter PTAS runs in time n O(1/ε). We would prefer an efficient PTAS (EPTAS); that is, running in time f(ε) n c, where c is independent of ε. Recent examples of improving a PTAS to an EPTAS are the Euclidean TSP (n O(1/ε) [1] to O(ε O(ε) + n log n) [18]) and the weighted planar graph TSP (n O(1/ε2) [2] to 2 O(1/ε2) n [15]). In the same vein, we present a linear time EPTAS for the 2-ECSS problem in unweighted planar graphs in Section 3. For this we use the diameter reduction approach which Klein [15] applied to the metric TSP in weighted planar graphs. For the 2-ECSS problem in weighted planar graphs, the best previously known polytime approximation ratio is 2, achieved by the algorithm for general weighted graphs [13]. There is also an approximation scheme running in time n O(log n log(1/ε)/ε) [3], a quasi-polynomial. In Section 4 we present the first PTAS for this problem. Again we use ideas developed for the metric TSP; however, our situation is more complicated because it is insufficient to replace the input with a spanner (a subgraph which approximates the original metric). Instead, at each step of the dynamic program we must consider using a small number edges which are inessential to the metric. The bounding of this number (Lemma 10) is a key novelty in our approach. 2 An Exact Algorithm for Graphs of Bounded Treewidth In this section we design an exact algorithm for the 2-ECSS problem in weighted graphs of bounded treewidth. We first review basic notions for treewidth. Definition 1. Let G = (V, E) be a graph. A tree decomposition of G is a pair T D = ({X i : i I}, T = (I, F )), where {X i : i I} is a family of subsets of V and T is a tree, satisfying: (i) i I X i = V, (ii) for any edge uv E there exists an i I such that u, v X i, and (iii) for any v V, T [{i I : v X i }] is a subtree of T. The width of the tree decomposition is max i X i 1. The treewidth tw(g) of G is the minimum width of a tree decompositions of G. Finding the treewidth of a graph is NP-hard. However, for a given integer k, one can find a tree decomposition of a graph G of width at most k or decide that tw(g) > k in time O(f(k) n) [5]. In [19] it was 2

3 shown that f(k) = 2 O(k3), which makes the algorithm in [5] impractical for large values of k. For algorithmic purposes a especially simple tree decomposition is very useful. Definition 2. A tree decomposition ({X i : i I}, T ) of a graph G = (V, E) rooted at some r I is called nice, if the following conditions are satisfied: 1. Every node of the tree T has at most two children. 2. If a node i I has two children j and k, then X i = X j = X k (in this case i is called a JOIN NODE). 3. If a node i has one child j, then either (a) X i = X j + 1 and X j X i (in this case i is called an INSERT NODE), or (b) X i = X j 1 and X i X j (in this case i is called a FORGET NODE). Lemma 1 ([17, Lemma ]). Given a tree decomposition of a graph G = (V, E) that has width k and O(n) nodes, where n = V, we can find a nice tree decomposition of G that also has width k and O(n) nodes in time O(kn). From now on we assume G = (V, E) is a weighted graph with weight vector w R E +, and we let n = V. Furthermore, let ({X i : i I}, (I, F )) be a nice tree decomposition of G of width k and order I = O(n), rooted at r I. For any i I we define V i to be the set of vertices contained in X i and all X j, for which j is a descendant of i in T. Moreover, we let G i = G[V i ]. Note that V r = V and G r = G. We now consider the problem of finding a minimum weight 2-ECSS of G using this tree decomposition. For each i I we want to solve the following subproblem. Given a set of edges S E(X i ) and a graph H = (V H, E H ) with V H V = X i, find a subgraph G(i, S, H) of G i of minimum weight, such that E(G(i, S, H)) E(X i ) = and such that G(i, S, H) S H is 2-edge-connected. We let w(i, S, H) denote the weight of the graph G(i, S, H). In particular, the graph G(r, S, H ) S, where S minimizes w(r, S, H )+ w(s ) among all choices of S E(X r ) (H = (X r, )), is a minimum weight 2-ECSS of G. To determine w(r, S, H ) for S E(X r ), we will work up the tree decomposition from the leaves to the root solving the aforementioned subproblems. The graph H in a particular subproblem describes how the vertices in a set X i may be connected outside the graph 3

4 G i. We therefore refer to H as an external type of G i. We can think of H as a subgraph of G[(V \ V i ) X i ], that is part of a solution to the 2-ECSS problem on G, complementing the solution on G i. In general, there is an exponential number of subgraphs of G[(V \V i ) X i ] that can appear as an external type of the graph G i. However, as we will show, the number of external types H of the graph G i, which must be considered for a subproblem at the vertex i I of the tree decomposition, can be bounded only in terms of X i. Given a graph H = (V H, E H ), and a subset X V H, there is an algorithm [8] returning a simplified graph ext(h, X), which has the same 2-edge-connected properties when used as an external type for another graph, which shares only vertices in X with H. The detailed properties ext(h, X) are given in Lemma 2 and Corollary 1. Lemma 2. Let H = (V H, E H ) be a graph and X V H. Moreover, let H = ext(h, X). Then, if H is another graph with V H V H = X, then H H is 2-EC if and only if H H is 2-EC. Furthermore, V H 3 X. Lemma 2 implies that w(i, S, H) = G(i, S, ext(h, X i )), and that ext(h, X i ) does not have too many vertices. To make the computation efficient, we need that the number of possible external types that can appear during the algorithm is small. We let T ext (X) = {ext(h, X) : H = (V H, E H ), X V H } denote the set of all possible graphs ext(h, X) for a fixed set of vertices X. The following corollary gives us the bound on the number of possible external types. Corollary 1. For any set X we have T ext (X) = 2 O( X log X ). The algorithm for the 2-ECSS problem on G will compute the graphs G(i, S, H) and the values w(i, S, H) for every possible choice of i I, S E(X i ) and H T ext (X i ), working up from the leaves of T to the root r. The following four lemmata will help us prove the correctness of our algorithm. Lemma 3. Let i I be a leaf in T. Moreover let S E(X i ) and let H T ext (X i ). Then G(i, S, H) = and w(i, S, H) = 0. Proof. This is true by the definition of G(i, S, H). Lemma 4. Let i I be a join node with children j 1 and j 2, S E(X i ) and let H T ext (X i ). Then [ ] w(i, S, H) = min w(j 1, S, H 1 )+w(j 2, S, ext(h G(j 1, S, H 1 ), X j2 )). H 1 T ext(x j1 ) 4

5 Proof. The idea of the proof is as follows. If i is a join node with children j 1 and j 2, then the subgraph G(i, S, H) of G i achieving the minimum weight w(i, S, H) is a disjoint union of two subgraphs G 1 and G 2 of G j1 and G j2, respectively.we show that then G 1 and G 2 must be solutions to one of the previously computed subproblems at the nodes j 1 and j 2, related by the formula in the lemma. Let H 1 T ext (X j1 ) be the graph that achieves the minimum value W on the right hand side of the equality. Furthermore, let G 1 = G(j 1, S, H 1 ) and G 2 = G(j 2, S, ext(h G 1, X j2 )). Now G 2 S (H G 1 ) is 2-EC and thus G 1 G 2 is a feasible solution to the subproblem (i, S, H). Note that G 1 and G 2 are edge-disjoint, since X j1 = X j2 and G 1 and G 2 both do not contain any edges between vertices in X j1. Hence w(i, S, H) w(g 1 G 2 ) = w(g 1 ) + w(g 2 ) = W. To prove the reverse inequality, let H opt be a subgraph of G i with weight w(h opt ) = w(i, S, H) such that H opt S H is 2-EC. Let G 1 = H opt [V j1 \ X j1 ] and G 2 = H opt [V j2 \ X j2 ]. Also let H 1 = ext(h G 2, X j1 ). Since H opt = G 1 G 2, we have that G 1 S H 1 is 2-EC and thus w(j 1, S, H 1 ) w(g 1 ). Next, we let G 1 = G(j 1, S, H 1 ), i.e. G 1 S (H G 2) is 2-EC. Hence G 2 is a feasible solution to the subproblem (j 2, S, ext(h G 1, X j 2 )) and therefore w(j 2, S, ext(h G 1, X j 2 )) w(g 2 ). We finally get W w(j 1, S, H 1 )+w(j 2, S, ext(h G 1, X j 2 )) w(g 1 )+ w(g 2 ) = w(h opt ) = w(i, S, H). The proofs of the following two lemmas are similar to that of Lemma 4 and we omit them in this abstract. Lemma 5. Let i I be a insert node with child j, v V \ X j such that X i = X j {v}, S E[X i ] and let H T ext (X i ). Then w(i, S, H) = w(j, S E(X j ), ext(h (S E(v, X j )), X j )). Lemma 6. Let i I be a forget node with child j, v V such that X i = X j \ {v}, S E(X i ) and let H T ext (X i ). Then [ ] w(i, S, H) = min w(j, S S 1, H 1 = (V H {v}, E H )) + w(s 1 ). S 1 E(v,X j ) Using Lemmas 3-6, the values w(i, S, H) as well as the graphs G(i, S, H) can be computed by using the previously computed values at the children of i in the tree decomposition. We now analyze the running time of the algorithm. The tree decomposition has O(n) vertices and for each i I we have 2 k2 k O(k) subproblems, 5

6 since there are at most k 2 edges in E(X i ) and at most k O(k) external types for X i (Lemma 1). The most time consuming type of subproblem to solve is the one described in Lemma 4, which takes 2 O(k log k) time. Thus the total running time of the algorithm is O(2 O(k2) n). Theorem 1. The 2-ECSS problem in a weighted graph of with n vertices and a tree decomposition of width k can be solved exactly in O(2 O(k2) n) time. The constant in the running time of the algorithm can be improved using the following observation. Since G(i, S, H) is a graph with no edges in E(X i ), we have that G(i, S, H) = G(i, S, H) for all S that we can obtain from S by removing chords. The detailed arguments for this claim are identical to those in the proof of Lemma 2. Therefore, during the whole algorithm, we only have to consider edge sets S E(X i ) for some subproblem (i, S, H), which have no chords. We state the following lemma without proof. Lemma 7. Let G = (V, E) be a graph with no chords. Then E 2 V 2. For a given i I, therefore, we only need to consider at most (k 2 ) k = 2 O(k log k) different subsets S E(X i ) and thus we have the following improved running time. Theorem 2. Let a 2-EC graph G = (V, E) be given with a tree decomposition of width k, and let w R E + be a non-negative weight vector. Then, in time O(2 O(k log k) n), one can find a 2-ECSS of G of minimum weight. 3 A Linear-Time PTAS for Unweighted Planar Graphs We will now apply the methods from the previous section to obtain a linear time PTAS for the 2-ECSS problem in unweighted planar graphs. (More generally, the approach here would also work in weighted planar graphs whenever w(g) = O(OPT).) As in [8], we apply cycle contractions. Suppose G = (V, E) is a graph, C is a simple cycle in G, and G/C is the result of contracting C to a vertex (we remove any self-loops but leave parallel edges). First observe that G is 2-EC iff G/C is 2-EC. If H is a 2-ECSS in G, then its image (H C)/C is a 2-ECSS in G/C. Conversely, if H is a 2-ECSS in G/C, we can lift it back to a 2-ECSS H = H C in G (so H /C = H). Thus if we can find a near-optimal 2-ECSS H in G/C, then we also have a 6

7 near-optimal 2-ECSS H in G, with at most w(c) more additive error. Furthermore, if the vertex v representing C in G/C is a cut vertex, then the 2-ECSS problem in G/C decomposes into independent subproblems, one per block around v. The strategy to design the PTAS will be as follows. Given an unweighted planar 2-EC graph G, let OPT denote the size of a minimum 2-ECSS of G. We will first use methods from [4] and [15] to decompose the graph into parts of small treewidth by contracting a set of low weight cycles and committing only a small error to the solution. The 2-ECSS problems on the parts of small treewidth will be solved exactly using the algorithm from Section 2. So, let G = (V, E) be an embedded planar 2-EC graph. Our algorithm will make use of the following lemma, which can be derived from methods in [15] and [4]. Lemma 8. Let G = (V, E) be an embedded planar graph, w R E +, and k a positive integer. Then one can find a set of simple, edge-disjoint cycles C, such that w(e(c)) w(g)/k and such that the 2-ECSS problem on G/C can be decomposed into independent 2-ECSS problems on k-outerplanar graphs. Algorithm 1 gives a general overview of the PTAS for the unweighted planar graph 2-ECSS problem. The subproblems considered in step 1 are 2-EC planar graphs of outerplanarity k. Let H be such a graph. Since H is k-outerplanar, it can be triangulated in such a way that the outerplanarity of H is still k and such that every vertex of H has distance at most k to some vertex on the infinite face. Let H be the triangulated graph. By choosing an arbitrary v 0 on the infinite face and doing a breadth-first search on H, we can obtain a spanning tree T of H of radius at most k. This spanning tree will help us define a tree decomposition of H. We let I be the set of triangles in H and two triangles i, j I form an edge ij F if and only if they share an edge in E(H ) \ E(T ). Moreover, for a triangle i = {u, v, w}, we let X i be the set of all vertices on the three paths from u, v, and w, respectively, to the root v 0 of T. It is shown in [11], that T D = ({X i : i I}, (I, F )) is indeed a tree decomposition of H of width at most 3k, and that it can be found time O(kn). Since H is a subgraph of H, T D is also a tree decomposition of H. We can now apply the 2-ECSS algorithm for graphs of bounded treewidth to H and T D and we conclude the following theorem. Theorem 3. Given an unweighted 2-EC planar graph G = (V, E) and ε > 0, Algorithm 1 finds an (1 + ε)-approximate 2-ECSS of G in time 2 O( 1 ε log 1 ε ) n, where n = V. 7

8 Proof. We first prove the approximation guarantee of the algorithm. In any graph a 2-ECSS must use at least n edges, and therefore OP T n. Moreover, since G is planar, E 3n 6. The only error committed by the algorithm is adding the edges in E(C), for which by Lemma 8 we have E(C) E /k (3n 6)/ 3 ε ε n ε OP T. The running time is dominated by the application of the algorithm from Section 2, and is at most 2 O( 1 ε log 1 ε ) n by Theorem 2. Algorithm 1 A PTAS for the 2-ECSS problem on unweighted planar graphs Input: An embedded planar 2-EC graph G = (V, E), ε > 0. Output: A (1 + ε)-approximate 2-ECSS of G. 1: Apply Lemma 8 to the G with k = 3/ε and obtain a set of cycles C. 2: for all 2-EC components H of G/C do 3: Find a tree decomposition T D of H of width at most 3k as described above. 4: Transform T D to a nice tree decomposition using Lemma 1. 5: Apply the algorithm from Section 2 to H and T D and obtain S H. 6: end forthe solutions S H and E(C). Moreover, for planar graphs we can derive a better bound on the number of external types that can occur during the dynamic program (cf. [8]). We conclude this section with the following theorem. Theorem 4. Given an unweighted 2-EC planar graph G = (V, E) and ε > 0, one can find a (1+ε)-approximate 2-ECSS of G in time 2 O(1/ε) n, where n = V. 4 A PTAS for Weighted Planar Graphs In this section we will sketch the PTAS for the 2-ECSS problem in weighted planar graphs. The main idea of the algorithm is similar to Algorithm 1, the PTAS for unweighted planar graphs. If Algorithm 1 is applied to a weighted graph G with weights w R E +, then it will still run in linear time, however the approximation ratio will be 1 + O(ε w(g)/op T ), which may be unbounded for general weights. To avoid this problem we first construct a spanning subgraph of the input graph whose weight is at most constant times the optimal solution OP T. Similarly to the algorithm for unweighted planar graphs, we would like to decompose the spanner according to Lemma 8 into independent 8

9 2-ECSS problems of bounded outerplanarity, and solve these using a dynamic program similar to the algorithm from Section 2. However, we have to consider that certain edges that were deleted during the construction of the spanner have to appear in any (1 + ε)-approximate solution. In the following we will concentrate on the construction of the desired spanning subgraph. For a weighted graph G = (V, E), a weight vector w R E + and some s 1, a s-spanner of G is a spanning subgraph H of G, such that for any two vertices u, v V we have dist H s dist G (u, v), i.e. in H the distances between any two vertices are approximately the same as the respective distance in G. Of course G itself is a s-spanner of G for any s 1, but in many applications one seeks to find spanners with particular properties, such as few edges or small weight, so that instead of solving the given problem on G one can instead solve the problem on a much smaller graph. One example for this is the metric-tsp PTAS for weighted planar graphs, which starts with replacing the input graph by a s-spanner while effectively bounding w(g)/op T for the remainder of the algorithm. Suppose now that G is a weighted embedded planar graph and H is a subgraph. A chord e of H is an edge of G not in H. Note that H and e inherit embeddings from G. For each chord e we define w H (e) as the length of the shortest walk connecting the endpoints of e along the boundary of the face of H containing e. The spanner algorithm Augment(G, w, A, S) from [3] has as inputs a weighted planar graph G = (V, E), weights w R E +, a subgraph A of G and some s 1. Its properties are as follows: Lemma 9 ([3, Theorem 2]). Let H = Augment(G, w, A, s), where G, w, A and s are as above. Then H is a s-spanner of G containing A. In particular, w H (e) s w(e) for any e E \ E(H). Moreover, if A is connected, then w(h) (1 + 2/(s 1)) w(a). Given a face f in H, the chords of f are the edges of G embedded inside f, according to G s embedding. A face-edge e of f is an abstract edge connecting two vertices of f; unlike a chord, a face-edge is not necessarily an edge of G. (If vertices appear more than once on f, we must specify which appearances we want as the endpoints of e.) We say the face edge e crosses a chord c if c is a chord of the same face f and their endpoints are distinct vertex appearances on f, and they appear in cyclic ecec order around the boundary of f. Note that we may embed e inside f so e intersects only the crossed chords. The following lemma shows that the spanner has nice intersection properties with a near-optimal solution 9

10 to the 2-ECSS problem; it is key in bounding the size of the dynamic program described later. Lemma 10. Let G = (V, E) be a planar graph and w R E +. Let A be a 2- approximate 2-ECSS of G and H = Augment(G, w, A, 2). Furthermore, let S be an optimal 2-ECSS of G and D a positive integer. Then there exists a 2-ECSS S of G with the following properties: 1. w(s) (1 + 2/D) w(s ) 2. For any face f of H, and any face-edge e of f, the number of edges in S crossing e is at most 2D. Proof. Consider a face f of H and let C be the set of chords of f which are also edges of S. Moreover, let E f be the set of edges on the boundary of f. There are potentially very many edges in S crossing a particular face-edge of f. We will remove some edges from S and replace them by walks on E f as to maintain a 2-edge-connected subgraph S of G, while not increasing the weight of S by too much and ensuring the bound on the number of edges in S crossing any face-edge of f. Let Sf be the set of edges in S embedded inside the face f, i.e. these are edges which are not in H. The dual edges of all edges in Sf form a tree T, and we will orient each such dual edge in the following way. Any e = uv S f divides the boundary of f, i.e. E f, into two walks P 1 and P 2, and we will orient the dual edge e towards that part of the boundary of f which has smaller total weight, and we will call this walk P e. Using the definition above this means w H (e) = w(p e ). In particular we know by Lemma 9, w(p e ) 2 w(e). Moreover, any vertex of T has indegree at most 1, since if a vertex had indegree at least two, then two disjoint parts of E f would both have total weight greater than w(e f )/2, which is a contradiction. Therefore there must exist a root vertex r of T, such that all edges in T are oriented away from r. We will now modify S as follows. Let TD be the set of dual edges at distance D from r in T. For any e TD, we will delete from S all edges which are dual to the edges descendant to e in T. Moreover, we add P e to S. We call the resulting graph S. Note that P e P e = for any two edges e, e TD. It is clear that S remains a 2-ECSS of G. Further, the dual tree of edges in the newly constructed S which are embedded in f has now depth at most D. This shows that any face-edge of f is now crossed by at most 2D edges in S. It remains to show that adding the walks P e for all e TD increases the weight of S by at most 2/D w(sf ). If e TD and an edge ẽ 10

11 is on the path from e to r in T, we say ẽ e. For any fixed ẽ E(T ) of distance at most D from r we then have that e TD :ẽ e P e Pẽ. Thus e TD :ẽ e w(p e) w(pẽ). In turn, for any fixed e TD there are exactly D edges ẽ with ẽ e. Now the total weight by which we increase the weight of the solution S is w(p e ) 1/D e T D w(p e ) 1/D w(p e ) 1/D w(pẽ) e TD ẽ e ẽ T e TD :ẽ e ẽ T 2/D w(s f ). When doing this procedure for all faces of H, we obtain the desired 2-ECSS S of G. For each face f of H we added edges to S of weight at most 2/D w(sf ), so the total weight added to S for all faces of H is at most 2/D w(s ). Algorithm 2 gives a high-level description of the PTAS for the weighted 2-ECSS problem in planar graphs. Algorithm 3 is a dynamic program based on the ideas from Section 2 and is discussed more thoroughly below. In order for Algorithm 2 to be a PTAS for the 2-ECSS prob- Algorithm 2 A PTAS for the 2-ECSS problem on weighted planar graphs Input: A planar 2-EC graph G = (V, E), edge weights w R E +, ε > 0. Output: A (1 + ε)-approximate 2-ECSS of G. 1: Find a 2-approximate 2-ECSS A of G (e.g. using [13]). 2: H = Augment(G, 2, A) 3: Apply Lemma 8 to H with k = 24 and obtain a set of cycles C. ε 4: for all 2-EC components F = (V F, E F ) of H/C do 5: Apply Algorithm 3 to F with D = 4 and obtain SF. ε 6: end for the union of the solutions S F and E(C). lem on weighted planar graphs, we have to make use of Lemma 10, which guarantees that there exists a 2-ECSS of G of weight at most (1 + 2/D) OP T (1 + ε/2) OP T, which crosses each face edge of the spanner H at most 2D times. Algorithm 3 finds a minimum weight 2-ECSS of G which satisfies this condition for only a certain subset of all face-edges, and therefore it also has weight at most (1 + ε/2) OP T. Theorem 5. Given a planar 2-EC graph G = (V, E) and ε > 0, Algorithm 2 computes an (1 + ε)-approximate 2-ECSS of G in time n O(1/ε2), where n = V. Proof. Let us first prove the approximation guarantee of the algorithm. By Lemma 9 we have that w(h) (1 + 1/( 2 1)) w(a) 12 OP T, 11

12 since A is a 2-approximate 2-ECSS of G. Applying Lemma 8 in step 2 with k = 24 ε yields a collection of cycles of weight at most w(h)/k 12 OP T/ 24 ε ε 2 OP T. Therefore, by committing the cycles in C to our solution, we add an error of at most ε 2 OP T to the solution. For each 2-EC component F of H/C, we find a (1 + ε/2)-approximate 2-ECSS of F as argued above, and all the 2-ECSS problems on those 2-EC components are independent as described in Section 3. In step 2 we combine the solutions and their total weight is at most (1 + ε/2) OP T. Together with C they comprise a 2-ECSS of the original input graph G, whose total cost is at most (1 + ε) OP T. The overall running time is dominated by the application of Algorithm 3 in steps 2-2 and is bounded by n O(kD) = n O(1/ε2). We will now sketch the ideas for Algorithm 3. Let F = (V F, E F ) be a 2-EC component of H/C. We denote by C F the set of edges from the original input graph G which were deleted during the construction of the spanner H and are now missing from F. Note that F C F is still a planar graph. We can triangulate each face of F (as described in Section 3), to obtain a planar graph F and a spanning tree T of F of radius at most k. Moreover, let E = E(F ) \ E F. Some of the edges in T may be edges in E F and some may be edges in E, and we let T = E(T ) E. We will make use of the same tree decomposition of F as used in the PTAS for unweighted planar graphs. For a vertex u V F let P u denote the path from u to the root v r of T, and for a triangle i = uvw, we denote by P i = P u P v P w the union of the three paths from u, v and w to v r. Remember, we can obtain a tree decomposition T D = ({X i : i I}, T T D ) of F by letting I be the set of triangles in F and by letting X i = V (P i ) for each i I. Two triangles i, j I are connected by an edge in T T D if and only if they share an edge which is not in T. T D is a tree decomposition of F and therefore also of F of width at most 3k, and every i I has degree at most 3 in T T D. We apply Lemma 1 to T D and change it to a nice tree decomposition, then each X i (i I) is still the vertex set of the union of three paths P u, P v and P w, but now u, v and w do not necessarily lie on a triangle of F. We now define the subproblems that we want to solve in our dynamic program. For an edge e T, let C e C F be the set of edges in C F crossing e. Let now i I be a node of the tree decomposition. The set of edges in C F crossing any edge in P i is denoted by C i = C e. e P i Moreover, we let S E F (X i ) be a set of edges of F with both endpoints 12

13 in X i. Furthermore, let C C i be a set of crossing edges, such that C C e 2D for any e P i T. Finally, for any subproblem we also have to define an external type on the vertices in X i and V (C), the endpoints of edges in C. Thus let also a Ĥ T ext(x i V (C)) be given. The subproblem (i, S, C, Ĥ) is now the following. Find a subgraph F = F (i, S, C, Ĥ) of F [V i] C F of minimum weight w(i, S, C, Ĥ),, such that F H is 2-edge-connected, F E F (X i ) = S and such that F C i = C. If we let (S, C ) = argmin{w(r, S, C, Ĥ ) : S E F (X r ), C C i : C C e 2D for all e P r T }, then F (r, S, C, Ĥ ) is a minimum weight 2-ECSS of F C F which crosses each edge in P r T at most 2D times. Therefore the weight of that solution, i.e. w(r, S, C, Ĥ ), is at most the weight of a minimum weight 2-ECSS of F C F which crosses all edges in T at most 2D times. This is the property we needed in the proof of Theorem 5. Computing the values w(i, S, C, Ĥ) and the graphs F (i, S, C, Ĥ) can be done using Lemmas similar to Lemmas 3-6. The running time is n O(kD) ; we omit its analysis, except to remark it is dominated by considering all subsets C of O(kD) crossing edges. Algorithm 3 Generalized 2-ECSS for planar graphs Input: A planar graph F, a set of edges C F and D (as described above). Output: A 2-ECSS of F C F. 1: Triangulate F and obtain F of radius k. 2: Find a spanning tree T of F of radius k. 3: Find a tree decomposition T D of F of width at most 3k using T. 4: Refine T D and obtain T D, a nice tree decomposition with root r. 5: Solve the dynamic program on T D as described above. 6: (S, C ) = argmin{w(r, S, C, Ĥ ) : S E F (X r), C C i : C C e 2D for all e P r T }. F (r, S, C, Ĥ ) 5 Future Work The PTAS results of this paper are likely to carry over to the 2-VCSS problem (minimum weight 2-vertex-connected spanning subgraph), as happened before [8]. A question motivated by the TSP is whether we can find a PTAS for the Steiner or subset version of the 2-ECSS problem. That is, along with the graph G we are given a subset R of required vertices; we must find a 2-EC subgraph spanning at least the vertices in R. For the subset 13

14 version of planar metric TSP, Klein s subset spanner construction [16] yields an EPTAS. A similar subset spanner construction yields another EPTAS for the Steiner tree problem in planar graphs [6]. We can apply Klein s subset spanner construction to generalize our Lemma 9; however, the generalization of Lemma 10 remains a roadblock. 14

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