Augmenting Outerplanar Graphs to Meet Diameter Requirements

Proceedings of the Eighteenth Computing: The Australasian Theory Symposium (CATS 2012), Melbourne, Australia Augmenting Outerplanar Graphs to Meet Diameter Requirements Toshimasa Ishii Department of Information and Management Science, Otaru University of Commerce, Otaru 047-8501, Japan, Email: ishii@res.otaru-uc.ac.jp Abstract Given a graph G = (V, E) and an integer D 1, we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D. It is known that no constant factor approximation algorithms to this problem with an arbitrary graph G can be obtained unless P = NP, while the problem with only a few graph classes such as forests is approximable within a constant factor. In this paper, we give the first constant factor approximation algorithm to the problem with an outerplanar graph G. We also show that if the target diameter D is even, then the case where G is a partial 2-tree is also approximable within a constant. Keywords: undirected graph, graph augmentation problem, diameter, outerplanar graphs, partial 2- trees, constant factor approximation algorithm 1 Introduction In communication networks, some transfer delay occurs when we send a message from one node to another node. The least number of links through which the message has to be transmitted is considered as one measurement of such a transfer delay. Thus, the diameter of a network, which is defined as the maximum distance between every two nodes in it, is an important parameter to measure the performance of the network. From this point of view, the problems of constructing a graph with a small diameter by adding new edges have been studied (Alon et al. 2000, Bilò et al. 2010, Chepoi and Vaxès 2002, Chepoi et al. 2006, Dodis and Khanna 1999, Ishii et al. 2006, Li et al. 1992, Schoone et al. 1987). Given an undirected graph G = (V, E) and an integer D 1, the augmentation problem with diameter requirements (for short, APD) is to augment G by adding a minimum set of new edges that reduces the diameter to at most D. Note that the case of D = 1 is trivial, because only the complete graph can have diameter one. In general, Schoone et al. (1987) showed that APD is NP-hard for any fixed This research was partially supported by the Scientific Grantin-Aid from Ministry of Education, Culture, Sports, Science and Technology of Japan. Copyright c 2012, Australian Computer Society, Inc. This paper appeared at the 18th Computing: Australasian Theory Symposium (CATS 2012), Melbourne, Australia, January- February 2012. Conferences in Research and Practice in Information Technology (CRPIT), Vol. 128, Julian Mestre, Ed. Reproduction for academic, not-for profit purposes permitted provided this text is included. D 3. Moreover, it was shown that there is a constant c > 0 such that APD is not approximable within a factor of c log n in polynomial time, unless P=NP, due to Li et al. (1992) for D 4, and due to Dodis and Khanna (1999) for D {2, 3}, where n = V. Recently, Bilò et al. (2010) showed that APD is approximable within a factor of O(log n) in polynomial time. Thus, APD is Θ(log n)-approximable. Also, about the bounds on the optimal value OP T (G, D) to APD with a graph G and an integer D, Alon et al. (2000) showed that OP T (G, 2) = n 1 and OP T (G, 3) n O( 3 ) hold for any graph G with the maximum degree and a sufficiently large number of vertices and that OP T (G, D) n/ D/2 holds for any connected graph G. There are also many studies on APD with some restricted classes of graphs. Erdős et al. (1998) investigated upper and lower bounds on OP T (G, D) in the case where a given graph and an augmented graph are restricted to be triangle-free. Alon et al. (2000) proved that OP T (C n, D) = n/(2 D/2 1) O(1) holds for any cycle C n of n vertices; it follows that a (1, c)-approximate solution can be obtained in polynomial time for a constant c. Here, let us call a solution an (a, b)-approximate solution if the number of edges in the solution is at most aop T (G, D) + b, and an algorithm that delivers such a solution an (a, b)- approximation algorithm. As for the case where G is a forest, Chepoi and Vaxès (2002) presented a 2- approximation algorithm for an even D, and Ishii et al. (2006) provided an 8-approximation algorithm for an odd D. The latter one has been improved to a (2 + 1/δ, δ 5 )-approximation algorithm for any integer δ > 0 due to Chepoi et al. (2006). Moreover, Chepoi and Vaxès (2002) also pointed out that their algorithm for forests and even diameters can be applied to a wider class of graphs G satisfying the following conditions (i) and (ii). (i) G is a Helly graph (Berge 1989). (ii) There exists a polynomial time algorithm to the l-dominating SET with G, the problem of finding a minimum set X of vertices such that the distance from each vertex to some vertex in X is at most l. Forests and dually chordal graphs (Brandstädt et al. 1998) are included in such a class of graphs. However, to our best knowledge, for any graph classes other than cycles or Helly graphs, there are only a few results about whether APD is approximable within a constant in polynomial time or not, while it is still left open whether APD with forests is NP-hard. In this paper, we consider the case where G is outerplanar, and give the first constant factor algorithm for APD; our algorithm delivers a (2, 3)- approximate (resp., (36, 42)-approximate) solution in O(n 5 ) (resp., O(n 8 )) time if D is even (resp., odd). For an even D, our algorithm is based on Chepoi and Vaxès (2002) s algorithm for a Helly graph; their algorithm finds a 2-approximate set E of edges so that in the augmented graph G = (V, E E ), ev- 123

CRPIT Volume 128 - Theory of Computing 2012 ery vertex v V is at distance at most D/2 from some vertex u. While an outerplanar graph is not a Helly graph, we show that such a set E with E 2OP T (G, D) + 3 can be found in O(n 5 ) time. For an odd D, our strategy follows Ishii et al. (2006) s approaches for a forest and an odd D; their 8-approximation algorithm first finds a set E 1 of edges with diam((v, E E 1)) D + 1 and E 1 2OP T (G, D + 1) 2OP T (G, D) by applying the above method for even diameters, and next reduces the diameter of (V, E E 1) from D+1 to D by adding a set E 2 of edges with E 2 2 E 1 +2L for some lower bound L on OP T (G, D). In order to obtain a constant factor approximation algorithm, we extend this lower bound L for forests to apply it to the case of outerplanar graphs. Also, we investigate the case where G is a partial k-tree. Notice that outerplanar graphs are partial 2- trees. While in this case, the above Chepoi and Vaxès approaches for even diameters do not work even if G is a 2-tree, we show that if G is a partial 2-tree and D is even, APD is (8,5)-approximable in O(n 5 ) time. The paper is organized as follows. In Section 2, we state our main results about APD with outerplanar graphs and partial 2-trees, after introducing some basic notation. In Section 3, we propose a (2,3)-approximation algorithm, named Outerplanar Even(D), to APD with an outerplanar graph G and an even D. In Section 4, we give a (36,42)-approximation algorithm, named Outerplanar Odd(D), to APD with an outerplanar graph G and an odd D, after inroducing a new lower bound on OP T (G, D). In Section 5, we discuss the case where G is a partial k-tree and show that if G is a partial 2- tree and D is even, then APD is (8,5)-approximable. In Section 6, we give concluding remarks. Some of the proofs are omitted due to space limitation. 2 Main theorems Let G = (V, E) stand for an undirected simple graph with a set V of vertices and a set E of edges. An edge with end vertices u and v is denoted by (u, v). We denote V by n and E by m. A singleton set {x} may be simply written as x, and implies proper inclusion while means or =. For a set V, a family V of pairwise disjoint subsets of V with V V i V i = V is called a partition of V. In G = (V, E), its vertex set V and edge set E may be denoted by V (G) and E(G), respectively. For a subset V V in G, G[V ] denotes the subgraph induced by V. For a set E of edeges with E E =, we denote the augmented graph (V, E E ) by G + E. For a set E of edges, we denote by V [E ] the set of all end vertices of edges in E. For a vertex set X V in a graph G, we denote by N G (X) the set of vertices in V X adjacent to some vertex v X, and denote N G (X) X by N G [X]. A set E of edges is called a star (with center v) if each edge in E has v as one of its end vertices. A path between two vertices u and v is called a (u, v)-path. The length of a path P is defined by the number of edges in P and is denoted as P (i.e., P = E(P ) ). For two vertices u, v V in G = (V, E), the distance between u and v is defined as the length of a path between u and v with the shortest length, and it is denoted by d G (u, v). For two sets X, Y V of vertices, let d G (X, Y ) = min{d G (u, v) u X, v Y }. The diameter of a graph G, denoted by diam(g), is defined as the maximum among distances between all pairs of vertices in G. For a vertex u V in a graph G and an integer k, let NG k (u) denote the set of vertices v with d G (u, v) = k. A set B G (u, k) = k k N G k (u) of vertices is called the ball centered at u of radius k. B G (u, k) may be simply called a k-ball (with center u). For a subset V V of vertices and a family B of balls, we say that B covers V if every vertex in V is contained in some ball in B. A set V of vertices is called a vertex cut (of G) if the deletion of V increases the number of components in G. In particular, when V = 1, the vertex v with V = {v} is called a cut vertex (of G). A bridge is an edge whose deletion increases the number of components in G. A graph is called planar if it can be drawn in the plane without generating a crossing by two edges, and a plane graph is a particular drawing of a planar graph. A plane graph divides the plane into regions, and a face of a plane graph is the maximal region of the plane that contains no vertex in that region. A face whose vertex set is {u 1, u 2,..., u k } with (u i, u i+1 ) E(G), i = 1, 2,..., k (where u k+1 = u 1 ) is denoted by [u 1 u 2 u k ]. We call a face consisting of k vertices a k-face. A planar graph G is called outerplanar if it can be drawn in the plane so that all vertices lie on the boundary of some face. Such a drawing is referred to as an outerplane graph. The face whose boundary contains all vertices of G is called the outer face and all other faces are called inner faces. We call an edge belonging to the boundary of the outer face an outer edge and all other edges inner edges. Note that each bridge is an outer edge. An outerplane graph G is called maximal if every inner face is a 3-face in G. A tree decomposition of G = (V, E) is defined as a pair (T = (V T, E T ), U = {U x V x V T }) of a tree T and a family U of subsets of V satisfying (a) x V T U x = V, (b) for every (v, v ) E, {v, v } U x for some x V T, and (c) for every v V, {x V T v U x } induces a connected component in T (Robertson and Seymour 1986). The width of a tree decomposition (T, U) is max{ U x 1 x V T }, and the treewidth of G is the minimum width over all tree decompositions of G. A graph with treewidth at most k is also called a partial k-tree. It is known that any outerplanar graph is a partial 2-tree. Throughout the paper, we shall use node to denote an element in V T for distiguishing it from an element in V. Here, we introduce the following basic properties about tree decompositions, which will be utilized in the subsequent sections. Lemma 2.1 Let (T = (V T, E T ), U) be a tree decomposition of G, (x 1, x 2 ) be an edge in T, and VT 1 and V T 2 be two components in T {(x 1, x 2 )} = (V T, E T {(x 1, x 2 )}). Then, U x1 U x2 is a vertex cut separating ( x V U T 1 x ) (U x1 U x2 ) from ( x V U T 2 x ) (U x1 U x2 ) in G. In this paper, we consider the following problem of augmenting an initial graph by adding a minimum set of edges to meet required diameter constraints. Problem 2.2 Augmentation Problem with Diameter Requirements (APD) Input: A graph G = (V, E) and an integer D 1. Output: A set E of edges with the minimum cardinality such that diam(g + E ) D holds. For APD, we show the following two theorems. Theorem 2.3 If G is outerplanar and D is even (resp., odd), then a (2, 3)-approximate (resp., (36, 42)-approximate) feasible solution to APD can be found in O(n 5 ) (resp., O(n 8 )) time. Theorem 2.4 If G is a partial 2-tree and D is even, then an (8, 5)-approximate feasible solution to APD can be found in O(n 5 ) time. 124

3 Outerplanar graphs and even diameters In this section, we consider the case where G is outerplane and D is even. After introducing tree decompositions of outerplane graphs in Section 3.1, we show that APD is (2,3)-approximable in O(n 5 ) time in Section 3.2. 3.1 Tree decomposition Let G = (V, E) be an outerplane graph. The treewidth of G is at most two, i.e., there exists a tree decomposition (T = (V T, E T ), U = {U x V x V T }) of G such that U x 3 for all U x U. While a tree decomposition of G is not unique, we here introduce a tree decomposition of G based on a maximal outerplane graph of G, which will be utilized for proving several theorems and lemmas in the subsequent sections. Let G M = (V, E M ) be a maximal outerplane graph with E E M, and F be the family of all inner faces of G M (note that each face in F is a 3-face). Then, we can construct a tree decomposition (T = (V T, E T ), U) in the following manner (I) and (II): (I) For each inner face F = [v 1 v 2 v 3 ] (resp., bridge e = (v 1, v 2 )) of G M, we introduce a new vertex x F (resp., x e ) with U xf = {v 1, v 2, v 3 } (resp., U xe = {v 1, v 2 }); let V T = {x F F F} {x e e B(G)} where B(G) denotes the set of all bridges in G. (II) E T consists of the following three sets ET 1, ET 2, and E3 T. An edge (x 1, x 2 ) ET 1 if and only if x 1 = w F1 and x 2 = w F2 for two 3-faces F 1 and F 2 in F which have a common inner edge. For each cut vertex v in G where V i, i = 1, 2,..., p denotes the connected component in G v which has a neighbor of v, we pick up one node x i for each i such that U xi U contains v and some vertex in V i ; let ET 2 (v) = {(x i, x i+1 ) i = 1, 2,..., p 1}. Let ET 2 = {ET 2 (v) v is a cut vertex in G}. If (V T, ET 1 E2 T ) is disconnected (i.e., consists of q components with q 2), then we add q 1 edges to make it connected; denote the set of these added edges by ET 3. It is not difficult to see that the resulting (T, U) is a tree decomposition of G. Throughout the paper, we call a tree decomposition of an outerplane G constructed based on its maximal outerplane graph G M with E(G) E(G M ) and the above procedure (I) and (II) a tree decomposition of G w.r.t. G M. 3.2 Algorithm Proceedings of the Eighteenth Computing: The Australasian Theory Symposium (CATS 2012), Melbourne, Australia Let G = (V, E) be an outerplane graph and D = 2R be an even integer with R 1. We then show that APD is (2,3)-approximable in O(n 5 ) time. First we start with a preparatory lemma. Lemma 3.1 Let E be an optimal solution to APD and B be the family of (R 1)-balls centered at vertices in V [E ]; B = {B G (v, R 1) v V [E ]}. Then every two vertices u 1, u 2 uncovered by B satisfy d G (u 1, u 2 ) D. Proof. Let u 1, u 2 be two arbitray vertices uncovered by B (i.e., u 1, u 2 / v V [E ] B G(v, R 1)). Note that we have d G (u i, V [E ]) R for i = 1, 2. Hence, the length of any (u 1, u 2 )-path containing some edge in E is at least 2R+1 = D+1. From diam(g+e ) D and this, it follows that there exists a (u 1, u 2 )-path with length at most D which contains no edge in E ; d G (u 1, u 2 ) D. Let W denote the set of all vertices uncovered by B. By utilizing this property, Chepoi and Vaxès (2002) proved that the case where G is a forest is 2- approximable. If G is a forest, then the set W can be covered by one R-ball B 1 from Lemma 3.1. Moreover, the set E 1 = {(v 1, v) v V [E ]} of new edges is a feasible solution with E 1 V [E ] 2OP T (G, D) where v 1 is the center of B 1, since every vertex v V is at distance at most R from v 1 in G + E 1. Based on this observation, Chepoi and Vaxès s algorithm delivers a 2-approximate solution by computing one R-ball and a minimum number of (R 1)-balls such that the family of these balls covers V. Furthermore, by extending these observations, they pointed out that if G is a Helly graph, then there exists a 2-approximate solution E 1 such that E 1 is a star with some center v 1 and d G+E1 (v 1, v) R for every v V. On the other hand, while an outerplanar graph is not a Helly graph, we can prove that W can be covered by one R-ball and at most three (R 1)- balls, as shown in Lemma 3.3 below. For proving this, we show the following property about partial k-trees, which is an extension of properties about diam(g)/2 -dominating sets shown by Gavoille et al. (2001), but its proof is omitted. Lemma 3.2 Let G = (V, E) be a partial k-tree and (T = (V T, E T ), U) be a tree decomposition of G. Let W be a set of vertices in G such that every two vertices w, w W satisfy d G (w, w ) d. Then, G has a set U x U of vertices such that every vertex w W satisfies d G (w, U x ) d/2. Lemma 3.3 Let G = (V, E) be an outerplane graph. For E and B as defined in Lemma 3.1, the set W of all vertices uncovered by B is covered by one R- balls and at most three (R 1)-balls. Proof. By Lemma 3.1, we have d G (w, w ) 2R for every pair of w, w W. (3.1) Let G M = (V, E M ) be a maximal outerplane graph of G with E E M, and consider a tree decomposition (T = (V T, E T ), U = {U x V x V T }) of G w.r.t. G M. Let x r be a node in T such that d G (w, U xr ) R all w W (3.2) (such a node x r exists by Lemma 3.2). We regard T to be rooted at x r V T. For each node x V T, let D(x; T ) be the set of descendants of x (including x), and U X = x X U x for X V T. If all vertices w W satisfy d G (w, U xr ) R 1, then we are done by U x 3. Assume that some vertex w 1 W satisfies d G (w 1, U xr ) = R, and it is contained in U D(x1;T ) for some child x 1 of x r. If the set V 1 = U D(x1 ;T ) U xr is adjacent to exactly one vertex v 1 U xr (i.e., v 1 is a cut vertex separating w 1 from U xr {v 1 }), then by (3.1) and (3.2), every vertex in W is at distance at most R from v 1 ; W can be covered by one R-ball centered at v 1. Suppose that V 1 is adjacent to exactly two vertices v 1 and v 2 in U xr (note that since G is outerplane, N G (V 1 ) {1, 2}). This also indicates that U xr = 3 and U xr forms a 3-face in G M ; let U xr = {v 1, v 2, v 3 }. Let x 2 (resp., x 3 ) be the child of x r such that V 2 = U D(x2;T ) U xr (resp., V 3 = U D(x3;T ) U xr ) satisfies N G (V 2 ) = {v 2, v 3 } (resp., N G (V 3 ) = {v 3, v 1 }) if exists. Observe that all other children x of x r satisfies N G (V ) = 1 where V = U D(x;T ) U xr since G is outerplane, and we can assume that for each such a V, every vertex w V W satisfies d G (w, U xr ) R 1 since otherwise we are done as 125

CRPIT Volume 128 - Theory of Computing 2012 observed above; all vertices in V W can be covered by {B G (v i, R 1) i = 1, 2, 3}. Assume that there exists a set W W of vertices uncovered by B = {B G (v 1, R), B G (v 2, R 1), B G (v 3, R 1)} (since otherwise we are done). We can observe that every vertex w V 3 W with d G (w, {v 3, v 1 }) = R satisfies d G (w, v 1 ) = R, since otherwise (i.e., d G (w, v 1 ) > R) we have d G (w 1, w) d G (w 1, {v 1, v 2 })+min{d G (w, v 1 ), d G (w, v 3 )+1} > 2R, contradicting (3.1) (note that {v 1, v 2 } and {v 3, v 1 } are both vertex cuts in G). It follows that W V 1 V 2. Then there are two possible cases (A) and (B): (A) W V 2 and (B) W V 2 =. (A) Let W 2 = W V 2. By d G (w 1, {v 1, v 2 }) = R, d G (w 2, {v 2, v 3 }) = R, and d G (w 1, w 2 ) 2R, it follows that d G (w 1, v 2 ) = d G (w 2, v 2 ) = R for every w 2 W 2. In a similar way, we can observe that every vertex w V 1 W with d G (w, {v 1, v 2 }) = R satisfies d G (w, v 2 ) = R. Hence, if d G (w, {v 1, v 3 }) R 1 for all w V 3 W, then {B G (v 1, R 1), B G (v 2, R), B G (v 3, R 1)} covers W. Assume that some vertex w 3 V 3 W satisfies d G (w 3, {v 1, v 3 }) = R. Then, by exchanging parts of the pair {w 1, V 3 } and {w 3, V 1 } in the above arguments, we can see that V 1 W =, i.e., W = W 2. Moreover, similarly to the above arguments for w 1 and w 2, we can observe that for every vertex w 2 W 2, we have d G (w 2, v 3 ) = d G (w 3, v 3 ) = R; d G (w 2, v 2 ) = d G (w 2, v 3 ) = R holds. We here claim that d G (w 2, v 23 ) R 1 for all w 2 W 2, where U x2 = {v 2, v 3, v 23 }. Indeed, this follows since d G (w 2, v 2 ) = d G (w 2, v 3 ) = R holds, and {v 2, v 23 } or {v 3, v 23 } is a vertex cut of G separating v 3 or v 2 from w 2, respectively. Thus, we can see that W is covered by B {B G (v 23, R 1)}. (B) Note that d G (w, {v 2, v 3 }) R 1 for all w V 2 W. Also note that d G (w, {v 3, v 1 }) R 1 for all w V 3 W, since otherwise it follows that d G (w, v 1 ) R for all w V 1 W as observed in the case (A), which contradicts W. Hence, if every vertex w V 1 W with d G (w, {v 1, v 2 }) = R satisfies d G (w, v 2 ) = R, then {B G (v 1, R 1), B G (v 2, R), B G (v 3, R 1)} covers W. Consider the remaining cases: there exist two vertices w 1, w 1 V 1 W with d G (w 1, {v 1, v 2 }) = d G (w 1, {v 1, v 2 }) = R, d G (w 1, v 2 ) > R, and d G (w 1, v 1 ) > R. Then, we move from x r to x 1 and apply the above arguments to x 1. Notice that d G (w, U x1 ) R still hold for all w W since by d G (w 1, {v 1, v 2 }) = R and (3.1), all w W U D(x1;T ) satisfy d G (w, {v 1, v 2 }) R. If we cannot still find a required family of balls for x 1, then we move to one child of x 1 in T or go back to x r. Here we remark that if we go back to x r, then we are done by adding the current family B of three balls centered at vertices in U x1 to B G (v 3, R 1). Indeed, in this case, all vertices in W uncovered by B are contained in V 2 V 3 {v 3 } or some component V of G U xr with N G (V ) = {v 3 }, and all such vertices are at distance at most R 1 from {v 1, v 2, v 3 } in G as observed above. Therefore, by repeating the above arguments to a leaf of T, we can find a required family of balls covering W. As a corollary of this lemma, we can observe that V can be covered by one R-ball and at most V [E ] + 3 ( 2OP T (G, D)+3) (R 1)-balls. Based on this, we can construct a (2,3)-approximate solution to APD by computing one R-ball and a minimum number of (R 1)-balls which cover V. This algorithm is described as Algorithm Outerplanar Even(D). Now we have the following lemma by a slight modification of Demaine et al. (2005) s algorithm which computes a minimum number of R-balls covering V in a graph with bounded branchwidth and the fact Algorithm 1 Algorithm Outerplanar Even(D) Input: An outerplane graph G = (V, E) and an even integer D = 2R 2. Output: A set E 1 of new edges such that diam(g + E 1 ) D and E 1 2OP T (G, D) + 3. 1: Choose a center c V for the R-ball and a set C 1 of centers for (R 1)-balls so that the family of these C 1 +1 balls covers V in G and the number C 1 + 1 of centers is minimized. 2: Halt after outputting the set E 1 = {(c, c) c C 1 }. that the branchwidth of any graph with treewidth at most two is at most two. Lemma 3.4 Let G be a partial 2-tree and R 0 be an integer. (i) For a set V V (G) of vertices, a minimum number of R-balls covering V can be computed in O( V (G) 4 ) time. (ii) For a family B of R-balls in G, a minimum set V of vertices hitting B (i.e., a minimum set V of vertices satisfying V B for all B B) can be found in O( V (G) 4 ) time. Notice that V hits a family B of R-balls if and only if {B(v, R) v V } covers the set of all centers in B. In Step 1 of Algorithm Outerplanar Even(D), c and C 1 can be obtained by computing a minimum number of (R 1)-balls covering V B G (v, R) for each v V ; it takes O(n 5 ) time. Thus, the case where D is even in Theorem 2.3 has been proved. Before closing this section, we show several properties of the set E 1 of edges obtained by Algorithm Outerplanar Even(D), which will be referred to when we deal with the case where the target diameter is odd. Lemma 3.5 (i) C 1 = E 1 2OP T (G, 2R) + 3 2OP T (G, 2R 1) + 3 holds. (ii) The family of (R 1)-balls with centers in C 1 N G (c ) covers V if R 2. (iii) Every vertex v V satisfies d G1 +E 1 (c, v) R. (iv) Every two vertices u 1, u 2 V with d G1 +E 1 (u 1, u 2 ) > 2R 1 satisfy d G1 +E 1 (u 1, u 2 ) = 2R and d G+E1 (c, u 1 ) = d G+E1 (c, u 2 ) = R; such a vertex u i satisfies d G (u i, c) = R 1 for some c C 1 N G (c ). 4 Outerplanar graphs and odd diameters In this section, we consider the case where G is outerplanar and D = 2R 1 is an odd integer with R 2 (note that the case of D = 1 is trivial). We will show that APD is (36, 42)-approximate in O(n 8 ) time. In this case, such a good property as Lemma 3.1 does not hold. Moreover, it is not difficult to see that OP T (G, D)/OP T (G, D+1) cannot be bounded from above by any constant even in the case of trees. Our algorithm consists of two steps; in the first step, we apply Algorithm Outerplanar Even(2R) to G to obtain a set E 1 of edges with diam(g + E 1 ) 2R and E 1 2OP T (G, 2R) + 3, and in the second step, we will reduce the diameter of G + E 1 by one with an addition of another edge set. Note that by Lemma 3.5(i), we have E 1 2OP T (G, D) + 3; ( E 1 3)/2 is a lower bound on OP T (G, D). For the second step, after introducing another lower bound on OP T (G, D) in Section 4.1, we discuss the details for the way of obtaining a constant factor approximate solution based on these lower bounds in Section 4.2. Here, we define some notations and derive several properties about H 1 = G + E 1. As for E 1 126

obtained in the first step, let c and C 1 be a vertex and a vertex set in G, respectively, such that {B G (c, R)} {B G (c, R 1) c C 1 } covers V and we have C 1 = E 1. In H 1, a vertex u V is called distant if d H1 (u, u ) > D (= 2R 1) holds for some other vertex u. Let S(H 1 ) denote the set of all distant vertices in H 1. By Lemma 3.5(iv), we see that every distant vertex u S(H 1 ) satisfies d G (u, c) = R 1 for some c C 1 N G (c ); let C(u) = N R 1 G (u) (C 1 N G (c )) for u S(H 1 ). Then, it is not difficult to see that the following properties hold. Lemma 4.1 Let u 1 and u 2 be two distant vertices in S(H 1 ). (i) If C(u 1 ) C(u 2 ), then d G (u 1, u 2 ) D. (ii) Let E be an arbitrary feasible solution to APD. If C(u 1 ) C(u 2 ) = and V [E ] (B G (u 1, R 2) B G (u 2, R 2)) =, then E E contains an edge connecting N R 1 G Proceedings of the Eighteenth Computing: The Australasian Theory Symposium (CATS 2012), Melbourne, Australia (u 1) and N R 1 G (u 2). Lemma 3.5(iv) also says that if two vertices u, u V satisfy d H1 (u, u ) > D, then both of u and u are distant. Hence, in the second step, we concentrate on reducing the distance between every two distant vertices u, u S(H 1 ) with d G (u, u ) > D. Our strategy for augmenting a graph is based on three types of lower bounds on OP T (G, D). One is ( E 1 3)/2 = ( C 1 3)/2, which follows from E 1 2OP T (G, 2R) + 3 2OP T (G, D) + 3 as observed in Lemma 3.5(i). The other two ones are given in the following lemma and the next subsection, respectively. Let C 2 be the set of vertices v N G (c ) such that v C(u) for some u S(H 1 ). Lemma 4.2 C 2 /4 1 is a lower bound on OP T (G, D). Notice that by (4.1) and (4.2), we have c i c j if S i S j. The following lemma is obtained similarly to Lemma 4.1. Lemma 4.3 Let E be an arbitrary feasible solution to APD and S be a partition of S satisfying (4.1) and (4.2). For two distnict members S i and S j of S, if B G (u, R 2) V [E ] = = B G (u, R 2) V [E ] for some u S i and u S j, then (i) d G (S i, S j ) D or (ii) E contains an edge connecting N R 1 G (u) and N R 1 G (u ). Let E be an arbitrary feasible solution to APD. Assume that for I {1, 2,..., p}, B G (u, R 2) V [E ] = holds for some u S i S if and only if i I. Note that Lemma 4.3 says that if S i and S j satisfy d G (S i, S j ) > D for {i, j} I, then at least one edge in E connects N R 1 R 1 G (u) and NG (u ) for some u S i and u S j with B G (u, R 2) V [E ] = = B G (u, R 2) V [E ]. Let E 1 E be the set of this type of edges in E. Then, we can see that E 1 I ( I 1)/2 q(i) where q(i) is the number of pairs S i, S j with {i, j} I and d G (S i, S j ) D. On the other hand, for i / I, B G (u, R 2) contains at least one end vertex of an edge in E for each u S i ; denote this type of set of edges by E 2 E. Since one edge can connect at most two pairwise disjoint sets, we can see that E 2 (1/2) i/ I h i holds, where h i is the cardinality of a minimum set hitting {B G (u, R 2) u S i } (i.e., h i = min{ Y Y B G (u, R 2) for every u S i }). Since E 1 E 2 = by (4.2), it follows that E E 1 + E 2 I ( I 1)/2 q(i) + (1/2) i/ I h i. Let g 1 (S, I) = I ( I 1)/2 q(i) + (1/2) i/ I h i. Proof. Assume that each vertex in C 2 is numbered c 1, c 2,..., c t in clockwise order on the boundary of its outerface and t 5. Also suppose that c is located before c 1 and after c t in clockwise order on it. Let S i denote the set of distant vertices u S(H 1 ) satisfying c i C(u). It was shown by Bilò et al. (2010) that OP T (G, D) V 1, if there is a set V of vertices such that every two distinct vertices v, v V satisfy d G (v, v ) > D. Hence, it suffices to show that whenever i j 4, any two vertices u S i and u S j satisfy d G (u, u ) > D. Let u S i and u S j be two vertices satisfying i j 4; assume that j i + 4 without loss of generality. We have d G (u, c i+1 ) R 1 since otherwise u would not be distant. Also, d G (u, c i+2 ) R holds since any (u, c i+2 )-path in G contains c or c i+1 by the structure of outerplane graphs and we have d G (u, {c, c i+1 }) R 1. Similarly, d G (u, c i+2 ) d G (u, c j 2 ) R, which means that d G (u, u ) 2R = D + 1. 4.1 Lower bound on OP T (G, D) Let S be a subset of V. We consider a partition S = {S 1, S 2,..., S p } of S and a set {c 1, c 2,..., c p } of vertices satisfying the following two conditions (4.1) and (4.2): For every u S i, we have d G (u, c i ) = R 1 and d G (u, c j ) > R 1 for all j i. (4.1) For every two distinct sets S i and S j, we have d G (S i, S j ) D. (4.2) Based on these observations, min{g 1 (S, I) I {1, 2,..., p}} is a lower bound on OP T (G, D). For 1 p p, let I (p ) be the set of p indices such that for every pair of i I (p ) and i / I (p ), we have h i h i (i.e., I (p ) is the set of p indices obtained from {1, 2,..., p} by choosing an index i in the nonincreasing order of h i ), and I (0) =. Hence, it follows that for p 2, p (p 1)/2 max{q(i) I {1, 2,..., p}, I = p } + (1/2) i / I (p ) h i min{g 1 (S, I) I {1, 2,..., p}, I = p }. Here, we can observe that q(i) 2 I 3 if G is outerplane and I 2. Claim 4.4 If G is outerplane, then max{q(i) I {1, 2,..., p}, I = p } 2p 3 for p 2. Proof. For each S i S, let Si = u S i B G (u, R 1). By (4.2), every two distinct sets Si and Sj are pairwise disjoint. Since Si contains all vertices of a (c i, u)-path with length R 1 for u S i by (4.1), Si induces a connected graph in G. From these, it is not difficult to see that the graph H obtained from G by contracting each Si into one vertex s i multiple edges with a single edge, and deleting, replacing selfloops is also outerplane. Let I be a subset of {1, 2,..., p} with I 2. If d G (S i, S j ) D, then it follows from (4.2) that d G (S i, S j ) = D and some edge e E connects N R 1 R 1 G (u) and NG (u ) for some u S i and u S j. This indicates that the edge e connects Si and Sj, and hence we have (s i, s j ) E(H) in H. Thus, q(i) is equal to the number of edges in the subgraph H 127

CRPIT Volume 128 - Theory of Computing 2012 of H induced by {s i i I}. Since any outerplane graph G has at most 2 V (G ) 3 edges, we have q(i) 2 I 3. Let and g 2 (S, p ) = p (p 1)/2 max{0, 2p 3} + (1/2) i / I (p ) h i, g(g, S) = min{min{g 1 (S, I) I {1, 2,..., p}, 2 I 7}, min{g 2 (S, p ) p {0, 1} or p 8}}. From the above arguments, we can see that g(g, S) is a lower bound on OP T (G, D), and moreover, we can reduce the distance between every two vertices in S to at most D in polynomial time by adding a set E of edges with E 2OP T (G, D); namely, we have the following lemma. Lemma 4.5 Let G be outerplane and S = {S 1, S 2,..., S p } be a partition of S satisfying (4.1) and (4.2). Then, the following (i) (iii) hold: (i) OP T (G, D) g(g, S). (ii) For a set E of edges chosen according to the following (a) and (b), we have E 2g(G, S) and d G+E (u, u ) 2R 1 for every two vertices u, u S. (a) Assume that g(g, S) = g 1 (S, I) holds for some I {1, 2,..., p} with 2 I 7. Let c i0 be a vertex with i 0 I. Let Y be a minimum set hitting {B G (u, R 2) u i/ I S i }. Let E = ({(c i, c j ) i, j I} E) {(c i0, v) v Y }. (b) Assume that g(g, S) = g 2 (S, p ) holds for some p with p {0, 1} or p 8. Let c i0 be a vertex with i 0 I (p ) if p 1, and an arbitrarily chosen vertex if p = 0. Let Y be a minimum set hitting {B G (u, R 2) u i/ I (p )S i }. Let E = ({(c i, c j ) i, j I (p )} E) {(c i0, v) v Y }. (iii) The edge set E in (ii) can be found in O( V (G) 8 ) time. Proof. (i) This follows from the discussion before this lemma. (ii) First consider the case of (a). Let u 1 S 1 and u 2 S 2. If i / I for i = 1, 2, then d G+E (c i0, u i ) d G+E (Y, u i ) + 1 R 1; if {1, 2} I =, then d G+E (u 1, u 2 ) 2R 2 < D. If i I for i = 1, 2, then we have d G (u i, c i ) = R 1 by (4.1), and d G+E (u i, c i0 ) R by (c i0, c i ) E E. Hence, if {1, 2} I = 1, then d G+E (u 1, u 2 ) d G+E (u 1, c i0 ) + d G+E (u 2, c i0 ) 2R 1 = D. Also, if {1, 2} I, then d G+E (u 1, u 2 ) d G+E (u 1, c 1 ) + d G+E (u 2, c 2 ) + 1 = 2R 1 = D by (c, c 2 ) E E. It follows that d G+E (u, v) 2R 1 for every two vertices u, v S. It is clear that E 2g 1 (S, I) = 2g(G, S) by the definition of g 1. Consider the case of (b). Similarly to the case of (a), we can observe that d G+E (u, v) 2R 1 for every two vertices u, v S. Also, it follows that E 2g 2 (S, p ), since p (p 1)/2 2{p (p 1)/2 (2p 3)} whenever p 8. (iii) For finding E, we need to find out which of the above (a) and (b) occurs. For this, we need to compute g 1 (S, I) for all I {1, 2,..., p} with 2 I 7, and g 2 (S, p ) for all p with p {0, 1} or 8 p V (G). By Lemma 3.4, we can compute all of h i for S i S in O( V (G) 4 ) time. Once all of h i are obtained, g 1 (S, I) can be computed in O( V (G) ) time for each I, and g 2 (S, p ) can be computed in O( V (G) ) time for each p. Hence, the running time for finding E is dominated by that for computing min{g 1 (S, I) I = 7}; it takes O( V (G) I +1 ) = O( V (G) 8 ) time. We remark that this lower bound is an extension of Ishii et al. (2006) s one for the case where G is a forest. We also remark that the arguments before Claim 4.4 do not need the assumption that G is outerplane, i.e., they hold for an arbitrary graph. 4.2 Algorithm In this subsection, we give an algorithm for reducing the distance between every two distant vertices in S(H 1 ) to at most D. If there exists a partition of S(H 1 ) satisfying (4.1) and (4.2), then we can make the diameter of H 1 at most D by adding at most 2OP T (G, D) edges, according to the discussion in the previous subsection. However, in general, there is a case where no such a partition exists; for example, some distant vertex is at distance R 1 from at least two vertices in C 1 C 2. Hence, roughly speaking, for eliminating such distant vertices, we first add a set F 1 of edges connecting c and several vertices located between some two vertices in C 1 C 2. Notice that a vertex at distance at most R 1 from c is not any more distant. Then, for the set S(H 1 + F 1 ) of the remaining distant vertices having a partition satisfying (4.1) and (4.2), we apply the discussion in the previous subsection to add a 2-approximate set F 2 of new edges. Below, we first show how to find such a set F 1 of edges with F 1 8 C 1 + 4 C 2 1. Let G M = (V, E M ) be a maximal outerplane graph of G with E E M and N G (c ) = N GM (c ). Here such a G M can be constructed from G by first choosing an edge connecting v and v as an edge in E M for each pair of v and v such that {v, v } N G (c ) and v and v are on the boundary of the same inner face. We remark that this assumption on G M will be utilized for proving F 1 8 C 1 + 4 C 2 1. Let (T = (V T, E T ), U = {U x V x V T }) be a tree decomposition of G w.r.t. G M. We add to T a new node r and one edge connecting r and some node x V T with c U x. We regard this resulting tree T = (V T {r}, E T {(r, x )}) to be rooted at r. For each node x V T, let p(x; T ) be the parent of x in T. Let c C 1 C 2. We denote the set of nodes x V T with c U x by X c. Let x c be the node in X c with c / U p(x c ;T ); x c is the nearest node to r among X c and uniquely determined from the construction of T. Let Xc 1 be the set of nodes x in X c {x c} such that for some c C 1 C 2 {c}, we have c U x or at least one child x / X c of x has c U D(x ;T ), and Xc 2 = X c Xc 1 {x c} (recall that D(x; T ) is defined in Section 3.2). Then, we define the following two sets C 3 (c) and C 4 (c): let C 3 (c) = U Xc N T1 [x c ] (C 1 C 2 {c }), and C 4 (c) = {U Xc N T1 [x] x Xc 1 } (C 1 C 2 {c }), where T 1 = (V T, ET 1 ) (recall the definition of E1 T given in Section 3.1). Let F 1 = {(c, v) v C 3 (c) C 4 (c), c C 1 C 2 }. Intuitively, the edges between c and C 3 (c) (resp., C 4 (c)) in F 1 are added for eliminating distant vertices between c and some vertices in C 1 C 2 corresponding to the ancestors of x c (resp., the descendants of some x Xc 1 ) in T. Then, we can observe that F 1 8 C 1 + 4 C 2 1, whose proof is omitted. Lemma 4.6 F 1 8 C 1 + 4 C 2 1. Let H 2 = G 1 + F 1 (= G + (E 1 F 1 )). For the set S(H 2 ) of distant vertices in H 2, we can observe that 128

Proceedings of the Eighteenth Computing: The Australasian Theory Symposium (CATS 2012), Melbourne, Australia if two vertices u, u V satisfy d H2 (u, u ) > D, then {u, u } S(H 2 ) holds, since F 1 is a star with center c and we can apply the same arguments for H 1 to H 2. Hence, for reducing diam(h 2 ) by one, it suffices to decrease the distance between every two distant vertices in H 2 to at most D. Here we show that we can do this by adding a set of at most 2OP T (G, D) new edges to H 2 by proving that there exists a partition of S(H 2 ) satisfying (4.1) and (4.2). Lemma 4.7 There exists a partition of S(H 2 ) satisfying (4.1) and (4.2). Proof. Consider a family S = {N R 1 G (c) S(H 2) c C 1 C 2 } of subsets of S(H 2 ). We show that S is a partition of S(H 2 ) satisfying (4.1) and (4.2). Let u 1 be a distant vertex in S(H 2 ) and c 1 be a vertex in C 1 C 2 with c 1 C(u 1 ) (note that such a c 1 exists by Lemma 3.5(iv)). We first claim that u 1 is contained in U D(y1 ;T ) for a child y 1 / X c1 of x 1 or u 1 U x1 for some x 1 Xc 2 1. Indeed, if u 1 U y holds for some ancestor y of x c 1, we have u 1 U D(y;T ) for a child y / X c1 of x for some x Xc 1 1, or we have u 1 U x for some x X c1 with U x (C 1 C 2 {c 1 }), then from the construction of C 3 (c 1 ) C 4 (c 1 ), C 3 (c 1 ) C 4 (c 1 ) separates u 1 from c 1 or includes u 1. This indicates that we would have d G (u 1, C 3 (c 1 ) C 4 (c 1 )) R 2 and d H2 (u 1, c ) R 1, contradicting that u 1 S(H 2 ). Moreover, we can observe that C 3 (c 1 ) C 4 (c 1 ) separates u 1 from any c C 1 C 2 {c 1 }; d G (u 1, c) > R 1 for any c C 1 C 2 {c 1 }. It follows that S is a partition of S(H 2 ) and satisfies (4.1). Let u 2 be a distant vertex in S(H 2 ) with C(u 2 ) = c 2 c 1. Then, we claim that d G (u 1, u 2 ) D holds, which indicates that S satisfies (4.2). We first consider the case where c 2 U x2 or c 2 is contained in U D(y2 ;T ) for a child y 2 / X c1 of x 2 for some x 2 Xc 1 1. If {c 1, c 2 } separates u 1 from u 2 in G, then it is clear since d G (u i, {c 1, c 2 }) R 1 for i = 1, 2, d G (u 1, c 2 ) > R 1, and d G (u 2, c 1 ) > R 1. Assume that G has a (u 1, u 2 )-path P which includes neither c 1 nor c 2. This indicates that U x1 = U x2 = 3, and T has a (x 1, x 2 )-path P T consisting of edges in ET 1. On the other hand, we have (x 1, x 2 ) / ET 1 since otherwise U x1 {c 1 } C 4 (c 1 ) would hold and C 4 (c 1 ) would separate u 1 from c 1. It follows that P has two vertices v 1 and v 2 in C 4 (c 1 ), where U x {c 1 } = {v 1, v 2 } for some node x with x 1 x x 2 on P T. Since G is outerplane, {c 1, v 1 } and {c 1, v 2 } are both vertex cuts in G separating u 1 from u 2. Thus, we have d G (u 1, u 2 ) d G (u 1, {c 1, v 1, v 2 }) + 1 + d G (u 2, {c 1, v 1, v 2 }) R 1 + 1 + R 1 = D. Finally, consider the case where x c 1 and x c 2 has the least common ancestor x with x c 1 x x c 2. Then, from the construction of C 3 (c 1 ) and C 3 (c 2 ) and the property that u i is contained in a set corresponding to the descendants of some x i Xc 2 i for i = 1, 2, we can observe that every (u 1, u 2 )-path not containing C 1 C 2 includes at least two vertices in C 3 (c 1 ) C 3 (c 2 ), in a similar way to the previous case. Thus, also in this case, we have d G (u 1, u 2 ) D. Now we are ready to give a constant factor approximation algorithm, named Outerplanar Odd(D), for APD with G and D = 2R 1. Lemma 4.8 For a set E = E 1 F 1 F 2 obtained by Algorithm Outerplanar Odd(D), we have E 36OP T (G, D) + 42. Algorithm 2 Algorithm Outerplanar Odd(D) Input: An outerplane graph G = (V, E) and an odd integer D = 2R 1 3. Output: A set E of new edges such that diam(g + E ) D and E 36OP T (G, D) + 42. 1: Compute a center c, a set C 1 of centers, and the edge set E 1 that satisfy Lemma 3.5. Let C 2 be the set of vertices v N G (c ) such that v C(u) for some u S(G + E 1 ). 2: Compute C 3 (c) and C 4 (c) for each c C 1 C 2. Let F 1 = {(c, v) v C 3 (c) C 4 (c), c C 1 C 2 }. 3: Let S be a partition of S(G + (E 1 F 1 )) as defined in the proof of Lemma 4.7. According to Lemma 4.5 with S, we add a set F 2 of edges with F 2 2OP T (G, D) to G + (E 1 F 1 ) so that diam(g + (E 1 F 1 F 2 )) D. 4: Halt after outputting the set E := E 1 F 1 F 2. Proof. By Lemmas 3.5 and 4.2, we have C 1 2OP T (G, D)+3 and C 2 4OP T (G, D)+4, respectively. It follows from Lemma 4.6 that E 1 + F 1 9 C 1 +4 C 2 1 34 C 1 +42. By Lemma 4.5, we have F 2 2OP T (G, D). Thus, E 36OP T (G, D) + 42. It is not difficult to see that the running time of Algorithm Outerplanar Odd(D) is dominated by that for computing F 2 ; it takes O(n 8 ) time as observed in Lemma 4.5(iii). Summarizing the argument given so far, Theorem 2.3 is now established. 5 Partial k-trees A partial k-tree can be defined as a subgraph of a k-tree, which is a graph from a complete graph with (k+1) vertices by repeatedly adding a new vertex who has exactly k neighbors forming a complete graph. Here, we consider the case where G is a partial k- tree and D is even, as an extension of the case of outerplanar graphs. As observed in Section 3.2, if G is outerplanar and D = 2R, then there exists a (2,3)-approximate solution E s which is a star with center v s satisfying d G+Es (v s, v) R for every v V. However, this is not the case even for a 2-tree; there is a 2- tree G for which the number of (R 1)-balls covering V B G (v, R) is not bounded by any constant for every vertex v V. For example, consider APD with a 2-tree G = (V, E) in Fig. 1 and D = 2; the graph G is constructed from the complete graph K 3 with V (K 3 ) = {u 1, u 2, u 3 } by adding p new vertices adjacent to both of u 1 and u 2, p new vertices adjacent to both of u 2 and u 3, p new vertices adjacent to both of u 3 and u 1, and one new vertex w adjacent to both of u 3 and w for some vertex w u 1 adjacent to {u 2, u 3 }. Then, {(u 1, w)} is an optimal solution. However, for any v V, we need at least p (R 1)-balls for covering V B G (v, R). On the other hand, for a partial k-tree and its tree decomposition (T = (V T, E T ), U = {U x V x V T }) of G, there exists a (2k + 2)-approximate solution E, each edge in which has a vertex in U x as one of its end vertices for some x V T, as observed in the following lemma. Lemma 5.1 Let G be a partial k-tree and (T = (V T, E T ), U) be its tree decomposition. For APD with D = 2R, there exists a (2k + 2)-approximate solution E such that each edge e E satisfies V [e] U x for some U x U. Proof. Let E be an optimal solution to APD, B = {B G (v, R 1) v V [E ]}, and W be the set of 129

CRPIT Volume 128 - Theory of Computing 2012 Figure 1: Illustration of a 2-tree G. While diam(g + {(u 1, w)}) 2 holds, the size of any star E s whose center v s satisfies d G+Es (v s, v) 1 for every v V is at least p. vertices uncovered by B. Lemma 3.1 says that every two vertices w 1, w 2 W satisfy d G (w 1, w 2 ) D. Then, by Lemma 3.2, there exists a U x U for some x V T such that d G (U x, w) R for every w W. Then we claim that for E = {(u, v) u U x, v V [E ]}, we have diam(g + E ) D. Indeed, for each v V W, d G+E (u, v) R holds for each u U x by v B G (v, R 1) for some v V [E ], and hence we have d G+E (v 1, v 2 ) D for each pair of v 1, v 2 V W. For each pair of w W and v V W, d G+E (v, w) d G+E (v, u w ) + d G (u w, w) D, where u w denotes a vertex in U x with d G (u w, w) R for w W (note that such u w exists by d G (w, U x ) R). Observe that E U x V [E ] 2(k + 1)OP T (G, D). Hence, for a fixed k, if we have a polynomial time c-approximation algorithm A for solving APD with a partial k-tree H satisfying the following condition (5.1) and D = 2R, then APD with a partial k-tree and D = 2R is (2k + 2 + c)-approximable in polynomial time, according to Algorithm Treewidth Even(D). For a tree decomposition (T, U) of H, some U U satisfies d G (U, v) R for all v V (H). (5.1) Here note that for a fixed k, a tree decomposition of a partial k-tree can be computed in linear time (Bodlaender 1996). Indeed, the correctness of Algorithm 3 Algorithm Treewidth Even(D) Input: A partial k-tree G = (V, E) and an even integer D = 2R 2. Output: A set E of new edges such that diam(g + E ) D and E (2k + 2 + c)op T (G, D). 1: Compute a tree decomposition (T = (V T, E T ), U = {U x V x V T }) of G. 2: Choose U x U and a set C 1 of centers for (R 1)-balls covering V ( v U x B G (v, R)) such that C 1 is minimized. Let E 1 := {(u, c) u U x, c C 1 }. 3: Let H be the subgraph of G induced by v U x B G (v, R). Compute a set E 2 of edges with E 2 cop T (H, D) cop T (G, D) so that diam(h + E 2 ) D by applying Algorithm A to H. 4: Halt after outputting the set E = E 1 E 2. this algorithm can be proved as follows. As observed in the proof of Lemma 5.1, for an optimal solution E, {B G (v, R 1) v V [E ]} covers V ( v U B x G(v, R)) for some U x U. From the minimality of C 1, we have C 1 V [E ], from which E 1 (k + 1) C 1 (k + 1) V [E ] 2(k + 1)OP T (G, D). As for Step 3, note that the graph H is also a partial k-tree and satisfies (5.1). Thus, E (2k+2+c)OP T (G, D). Also, we can observe that d G+E (u, v) D for every pair of u, v V with {u, v} V (H) in a similar way to the proof of Lemma 5.1. By diam(h + E 2 ) D and this, we have diam(g + E ) D. We here remark that if in H + E 2, every vertex v V (H) is at distance at most R from U for some subset U of U x, then an addition of E 1 := {(u, c) u U, c C 1 } instead of E 1 is sufficient in Step 2; diam(g + (E 1 E 2 )) D and E 1 (2 U + c)op T (G, D). Also, we can observe that for a partial 2-tree G, there exists such an algorithm A, which we will show in the next subsection. 5.1 Partial 2-trees In this subsection, we show that if k 2, then there exists such an algorithm A which delivers a (4,5)- approximate solution in O( V (G) 5 ) time, and as its corollary, that APD with a partial 2-tree and D = 2R is (8,5)-approximable in O( V (G) 5 ) time. Let G be a partial 2-tree satisfying (5.1) and U x = {u 1, u 2, u 3 } be a member of U such that d G (v, U x ) R for all v V (G) for a tree decomposition (T = (V T, E T ), U = {U x V x V T }) of G (u i = u j may hold). Let G 1 be the graph obtained from G by adding a set E 1 of edges so that U x forms a complete graph; E 1 3. Observe that if two vertices v 1 and v 2 satisfy d G1 (v 1, v 2 ) > D, then d G (v 1, U x ) = d G (v 2, U x ) = R and N R G (v 1) N R G (v 2) = hold, since every vertex v V with d G (v, U x ) R 1 satisfies d G+E1 (v, u i ) R for each i = 1, 2, 3. Let S be the set of vertices v with d G (v, U x ) = R, and for I {1, 2, 3}, S I = {v S N R G (v) U x = {u i i I}}. Note that we have only to consider the distance between every two vertices v 1 and v 2 for v 1 S I1 and v 2 S I2 with I 1 I 2 =. Here, for a vertex cut U in G, we call a component in G U a U-component. For I {1, 2, 3}, we denote by V I the family of U x -components V such that N G (V ) = {u i i I}. Notice that since G is a partial 2-tree, we have V 1,2,3 =. Let E be an optimal solution to G and D. There are the following three possible cases (I) (III): (I) B G (v, R 1) V [E ] holds for all v S or there exists a vertex v S V such that B G (v, R 1) V [E ] = for some U x -component V with N G (V ) = 1, (II) (I) does not hold and for some U x -component V with N G (V ) = 2, there exists a vertex v S I V with I = 1 such that B G (v, R 1) V [E ] =, and (III) neither (I) nor (II) holds and for some U x - component V with N G (V ) = 2, there exists a vertex v S I V with I = 2 such that B G (v, R 1) V [E ] =. In each case, we have the following lemmas. Lemma 5.2 In Case-I, in the latter case, let V be a U x -component with N G (V ) = 1 such that there exists a vertex v S V with B G (v, R 1) V [E ] =, and assume without loss of generality that V V 1. Then, we have V [E ] V H for a minimum set V H hitting {B G (v, R 1) v S 2 S 3 S 2,3 }. For E 2 = 130

Proceedings of the Eighteenth Computing: The Australasian Theory Symposium (CATS 2012), Melbourne, Australia {(u 1, v) v V H }, we have diam(g+(e 1 E 2 )) D and E 2 V H 2OP T (G, D). Proof. Lemma 3.1 indicates that B G (v 1, R 1) V [E ] or B G (v 2, R 1) V [E ] for each pair of v 1, v 2 S with d G (v 1, v 2 ) > D. Hence, even in the case where V exists, for each v S 2 S 3 S 2,3, we have B G (v, R 1) V [E ]. From the minimality of V H, it follows that V [E ] V H. Observe that E 2 V H V [E ] 2OP T (G, D). In G + E 2, every vertex in S is at distance at most R from u 1. It follows that diam(g + (E 1 E 2 )) D. In Case-II, let V V 1,2 and I = {1} without loss of generality. Then, we divide into the following two cases: (II-1) V [E ] B G (v, R 1) for all v S 2 V and (II-2) V [E ] B G (v, R 1) = for some v S 2 V. Then, the following lemma holds. Lemma 5.3 In Case-II-1, we have V [E ] V H for a minimum set V H hitting {B G (v, R 1) v S 2 S 3 S 2,3 }. For E 2 = {(u 1, v) v V H }, we have diam(g + (E 1 E 2 )) D and E 2 V H 2OP T (G, D). In Case-II-2, let S i = S i V for i = 1, 2. Now by the structure of partial 2-trees, we can observe that V has a vertex u 4 such that {u 1, u 2, u 4 } is a vertex cut in G and every vertex in V {u 4 } is separated from u 2 by {u 1, u 4 } or from u 1 by {u 2, u 4 }. Then, let VH 1, V H 2, and V H 3 be minimum sets of vertices hitting {B G (v, R 1) v S 1 S 3 S 3,1 B G (u 4, R 1)}, {B G (v, R 1) v S 2 S 3 S 2,3 B G (u 4, R 1)}, and {B G (v, R 1) v S 1 S 2 S 3 S 2,3 S 3,1 B G (u 4, R)}, respectively. Lemma 5.4 In Case-II-2, we have V [E ] min{ VH 1, V H 2, V H 3 }. Let E 2 = {(u 2, v) v VH 1 } {(u 2, u 4 )} if VH 1 min{ V H 2, V H 3 }, E 2 = {(u 1, v) v VH 2 } {(u 1, u 4 )} if VH 2 min{ V H 1, V H 3 }, and E 2 = {(u 1, v) v VH 3 ( {B G (u, R 1) u S 2 S 3 S 2,3 B G (u 4, R)})} {(u 2, v) v VH 3 ( {B G (u, R 1) u S 1 S 3 S 3,1 B G (u 4, R)})} {(u 1, u 4 ), (u 2, u 4 )} otherwise. Then, we have diam(g + (E 1 E 2 )) D and E 2 2 min{ VH 1, V H 2, V H 3 } + 2 4OP T (G, D) + 2. Proof. By Lemma 3.1 and the assumption on V, we can observe that B G (v, R 1) V [E ] for all v S 3 S 2,3 S 3,1 (S 2 S 2) (S 1 S 1). There are the following three possible cases: (a) B G (v, R 1) (V [E ] {u 4 }) for all v S 1, (b) B G (v, R 1) (V [E ] {u 4 }) for all v S 2, and (c) otherwise. First consider the case of (a). From the minimality of VH 1, we have V [E ] VH 1. In G + E 2 where E 2 = {(u 2, v) v VH 1 } {(u 2, u 4 )}, every vertex in S is at distance at most R from u 2. Hence, we have diam(g + (E 1 E 2 )) D. Also, observe that E 2 VH 1 + 1 V [E ] + 1 2OP T (G, D) + 1. Next consider the case of (b). From the minimality of VH 2, we have V [E ] VH 2. In G + E 2 where E 2 = {(u 1, v) v VH 2 } {(u 1, u 4 )}, every vertex in S is at distance at most R from u 1. Hence, we have diam(g + (E 1 E 2 )) D. Also, observe that E 2 VH 2 + 1 V [E ] + 1 2OP T (G, D) + 1. Finally, consider the case of (c). Let S i be the set of vertices v S i such that B G(v, R 1) (V [E ] {u 4 }) = for i = 1, 2; by assumption on (c), we have S 1 S 2. Then, we first claim that each v S i is included in some {u i, u 4 }-component W i in G with u j / W i for {i, j} = {1, 2}. Indeed, if a vertex v S 1 is not included in any {u 1, u 4 }-component not containing u 2 without loss of generality, then v is separated from u 1 by the vertex cut {u 2, u 4 }. In this case, it follows by d G (v, {u 1, u 2 }) = R and v / B G (u 4, R 1) that d G (v, u 1 ) d G (v, {u 2, u 4 })+1 > R would hold, contradicting v S 1. By this claim, we can observe that in G[V ], every two vertices v 1 S 1 and v 2 S 2 are separated by the cut vertex u 4. On the other hand, Lemma 3.1 indicates that every two vertices v 1 S 1 and v 2 S 2 satisfy d G (v 1, v 2 ) D. Hence, each v S 1 S 2 satisfies v NG R(u 4). It follows that for every v S 1 S 2 S 3 S 2,3 S 3,1 NG R(u 4), we have (V [E ] {u 4 }) B G (v, R 1). By the minimality of VH 3, we have V [E ] VH 3. Let E 2 = {(u 1, v) v VH 3 ( {B G (u, R 1) u S 2 S 3 S 2,3 B G (u 4, R)})} {(u 2, v) v VH 3 ( {B G (u, R 1) u S 1 S 3 S 3,1 B G (u 4, R)})} {(u 1, u 4 ), (u 2, u 4 )}. For G + E 2, every vertex in S (S 2 NG R(u 4)) is at distance at most R from u 1, and every vertex in S (S 1 NG R(u 4)) is at distance at most R from u 2. Since every two vertices v, v NG R(u 4) satisfy d G (v, v ) D, it follows that diam(g + (E 1 E 2 )) D. Also, we can observe that E 3 2 VH 3 + 2 2 V [E ] + 2 4OP T (G, D) + 2 since for each v VH 3 ( {B G (u, R 1) u S 3 }), we have {(v, u 1 ), (v, u 2 )} E 2. Lemma 5.5 In Case-III, let V V 1,2 without loss of generality. Then, we have V [E ] V H for a minimum set V H hitting {B G (v, R 1) v S 1 S 2 S 3 }. For E 2 = {(u 1, v) v V H ( {B G (u, R 1) u S 2 S 3 })} {(u 2, v) v V H ( {B G (u, R 1) u S 1 })}, we have diam(g + (E 1 E 2 )) D and E 2 2 V H 4OP T (G, D). Proof. Since neither (I) nor (II) holds, we have B G (v, R 1) V [E ] for all v S 1 S 2 S 3. From the minimality of V H, it follows that V [E ] V H. Since for V V i,j, each vertex v V H V may be included in B G (v, R 1) B G (v, R 1) for some v S i and v S j, we have E 2 2 V H 2 V [E ] 4OP T (G, D). For G + E 2, every vertex in S S 2,3 is at distance at most R from u 1. On the other hand, for G + E 2, every vertex in S 1 is at distance at most R from u 2. Hence, it follows that diam(g + (E 1 E 2 )) D. Now we are ready to describe a (4,5)- approximation algorithm for solving APD with G and D (see Algorithm 4). Observe that according to the proofs of Lemmas 5.2 5.5, such a set E 2 of edges can be found in O( V (G) 5 ) time, because we need to compute minimum sets hitting some family of (R 1)-balls O( V (G) ) times and one minimum hitting set can be found in O( V (G) 4 ) time by Lemma 3.4. We here remark that from the construction of E 2, there exist at most two vertices u, u U x such that every vertex in S is at distance at most R from u or u in U x in G + E 2. Hence, from the arguments in the last paragraph in the previous subsection, we can find an (8,5)-approximate solution in O(n 5 ) time for APD with a partial 2-tree G and D = 2R by applying Algorithm Treewidth Even(D) to an arbitrary partial 2-tree while utilizing Algorithm 4 as Algorithm A. Summarizing, Theorem 2.4 is established. Finally, we consider the case where D is odd. We can apply approaches similar to Step 1 of Algorithm 131