A 1.8 approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2.

Transcription

1 A 1.8 approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2. Guy Even Jon Feldman Guy Kortsarz Zeev Nutov May 14, 2006 Abstract We present a 1.8-approximation algorithm for the following NP-hard problem: given a connected graph G = (V, E) and an additional edge set E, find a minimum size subset of edges F E such that (V, E F) is 2-edge-connected. Our result improves and simplifies the approximation algorithm of ratio ε [17]. Preliminary version appeared in Proceedings of APPROX 2001, Lecture Notes in Computer Science 2129, Springer, pp , Dept. of Electrical Engineering-Systems, Tel-Aviv University, Tel-Aviv 69978, Israel. guy@eng.tau.ac.il. Columbia University, New York, NY. jonfeld@ieor.columbia.edu. Part of this work was done while the author was visiting Tel-Aviv University. Rutgers University, Camden. guyk@crab.rutgers.edu. Part of this work was done while the author was at the Open University of Israel. Open University of Israel, 16 Klauzner St, Tel-Aviv, Israel. nutov@shaked.openu.ac.il.

2 1 Introduction Graph connectivity problems are fundamental to the study of robust network design. A multigraph (i.e., a graph in which parallel edges are allowed) is said to be k-edge-connected if every pair of nodes in the graph is connected by k edge-disjoint subpaths. The edge-connectivity of a graph is the largest value k for which the graph is k-edge-connected. High edge connectivity is useful in networks since connectivity is not lost even if some of the edges fail. In this paper we study one of the most basic connectivity augmentation problem: We are given a connected undirected graph G(V, E) and a set of additional edges (called links ) E V V. Consider a subset of links F E. Let E F denote the multiset union of edges obtained by adding the links in F to the edges in E (i.e., if (u,v) is both a link in F and an edge in E, then it appears twice in E F). An augmentation of G is a subset of links F E such that the edge-connectivity of G(V, E F) is higher than that of G(V, E). In our problem, the goal is to find a minimum size augmentation F such that G(V, E F) is 2-edge-connected. The 2-edge-connected components of the given graph G form a tree. It follows that by contracting these components, one may assume that G is a tree. Hence, our problem is equivalent to the Tree Augmentation Problem (TAP), defined as follows: Given a tree T(V, E) and a set of links E V V, find the smallest subset F E such that G(V, E F) is 2-edge-connected. A ρ approximation for this problem gives an approximation for a more general problem with the same ratio: augmenting the connectivity from k to k + 1 for any odd k. The equivalence of these two problems is due to the cactus model (see for example [4]). The Tree Augmentation Problem is NP-Complete [2], and not approximable within some constant c > 1 [13]. In [5] we presented a 1.5 ratio approximation algorithm for the problem. The full description of this result is now 40 pages long, uses a complicated credit scheme, and contains a long case analysis for trees with at most 5 leaves. Here we present a weaker result with the following advantages: (1) it improves the ratio of ( ε) [17]; (2) we believe it is simpler than [17] (i.e., fewer cases, fewer structures, shorter), and (3) the major techniques of the 1.5 approximation ratio algorithm are presented. Breaking the 2-approximation barrier. Every leaf of the input tree T must be incident to a link in the set F. Since each link can be incident to at most two leaves, it must be that F has size at least half the number of leaves in T. This simple lower bound can be used to design a 2-approximation algorithm for TAP (see Sec. 3). The first 2-approximation for TAP (which also holds for the weighted version) dates back twenty years previous to this result, and was given by Frederickson & Jájá [8]. Improving the approximation ratio below 2 was a long-standing open problem, and was characterized by Khuller [18] as the main open problem in graph augmentation and one of the main open problems in graph connectivity. The first approximation algorithm that achieved an approximation ration below 2 was presented by Nagamochi [17, 16]. He reported a ( ε)-approximation ratio. The running time of the algorithm depends exponentially on 1/ǫ. One could avoid the dependence of ε by setting, for example, ε = to obtain a 1.9 approximation algorithm. Several of the ideas introduced in [17, 16] are used in this paper (e.g., computing maximum leaf-closed trees and stems and the lower bound). Our approximation algorithm is based on a lower bound, called the unmatched leaves lower bound (see Claim 22). A similar lower bound was proven in [17, 16]. Key to our analysis is a 1

3 credit scheme for using the different parts of the lower bound. This credit system has a static component and a dynamic component. The static credit is computed in the beginning of the algorithm and is distributed to different parts of the graph. The dynamic credit is available only after the algorithm reveals it by proving that the optimum solution had to be larger than the initial static estimate. We believe that this technique has merit of its own and may be useful for other augmentation problems. Related Results. The weighted version of augmenting a graph so that it becomes 2-edge connected is called Bridge-Connectivity Augmentation (BRA) in [8]. In the BRA problem, the input consists of a complete graph G = (V,E) and edge weights w(e). The goal is to find a minimum weight subset of edges F such that G = (V,F) is 2-edge-connected. The TAP problem is simply the case in which the edges have {0,1, } weights; the edges of the connected graph have zero weight, links have unit weight, and all other edges have infinite weight. There are a few 2-approximation algorithms for the BRA problem. The first algorithm, by Frederickson and Jájá [8] was simplified later by Khuller and Thurimella [19]. These algorithms are based on constructing a directed graph and computing a minimum weight arborescence. The primal-dual algorithm of Goemans & Williamson [10] and the iterative rounding algorithm of Jain [11] are LP-based 2-approximation algorithms. The approximation ratio of 2 for all these algorithms is tight. Fredrickson and Jájá [8] showed also that when the edge costs satisfy the triangle inequality, the Christofides heuristic leads to a 3/2-approximation algorithm. Cheriyan et. al. [3] presented a 17/12-approximation algorithm for the special case of BRA in which edges have {1, } weights. A 4/3-approximation algorithm was presented for {1, } weights by Vempala & Vetta [20]. The best known ratio for the {1, } version is 5/4 [12]. Eswaran & Tarjan [6] presented a linear time algorithm for the special case of BRA in which all edges have unit weights. Surveys for broad classes of augmentation problems can be found in [18, 7, 14]. The TAP problem is sometimes posed equivalently as the problem of covering laminar families of subsets (see e.g., [2])). Namely, given a laminar family S of proper subsets of a ground set U, and an edge set E on U, find a minimum subset F E such that for every S S, there is an edge in F with one endpoint in S and the other in V S. Garg et al. [9] presented a 5/3 approximation algorithm for a special case when all the candidate edges are incident to the leaves of the tree. 2 Preliminaries Throughout the paper, the TAP instance consists of the input tree T = (V, E) and the set of links E. In this section, we define the basic graph terminology used throughout the paper. We also describe the basic operation of our augmentation algorithm, which is to add a link from E to the solution F, and then contract the resulting 2-edge-connected component of the tree T. We describe how the connectivity augmentation problem can then be seen as a covering problem, where links in E cover certain parts of the tree. We also give some simplifying assumptions that will be essential for the approximation ratio arguments in later sections of the paper. 2.1 Rooted Trees Let T = (V, E) denote a tree defined over a vertex set V and edge set E. A rooted tree is a tree T with a node r V designated as a root. The root induces a partial order on V as follows. For 2

4 u,v V, we say that u is a descendant of v, and that v is an ancestor of u, if v is an interior node in the path from r to u in T; if, in addition, (u,v) is an edge of T, then u is a child of v, and v is the parent of u. We denote the parent of u by par(u). The least common ancestor, lca(u,v), of u and v is the common ancestor of u and v with the largest distance to the root. Assumption 1: The input tree T = (V, E) is a rooted tree (T,r), where an arbitrary node r V is designated as the root. The leaves of a rooted tree T = (V, E) are the nodes in V r that have no descendants. We denote the leaf set of T by L(T), or simply by L, when the context is clear. We refer to a rooted tree as a pair (T,r), and do not mention the root when it is clear from context. Consider a rooted tree (T,r) and a vertex v V. The rooted subtree of T induced by v and its descendants is denoted by T v. Note that v is the root of T v. A subtree T of T is called a rooted subtree if T = T v, for some vertex v V. Definition 2: Let P be some property. A rooted subtree T of rooted tree T is P-minimal if T satisfies property P, but no proper rooted subtree of T satisfies property P. 2.2 Contraction of links To contract a subset X of vertex set V is to combine all nodes in X into a single new node x. All edges and links with both endpoints in X are deleted. All edges and links with one endpoint in X now have x as their new endpoint, but retain their correspondence with the edge or link of the original graph. If the root r is contained in X, then x becomes the root of the new tree. For simplicity, we say that the new node x contains a node v if v X. Let p(u,v) denote the path between u and v in the tree T = (V, E). Consider a link (u,v) E. If we decide to add (u,v) to the solution F, then the vertices along the path p(u,v) belong to the same 2-edge-connected component of the augmented graph (V, E F). Hence, we may contract the vertices along p(u,v). For a set of links F E, let T/F denote the tree obtained by contracting every 2-edge-connected component of T F into a single node. Since all contractions are induced by subsets of links, we refer to the contraction of every 2-edge-connected component of T F into a single node simply as the contraction of the links in F. Instead of defining a new node into which the vertices are contracted (as in the definition of contraction above), it is convenient to regard this contraction as a contraction of the vertices along p(u, v) into the least common ancestor, namely, lca(u, v). The advantage of this convention is that it enables a canonical representation of T/F: simply contract one by one all the paths in T corresponding to the links of F. The order in which the links are contracted does not affect the resulting tree. The contracted tree is a subtree of T. 2.3 TAP as a Covering Problem We say that a link (u,v) covers all the edges along the path p(u,v). This terminology enables us to formulate TAP as a covering problem. Namely, find a minimum size subset of links in E that covers the edges of T; we refer to such a link set as a cover of the tree T. We extend this terminology and say that a link (a,b) covers a node v if (v,par(v)) p(a,b). Note that any link set that covers every edge of T also covers every node of T, besides the root. 3

5 Assumption 3:(Feasibility) The link set E contains a cover; or, equivalently, the graph (V, E E) is 2-edge-connected. Note that feasibility can be tested beforehand by contracting every link of E. 2.4 Shadows The following definition will allow us to make some very important assumptions about the input link set E and the tree T that simplify many of the arguments we make. Definition 4: A link (u,v ) is a shadow of a link (u,v) if p(u,v ) p(u,v), and (u,v ) is a proper shadow of (u,v) if the inclusion is proper, i.e., if p(u,v ) p(u,v). A link is maximal if it is not a proper shadow of any other link. Assumption 5: The set of links E is closed under shadows, that is, if (u,v) E and p(u,v ) p(u,v) then (u,v ) E. Every instance can be rendered closed under shadows by adding shadows of existing links as new links. We refer to the addition of all shadows as shadow completion. Shadow completion does not affect the optimal solution size, since every shadow can be replaced by a maximal link. (However, shadow completion may increase the size of the set of links so that there are Ω(n 2 ) links.) The following examples illustrate how shadow completion helps in the analysis of the TAP problem. Example 1: Consider a rooted subtree T v and two nodes w T v and z T v. Assume that (w,z) is a link. Obviously, v is a node along the path p(w,z). Shadow completion implies that (w,v) is a link, as well as the links joining w to all its other ancestors along the path p(w,v). Example 2: If the problem instance is feasible, that is, the graph (V, E E) is 2-edge-connected, then shadow completion implies that (v,par(v)) is a link, for every non-root v. The reason is that an edge (v, par(v)) E must be covered by some link in E. (Recall that par(v) denotes the parent of v.) Definition 6: A cover F of T is shadows minimal if, for every link (u, v) F, replacing (u, v) by a proper shadow of (u,v) results in a set of links that does not cover T. 2.5 Reducible Edges Further simplifying assumptions can be made when we consider the relationship between pairs of edges in T. Definition 7: An edge e of a tree T is reducible with respect to a link set E if e is covered by a unique maximal link or if there exists a tree edge e e such that every maximal link that covers e also covers e. If an edge (u,v) is covered by a unique maximal link (a,b), then adding (a,b) to the cover is an optimal step (i.e., the size of an optimal solution in the residual problem is smaller by one). If an edge e is a shadow of every maximal link that covers e e, then e may be contracted without affecting the sizes of the optimal solution and the computed solution. Hence, we may preprocess the instance so that there are no reducible edges. 4

6 Assumption 8: There are no reducible edges in the tree T = (V, E) with respect to the link set E. The following proposition shows that if there are no reducible edges, then leaves cannot hang from degree-2 vertices. Proposition 9: If there are no reducible edges, then every parent of a leaf has at least two children. Proof: Consider a leaf l, its parent v and its grandparent u. There are at least two maximal links emanating from l. Hence, every maximal link emanating from l must cover the tree edge (u,v). Hence, (u,v) is reducible, a contradiction. Although there cannot be a path (of length 2 or more) consisting of degree-2 vertices ending at a leaf, the tree may contain long paths consisting of degree 2 nodes. Such paths may not end at leaves if there are no reducible edges. 3 Motivation: a 2-approximation algorithm We start the discussion with a 2-approximation algorithm. This rather simple algorithm introduces the notions of leaf-closed subtrees, up-links, local-ratio[1], and disjointness of credit. Our 1.8- approximation algorithm is based on these ideas. 3.1 Preliminaries Recall that L(T) refers to the set of leaves of T. The 2-approximation algorithm is based on the following leaf lower bound. Claim 10: If link set F E is a cover of the tree T, then F L(T). 2 Proof: Every leaf u L(T) has at least one link of F incident to it, since any link that covers the edge (u,par(u)) must be incident to u. Each link of F is incident to at most two leaves, and hence F L(T) /2. The ratio between the value of an optimal solution and the leaf lower bound can be arbitrarily large. For example, if T is a path, and E consists only of links parallel to the edges of T, then L(T) /2 = 1, but the value of an optimal solution is V 1. To obtain an approximation algorithm from the leaf lower bound, we partition the tree so that a local ratio argument can be applied. Let opt denote an optimum cover. We partition the input tree edges E into disjoint parts E 1,...,E k. Even though opt is unknown, each part E i is covered by a subset opt i opt such that the subsets opt 1,...,opt k are disjoint. If we are able to cover each part E i by at most 2 opt i edges, then a 2-approximate algorithm follows. Example 3: Suppose that E 1 is the set of edges emanating from leaves and that E 2 = E \ E 1. Covering E 1 is easy; simply pick a set of links F with one link per leaf. The cost of covering the leaves is L(T), and the leaf lower bound is L(T) /2, so the local approximation ratio for solving the subproblem E 1 is 2. However, what can we say about the local approximation ratio for solving the subproblem E 2? 5

7 Suppose we recurse now on T/F by covering the leaves of T/F. Unfortunately, this does not yield a 2-approximation algorithm. The problem lies in the fact that a link in opt can cover a leaf in L(T) as well as a leaf in L(T/F). Hence opt 1 and opt 2 are not disjoint. The example above motivates the following definition and lemma, which appeared in the paper of Nagamochi and Ibaraki [15] for U = L(T). Definition 11: Let U V. A rooted subtree T of T is U-closed (with respect to E) if there is no link between U V (T ) and V \ V (T ). A minimally U closed subtree can be found in polynomial time (see [15]). For any U V, the set of rooted subtrees that are U-closed is not empty since the whole tree T is trivially U-closed. This implies that, for every U V, there exists a minimal U-closed subtree (as in Definition 2). The highest link incident to a node u is defined as the link (u,v) such that v is as close as possible to the root. Note that, for every link (u,v ), v is not a proper ancestor of v. We denote the highest link incident to u by up(u). We often refer to up(u) as the up-link emanating from u. For a node set U V, we let up(u) = {up(u) : u U}. Lemma 12: Let U V, and let T be a U-closed minimal rooted subtree of T. Then up(u V (T )) covers T. Proof: Suppose that up(u V (T )) does not cover an edge (u,v) of T, where u is a child of v. Then T u is U-closed. This contradicts the minimality of T. A rooted subtree T of T is leaf-closed if it is L(T )-closed. Lemma 12 implies the following claim, that appears in [15]. Claim 13: [15] A leaf-closed minimal subtree T of T is covered by up(l(t )). 3.2 The 2-Approximation Algorithm We now present a simple 2-approximation algorithm. To simplify the correctness proof, we present a recursive version of the algorithm. The algorithm starts with the whole tree T. It finds a leafclosed minimal subtree T and returns the cover consisting of up(l(t )) and a (recursively found) cover of T/up(L(T )). The algorithm stops when T is contracted into a single node. A listing of the algorithm appears as Algorithm 1. Algorithm 1 cover(t) - A 2-approximate algorithm. 1: If V (T) = 1, then return and stop. 2: Compute a leaf-closed minimal subtree T of T. 3: Return up(l(t )) cover(t/up(l(t )). Since at least one edge is covered in each recursive call, the algorithm terminates. The algorithm returns a feasible cover since, by Claim 13, up(l(t )) covers the minimal leaf-closed subtree T. The following claim establishes the approximation ratio of the algorithm. Claim 14: The approximation ratio of Algorithm cover(t) is 2. 6

8 Proof: We prove that the size of the computed cover is 2-optimal by induction on k, the number of recursive calls of the algorithm. The induction basis, for k = 0, is trivial since T contains a single node. For the induction step, consider a leaf-closed minimal subtree T of the first iteration of the algorithm. Let F be an arbitrary optimal cover of T. Let FL(T ) be the links in F incident to the leaves of T, and let Fres be the links in F that cover at least one link not in T. The leaf lower bound implies that FL(T ) L(T ) /2. Since T is leaf-closed, both endpoints of every link in FL(T ) are in V (T ). Hence, FL(T ) and F res are disjoint. Moreover, F res covers T/up(L(T )). By the induction hypothesis, the recursive call for covering T/up(L(T )) produces a solution F res with at most 2 Fres links. Hence, the total number of links in the solution produced by the algorithm satisfies: up(l(t )) + F res = L(T ) + F res 2 FL(T ) + 2 F res 2 F, and the claim follows. One way to interpret the leaf lower bound is that we are given a credit of one coupon per leaf, where each coupon can pay for one link. Since the optimal solution uses at least L(T)/2 links, our goal in the 2-approximation algorithm is to pay for a cover of the graph using the coupons. In the algorithm above, we use L(T ) leaf coupons to cover the leaf-closed minimal subtree T using L(T ) links. Then, by showing that FL(T ) and F res are disjoint, we prove that the optimal cover F is forced to contain a link that is not a leaf-to-leaf link; this reveals an extra source of credit, due to the contribution of such a link to the degree-sum of F. It is useful to interpret the recursion step as an assignment of an extra coupon to the new leaf in T/up(L(T )) created by the contraction of the tree T. In the following section we present a lower bound that strengthens the leaf lower bound, as well as a generalization of the idea of leaf-closed minimal subtrees, in order to improve the approximation ratio. 4 The unmatched leaves lower bound In this section we present a new lower bound on the size of a minimum cover, called the unmatched leaves lower bound (see Coro. 23). This lower bound is based on a maximum matching of links. 4.1 Notation and preliminaries The following definitions are depicted in Figure 1. Definition 15: A non-root node s V r is a stem of a tree T if it has exactly two children and both of the children are leaves. The two children of a stem are called a twin-pair, and a link between them is called a twin-link. The edge connecting a stem s and its parent is called a stem-edge. The following claim proves the existence of twin-links if no edge is reducible. Claim 16: If no edge is reducible, then there is a twin-link between every twin-pair. Proof: Consider a twin-pair depicted in Figure 1. If (l 1,l 2 ) is not a link, then since there are at least two maximal links that cover (l 1,s), every maximal link that covers (l 1,s) also covers the edge (s,v). Hence, (s,v) is reducible, a contradiction. 7

9 a stem-edge a stem a leaf v s l 1 l 2 a twin link Figure 1: A stem, a twin-pair, a twin-link, and a stem-edge. (The node s is the parent of two leaves l 1 and l 2.) We denote the the set of stems of a tree T by St(T). Leaves and stems play a special role in our algorithm. We say that a node v of T is special if v is a leaf or a stem. We denote the set of special nodes in T by Sp(T). We denote the set of non-special nodes in T by nsp(t). We often use L,St,Sp for short to denote the set of leaves, the set of stems, and the set of special nodes, respectively. We say that a rooted subtree T of T is leaf-stem-closed if it is Sp(T)-closed (see Definition 11). For X,Y V and a set of links F, let F X,Y denote the be the set of links in F that have one endpoint in X and the other in Y. The following lemma shows that leaf-to-leaf links in a shadows minimal solution constitute a matching. Lemma 17: Let F be a shadows minimal cover of T (see Definition 6). Then, (i) every two links in F that share an endpoint cover disjoint sets of edges, (ii) there is exactly one link in F emanating from every leaf v of T, and (iii) F L,L is a matching. Proof: We begin by proving (i). Suppose that (v,u) and (v,w) are links in F. Since T is a tree, if the paths p(v,u) and p(v,w) have an edge in common, then the first edge (emanating from v) along these paths is identical. Let (v,q) be the first edge in these paths. The edge (v,q) is covered twice, hence F is not shadows minimal. (We can replace the link (v, u) F by its proper shadow (q,u) and still cover the edges of T.) Part (i) implies the other two parts. For a set of links F and a vertex v, let deg F (v) denote the number of links in F incident to v. For a subset U V of nodes, we let deg F (U) = v U deg F(v). Figure 2 depicts the terms defined in the following definition. Definition 18: Consider a matching M that consists of leaf-to-leaf links. A leaf l L is unmatched with respect to M if deg M (l) = 0. The set of unmatched leaves in T with respect to M is denoted by UL T (M). A stem s S is uncovered with respect to M if stem-edge (s,par(s)) is not covered by the links of M. The set of uncovered stems in T with respect to M is denoted by US T (M). Consider some optimum cover F, and assume that F is shadows minimal (we use F to denote a shadows minimal optimal cover for the remainder of the paper). The optimality of F implies that if s is a stem and l 1 and l 2 are the children of s, then it is not possible that both the links (s,l 1 ) and (s,l 2 ) are in F. The reason is that these two links can be replaced by the link (l 1,l 2 ). (Recall that the twin-link is always present by Claim 16.) The following assumption shows that we may exclude the case that a stem is linked in F to one of its children. 8

10 v s u b 2 v s l 1 l 2 b 1 a 1 Figure 2: A matching M consisting of two leaf-to-leaf links: (b 1,b 2 ) and (l 1,l 2 ). The leaf a 1 is the only unmatched leaf with respect to M. The stem s uncovered with respect to M while the stem s is covered by M. Assumption 19: Without loss of generality, an optimal shadows minimal cover F of T lacks links from a stem to one of its children. Proof: Assume that s is a stem with children v 1 and v 2. Suppose that (s,v 1 ) F and let u be in V such that (v 2,u) F. Then, we may replace the two links (s,v 1 ) and (v 2,u) by the links (v 1,v 2 ) and (s,u). (The link (s,u) is a shadow of (v 2,u).) This replacement maintains feasibility as well as shadows minimality. The following claim is a consequence of Assumption 19. Claim 20: Let s denote a stem. If F is a shadows minimal cover that satisfies Assumption 19, then deg F (s) 1. Moreover, deg F (s) = 1 if and only if s is uncovered with respect to F L,L. Note that Claim 20 also implies that a stem s is uncovered with respect to FL,L the twin-link between the children of s is in FL,L. if and only if 4.2 The lower bound In this section T is an arbitrary leaf-stem closed subtree of T. Recall that F denotes a shadows minimal optimal cover of T that satisfies Assumption 19. The following claim proves a lower bound on the number of links of F that have both endpoints inside T ; the lower bound is in terms of the number of leaves, uncovered stems, unmatched leaves and links incident to non-special nodes in T. It is useful to view this as a local bound that tells us how much effort has to be spent to cover a subtree T. This bound holds for any subtree T as long as it is leaf-stem closed. Notation. We use the following notation: V = V (T ) denotes the vertices in T, L = L(T ) denotes the set of leaves in T, St = St(T ) denotes the set of stems in T, nsp = nsp(t ) denotes the set of non-special nodes in T, 9

11 F L,L denotes the leaf-to-leaf links in F that are contained in T, F V,V denotes the links in F with both endpoints in T, US T denotes the stems in T that are not covered by F L,L, UL T denotes the leaves in T that are not covered by F L,L, deg F (V,V )(nsp ) denotes the sum of the degrees of non-special nodes in T when only links in F (V,V ) are considered. Claim 21: The number of links of F that are contained in T satisfies, F V,V 1 2 L US T UL T deg F V,V (nsp ). (1) Proof: The left hand side equals the number of links in F that have both endpoints inside T. We partition FV,V into all possible combinations of endpoints in L, St and nsp to get: FV,V = F L,L + F L,St + F L,nSp + F St,St + F St,nSp + F nsp,nsp. (2) By shadows-minimality and Assumption 19, every leaf has exactly one link of F incident to it, and every stem has at most one link of F incident to it. A link covering a leaf or a stem in T must be in FV,V since T is leaf-stem closed. By counting degrees we obtain L = 2 FL,L + F L,St + F L,nSp. (3) By Claim 20, a stem is uncovered with respect to FL,L if and only if there is a link of F emanating from it. Therefore, US T = F L,St + 2 F St,St + F St,nSp (4) Since FL,L is a matching, it follows that a leaf l is unmatched if and only if it is covered by a leaf-to-stem link or leaf-to-non-special link. Hence, UL T = F L,St + F L,nSp. (5) Multiply Eq. 3 by 1 2, multiply Eq. 4 by 1 3, multiply Eq. 5 by 1 6, and add them to get 1 2 L US T UL T = F L,L + F L,St F L,nSp F St,St F St,nSp F L,L + F L,St + F L,nSp + F St,St + F St,nSp + F nsp,nsp 1 3 deg F V,V (nsp ). (6) The claim now follows from Eq. 2. Let M denote the set of leaf-to-leaf links in F that are not twin-links. We use MT the subset of links in M that are incident to leaves in T. to denote 10

12 Claim 22: L 2 M T = UL T + 2 US T. Proof: The left hand side equals the number of unmatched leaves in T with respect to M. Such an unmatched leaf l can be unmatched due to exactly one of the following reasons: (i) l is unmatched with respect to FL,L or (ii) l is matched by a twin-link in F L,L. The two terms on the right hand side account for these two reasons. Namely, UL T is the number of unmatched leaves with respect to FL,L. For every pair of leaves matched by a twin-link, we have one stem in US T. Hence, the claim follows. Let M denote a maximum matching of leaf-to-leaf links excluding twin-links. Corollary 23: [The unmatched leaves lower bound] F (V,V ) 2 3 L 1 3 M T deg F V,V (nsp) 2 3 L 1 3 M T deg F V,V (nsp). Proof: The first part follows directly from Claims The second part follows from the maximality of M and the fact that T is leaf-closed. Let us compare the unmatched leaves lower bound with the leaf lower bound (we ignore the last term in the unmatched leaves lower bound in this comparison). Suppose that the matching M is a perfect matching of the leaves. Then, M T = L /2, and 2 3 L 1 3 M T = L /2, and the two lower bounds are equal. On the other hand, as M T decreases (i.e., when M is not a perfect matching), the gap between the unmatched leaves lower bound and the leaf lower bound increases. Another way to compare the lower bounds is to count leaves. In the leaf lower bound, every leaf contributes 1/2. In the unmatched leaves lower bound, an unmatched leaf contributes 2/3 and a matched leaf contributes 1/2. 5 A 9/5-approximation algorithm The 1.8-approximation algorithm distributes credit according to the unmatched leaves lower bound of Claim 22. This credit is used by the algorithm to pay for links that are added to the cover and then contracted to reduce the tree. The amount of credit never exceeds 1.8 F. Hence, if the algorithm is able to cover the tree without over-spending credit, then an approximation ratio of 1.8 is obtained. In Sec. 5.1, we begin by presenting the credit scheme and how accounting of credit is managed by using coupons and tickets. The Coupons Invariant specifies how many coupons should be assigned to nodes and links. In Sec. 5.2, we present a greedy step in which links the contraction of links is paid by coupons and tickets while preserving the Coupons Invariant. The intuition of the algorithm is presented in Sec The algorithm itself is presented in Sec Coupons and tickets The justification for coupons and tickets is provided by the following rearrangement of the unmatched leaves lower bound. 1.8 F (V,V ) 1.2 ( L 2 M T ) M T deg F V,V (nsp). (7) 11

13 After the maximum matching M (that excludes twin-links) is computed, the algorithm distributes coupons as follows. Unmatched leaves. Every leaf that is unmatched by a link in M is given 1.2 coupons. The total number of coupons distributed to unmatched leaves equals the first term in the right hand side of Eq. 7. Links in M. Every link in M is given 1.8 coupons. Note that the total number of coupons distributed to links in M equals the second term in the right hand side of Eq. 7. Consider the last term on the right hand side of Eq. 7. If we can prove that an optimal cover F contains a link that emanates from a non-special node v, then we may give 3 5 coupons to v. We refer to such an event as finding a ticket since coupons are known after the matching M is computed while tickets are not. Even though a ticket is worth 0.6 coupons, to simplify the presentation we say it is worth half a coupon. (In fact, the analysis works even if a ticket is worth 0.2 coupons since we are presenting a 1.8 approximation ratio rather than the tighter 1.5-approximation.) Note that the total number of tickets is bounded by deg F V,V (nsp), so the purchasing power of tickets does not exceed the third term in Eq. 7. We emphasize that the problem with using tickets is the burden of proving that F contains a link emanating from a non-special node. The coupons invariant. The algorithm maintains a partial cover J that is simply a set of links that were selected so far. Recall that the nodes of the contracted tree T/J are 2-edgeconnected components of T J. A compound node in T/J is a node that corresponds to a 2-edgeconnected component in T J that contains more than one node. Moreover, recall that we follow the convention that the compound node is named after the lca of the contracted 2-edge-connected component. We often identify a compound node v in T/J with the 2-edge-connected component in T J that is contracted into v. Invariant 24:[Coupons Invariant] 1. Every link in M owns 1.8 coupons. 2. Every leaf in T/J that was unmatched by M owns 1.2 coupons. 3. Every compound node in T/J owns one coupon. Active nodes. The algorithm does not maintain explicit lists of how many coupons are owned by each link and each node. Instead, the algorithm holds a partial cover J E and a subset of nodes A V called the active nodes. The set of active nodes is initialized to the set of unmatched leaves. After each contraction, the resulting compound node is activated while the active nodes that are contracted into the compound node are deactivated. Hence, every leaf in T/J that is not matched by M and every compound node is active. The Coupons Invariant requires that every active node own a coupon if it is compound and 1.2 coupons if it is an unmatched leaf. 5.2 Greedy Contractions Consider a subtree T T. Let coupons(t ) denote the number of coupons that reside in T. Note that a link in M contributes 1.8 coupons to coupons(t ) only if both endpoints are in T. 12

14 A subtree T is α-contractible if T is the union of α paths {p(u i,v i )} α i=1, where (u i,v i ) is a link. Note that these paths are not disjoint since T is connected. Definition 25: Let T denote an α-contractible subtree. The contraction of T into a compound node using α links is α-greedy if coupons(t ) α + 1. Implementation of a greedy contraction. Let J denote a partial cover of T, and let A denote the set of active nodes in T/J. An α-greedy contraction of a subtree T T/J is implemented by the algorithm as follows. Let F denote the set of α links that are used to contract T. Let v denote the lca of the nodes in T. The algorithm updates J and A as follows: J J F A (A \ T ) {v}. (8) The following claim shows that the Coupons Invariant is preserved by a greedy contraction. Claim 26: If the Coupons Invariant holds before a greedy contraction, then it also holds after it. Proof: After an α-greedy contraction is used to contract T, we are left with at least one coupon. The resulting compound node (i.e., the lca of the nodes contracted by the α-greedy contraction) is added to the set of active nodes and is granted a coupon. Hence the invariant that every compound node owns a coupon is maintained. Let coupons(t ) denote the number of tickets that reside in T. We also consider a greedy contractions that use tickets (here, a ticket is worth half a coupon). Definition 27: Let T denote an α-contractible subtree. The contraction of T into a compound node using α links is α-greedy with tickets if coupons(t ) tickets(t ) α + 1. An α-greedy contraction, if one exists, can found in polynomial-time as long as α is constant. One of the steps of the algorithm is to apply all possible 1- and 2-greedy contractions exhaustively. The following examples of greedy contractions illustrate various possible scenarios that may arise during this step of the algorithm. (A) (B) (C) (D) u s a v b c d a 1 v b 1.8 c d 1 b 2 v s b 1 a 1 Figure 3: Greedy contractions: (A) A 1-greedy contraction of a twin-link. (B) A 2-greedy contraction of two links in the matching. (C) A 2-greedy contraction of two links not in the matching. (D) A 2-greedy contraction using an original leaf and a matching link. 13

15 Example 4: Figure 3 depicts some greedy contractions. Note that there may be additional subtrees hanging from descendants of v in examples (B) and (C) and from descendants of u in example (D). Such additional subtrees are not depicted. (A) A 1-greedy contraction is depicted in which the twin-link between two unmatched twin-leaves is contracted. The contracted tree T consists of the stem s and the twin-leaves. Note that coupons(t ) = 2.4 since there are two unmatched leaves in T. Note also that T is 1- contractible. One coupon is used to pay for the contraction of the twin-link; another coupon is given to the compound node s created by the contraction of the twin-link. The remaining fraction of a coupon is lost. (B) A 2-greedy contraction is depicted in which two links (a,c) and (b,d) in M are contracted. (The important property in this example is that the paths p(a,c) and p(b,d) share a node.) In this example, the 2-contractible tree T equals the union of the paths p(a,c) p(b,d). (There could be additional subtrees hanging from the parents of the leaves that are not depicted; these subtrees are not contracted, so after the contraction they hand from the lca, namely from v). Note that coupons(t ) = > 3 since there are 2 links in M T. Note that T is 2-contractible. Two coupons are used to pay for the contraction of T ; a third coupon is given to the compound node v created by the contraction of the two links. (C) A 2-greedy contraction is depicted in which two links not in M are contracted. The 2- contractible tree T equals p(a,c) p(b,d). The number of coupons is at least > 3 since there is one link (b,c) M T and two unmatched leaves a,d UL T (M). The tree T is 2-contractible using the links (a,c) and (b,d) that are not in M. Two coupons are used to pay for the contraction of T, a third coupon is given to the compound node v created by the contraction of the two links, and the remaining coupons are lost. (D) A 2-greedy contraction of a subtree T. The 2-contractible tree T equals p(b 1,b 2 ) p(a 1,u), where u is the other endpoint of the up-link emanating from the original leaf a 1. The subtree T is contracted by contracting the link (b 1,b 2 ) M and (a 1,u). The number of coupons in T is = 3 since (b 1,b 2 ) owns 1.8 coupons and a 1 owns 1.2 coupons Basic Covers The following claim is analogous to Lemma 12, and follows from the same argument. Claim 28: If (T/(J M)) v is minimally A-closed, then (M T v ) up(a T v ) covers (T/J) v. Claim 28 justifies the following definition. Definition 29: The basic cover of (T/J) v is the set of links (M T v ) up(a T v ). We denote the basic cover of (T/J) v by basic-cover((t/j) v ). 5.3 Algorithm: intuition We now present an intuitive description of the approximation algorithm. Although the description of Algorithm i-cover(t) (listed as Algorithm 2) mimics the 2-approximation algorithm (listed as Algorithm 1), there are two major differences. 14

16 1. The partitioning now is done by a hypothetic oracle that is somehow able to identify a good leaf-stem-closed subtree T = T v. Recall that the unmatched leaves lower bound applies only to leaf-stem closed subtrees. The important property that subtree T satisfies is that the set of links J used to cover T charges only links in F V,V, where V denotes the nodes in T. This is important because the recursive step that covers T/J charges a disjoint set of links in F, namely, links in F \ F V,V. Obviously, this hypothetic oracle is not available, and we describe later how T is computed in reality. 2. The up-links from the leaves of T may not cover T. This is why the procedure bg-cover(t ) is used (listed as Algorithm 3). The procedure bg-cover covers T by its basic cover if it is indeed a feasible cover. If the basic cover of T is not a feasible cover, then a greedy contraction F is found. The procedure then recurses on T /F. We are left with the task of proving that if basic-cover(t ) is not feasible, then there exists a greedy contraction that can be efficiently found. We later prove that this is true, except for one special case in which T has only 3 leaves. We now note a few remarks about this intuitive algorithm: 1. We omitted the preprocessing needed to eliminate reducible edge in T. Such a preprocessing should take place in the first line of i-cover(t), and the contracted links (if any) should be added to the solution. 2. The matching M is computed by i-cover(t) so that coupons can be distributed to make greedy contractions well defined. Note that after T = T v is contracted, the recursive call recomputes the matching (from scratch) so that the root of v of T can participate as a leaf. 3. A link in M is not split by T. Namely, either both endpoints of the link are in T or outside T. This is true because T is leaf-closed. 4. The two important characteristics presented in this intuitive presentation are disjointness and progress by greedy contractions. Disjointness means that credit for covering the subtree T = T v originates only from links of F, both endpoints of which are in T. To achieve disjointness T must be leaf-stem closed. Progress by greedy contractions means that T is contracted by a sequence of greedy contractions that may end with a basic-cover. In each greedy-contraction, the number of coupons in the contracted subtree is greater than the number of links used in the contraction. The surplus (consisting of at least one coupon) is placed in the lca of the contracted subtree. Algorithm 2 i-cover(t) - An intuitive approximation algorithm. 1: If V (T) = 1, then return and stop. 2: M a maximum matching of leaf-to-leaf links in T, excluding twin-links. 3: Ask oracle for a good leaf-stem-closed subtree T = T v of T. 4: J bg-cover(t ). 5: Return J i-cover(t/j). 15

17 Algorithm 3 bg-cover(t ) - cover a leaf-stem-closed subtree using its basic-cover or greedy contractions. 1: If basic-cover(t ) covers T, then Return(basic-cover(T )) and stop. 2: Else, {apply a greedy contraction and recurse} 3: F a subset of links in T whose contraction is a greedy contraction. 4: Return(F bg-cover(t /F)). 5.4 The algorithm In this section we describe the approximation algorithm, listed below as Algorithm 4. (One particular case of a subtree with 3 leaves is deferred to Sec. 6.4.) The main issue that we address here is how to find the good leaf-stem-closed subtree that the oracle computed for us in the intuitive algorithm. The progress of bg-cover, namely, justifying the intermediate contractions as greedy contractions is addressed in Sec. 6. Algorithm 4 Tree-Cover(T) - a 1.8-approximation algorithm. 1: Notation: A - a set of active nodes, J - partial cover. 2: Assumption: There are no reducible edges in T. If not, reduce and add contracted links to final solution. 3: If T contains a single node, then Return( ) and halt. 4: Compute matching: M maximum matching of leaf-to-leaf links that are not twin-links. 5: Initialize: A all leaves in T that are not not matched by a link in M. J. 6: Exhaust 1- and 2-greedy contractions: (A, J) exhaust all possible 1-greedy, 2-greedy contractions in T/J 7: Find a node v T such that (T/(J M)) v is a minimally A-closed subtree of T/(J M). 8: while (T/J) v is not Sp(T)-closed do 9: Process: contract (T/J) v by its basic cover (special case if contains 3 leaves). J J basic-cover((t/j) v )) A (A \ (T/J) v ) {v}. 10: (A,J) exhaust all possible 1-greedy, 2-greedy contractions in T/J 11: Find a node v T such that (T/(J M)) v is a minimally A-closed subtree of T/(J M). 12: end while 13: Find a cover only of T v : J (J T v ) basic-cover((t/j) v ). 14: Contract T v and recurse: Return (J Tree-Cover(T/J)). The algorithm accepts a tree T to be covered (the set of links is implied). The algorithm maintains two sets: A (the set of active nodes) and J (a partial cover). We first assume that there are no reducible edges in T. If not, then tree edges are eliminated and maximal links are contracted 16

18 until all edges are not reducible achieved (see Sec. 2.5). Links contracted in this stage are added to the computed cover. The stopping condition of the algorithm is when T contains a single node. If the stopping condition does not hold, then the following preprocessing takes place (lines 4-6). A maximum matching M of leaf-to-leaf links not including twin-links is computed in line 4. Initialization takes place in line 5. The initial set of active nodes is the set of unmatched leaves, and the initial partial cover is empty. In line 6, all possible 1- and 2-greedy contractions in T are performed (see Sec. 5.2). Each greedy contraction requires updating the set A of active nodes and the partial cover J (see Eq. 8). In Line 7, a node v is found such that the subtree (T/(J M)) v is a minimally A-closed subtree of T/(J M). We refer to the subtree (T/J) v in the beginning of an iteration of the while-loop as the current subtree. The stopping condition of the while-loop in lines is that the current subtree (T/J) v is leaf-stem-closed. The criteria for selecting the current subtree may seem awkward; namely, after J M is contracted the subtree should be active-closed. As pointed out in the previous section, to achieve disjointness, we do not want to split links in M. So we must consider a subtree that does not split matching links. This explains why the links in M are contracted before the current subtree is chosen. Since we do not have enough credit to contract the up-links from matched leaves (because every matched link owns less than 2 coupons), a cheaper cover is used. By Claim 28, the current subtree is covered by its basic-cover. However, if the current subtree is not leaf-stem-closed, then disjointness might not be satisfied. Hence, we have not reached a good tree yet, and a coupon must given to the root of the current subtree after it is contracted. Progress is achieved in the case that the current subtree is not leaf-stem-closed by showing that the basic-cover is a greedy cover of the current subtree, except for one special case (see Theorem 33). The contraction of the links in the basic cover may enable new 1- and 2-greedy contractions before a new current subtree is found and a new iteration of the while-loop takes place. If the current subtree (T/J) v is leaf-stem-closed, then the subtree T v is a good tree. There is no need to recompute a cover of T v ; in line 13 the algorithm simply takes the basic cover of (T/J) v and the links in J T v. All other links in the partial solution are discarded because these links are not contributing to the contraction of the good tree. In this way the partial cover computed by the algorithm is identical to the one computed by the intuitive algorithm in which the good tree is given in advance by the oracle. To see that the algorithm terminates with a cover it suffices to prove that the while-loop terminates. Note that if v is the root of the current subtree in the beginning of an iteration of the while-loop, then at the end of the iteration, v and all its descendants can not be selected as the root of the new current subtree. Indeed, if (T/J) v is contracted by its basic cover, then v becomes active, and is not selected again as the root of the current subtree. In Section 6.4 we show that this is also true if the current subtree has 3 leaves. 17

19 6 Analysis of the algorithm 6.1 The AMJ-invariant In this section we present an invariant that is satisfied by the algorithm after 1- and 2-greedy contractions are exhausted. This invariant plays a crucial role in proving the correctness of the algorithm. Invariant 30: [The AMJ-invariant] Let A denote the set of active nodes. Let M denote a matching between leaves of a tree T. Let J E denote a partial cover. We say that the triple (A,M,J) satisfies the AMJ-invariant if the following seven conditions are satisfied. 1. The set of active nodes is independent; namely, there does not exist a link between two active nodes. 2. For every pair of links (a 1,b 1 ) and (a 2,b 2 ) in M, the paths p(a 1,b 1 ) and p(a 2,b 2 ) are vertexdisjoint. 3. Active nodes do not belong to paths covered by matching links. 4. Every stem s St(T) (T/J) has exactly one child incident to a link in M; the other child is an unmatched leaf. 5. If the subtree (T/J) v contains a link in M then it has at least three leaves (one of the leaves must be a compound node or an unmatched leaf in T). 6. Every leaf in T/(J M) is an active leaf in A (T/J). In particular, the contraction of a link in M does not introduce a new leaf in T/J. 7. Consider a node v T such that T = (T/(J M)) v is minimally active-closed. Then every active node u A (T/J) v is a leaf in T/(J M). 8. Every matched vertex has exactly one link in F emanating from it, i.e., if deg M (v) = 1, then deg F (v) = Satisfying the AMJ-invariant Claim 31: Suppose that M does not contain a twin-link in T/J and that the Coupons Invariant holds. If no 1- or 2-greedy contractions are applicable, then the AMJ-invariant holds. Proof: Proof of part 1: if (a 1,a 2 ) is a link between two active nodes, then the contraction of (a 1,a 2 ) is a 1-greedy contraction (as in part (1) of example 4). Proof of part 2: if p(a 1,b 1 ) and p(a 2,b 2 ) share a vertex v, for (a 1,b 1 ),(a 2,b 2 ) M, then the contraction of (a 1,b 1 ) and (a 2,b 2 ) is a 2-greedy contraction (as in part (2) of example 4). Proof of part 3: If an active node a belongs to a path p(b 1,b 2 ) covered by a matching link (b 1,b 2 ) M, then the contraction of (b 1,b 2 ) is a 1-greedy contraction. Proof of part 4: If both twin-leaves are not matched, then they are active. But then the twinlink connects them, in contradiction to part 1. If both leaves are incident to links in M, then the the stem is covered by both links, contradicting part 2. 18