Andr'as Frank. Augmenting graphs to meet edge--connectivity requirements. SIAM J. Disc. Math., 5(1):25--53, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... graph H F satis es H F (x; y) k for all terminal pairs x; y 2 T (i.e. we increase sizes of all cuts dividing T up to k) The edge connectivity problem with a terminal set T has been studied in [16, 18] The problem can be described as the local edge connectivity augmentation problem [3], and can be solved in O(jV jjEjjT j log (jV j =jEj) time by Gabow s algorithm [7] In this subsection, we consider the case in which a multigraph H = V = T [ S; E) with a terminal set has a bi level structure such that H is obtained from a multigraph (T; E ) jT j 2) by adding a ....

....this problem is an extension of the k edge connectivity augmentation problem, since it becomes the uniform edge connectivity augmentation problem if r(u) k for all u 2 V . The above augmentation problem (without edge costs) can be formulated as the local edge connectivity augmentation problem [3], and can be solved by Gabow s algorithm [7] In this section, as another generalization of the uniform edge connectivity augmentation problem, we consider the problem of augmenting H by a set F of the smallest number of new edges such that the resulting graph H F = V; E [ F ) satis es c H F ....

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Mathematics, 5 (1992) 25-53.

....connectivity. The number of nodes, jV j, is denoted by n. A preliminary version of this paper has appeared in the Proc. of the 28th ACM S.T.O.C. 1996) pp.3. DepartmentofCombinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3L Supported in part by NSERC grant no.OGP013 (NSERC code OGPIN 007) email: jcheriyan dragon.uwaterloo.ca Department of Mathematics and Computer Science, UniversityofDenver, 23 S. Gaylord St. Denver CO 80208. Supported in part by NSF Research Initiation Award grant CCR 9210604. email: ramki cs.du.edu URL: http: www.cs.du.edu ramki ....

....Waterloo, Ontario, Canada N2L 3L Supported in part by NSERC grant no.OGP013 (NSERC code OGPIN 007) email: jcheriyan dragon.uwaterloo.ca Department of Mathematics and Computer Science, UniversityofDenver, 23 S. Gaylord St. Denver CO 80208. Supported in part by NSF Research Initiation Award grant CCR 9210604. email: ramki cs.du.edu URL: http: www.cs.du.edu ramki We presentanO(k ) time (deterministic) algorithm for finding all the k shredders of a k node connected graph. This solves an open question raised byJord an [J 95] effi tly find a k separator of a k node connected graph whose ....

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A. Frank, "Augmenting graphs to meet edge-connectivity requirements," SIAM J. Disc. Math. 5 25--53.

....the problem remains NP hard, even when all demands are also uniform. The best known approximation in this case is 2[ connectivity requirement is [23] If, in addition, the underlying graph is the complete graph and multiple edges are allowed, then the problem is solvable in polynomial time [8, 28]. Other researchers have considered approximation algorithms for the capacitated network design problem when the objective is to design a network with enough capacity to route all demands simultaneously, without any restriction on the number of copies of edges allowed [1, 6, 26, 31] There has ....

A. Frank. Augmenting graphs to meet edge connectivity requirements. SIAM J. Discr. Math., 5(1):25--53, 1992.

.... and weights are arbitrary [6] and 2EC and 2VC under the same restrictions [22] ffl ECAP in the case where G is a complete graph with arbitrarily high edge multiplicity (i.e. any edge is allowed to be used arbitrarily many times) all weights are 1, and connectivity requirements are arbitrary [8]. We now state the main result of this paper. Main Theorem The ECAP and VCAP problems can be solved in O(n ) time for any instance (G; w; S; r) with G an n vertex planar graph, S a k vertex connected subgraph, and r ij 2 for all i; j 2 V S . The first five sections of the paper are ....

A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Disc. Math. 5 (1992), 25--53.

....for k vertex connectivity. We remark that the other three basic augmentation problems (where one wants to make G k edge connected or wants to make a digraph k edge or k vertex connected) have been shown to be polynomially solvable. These results are due to Watanabe and Nakamura [20] Frank [5], and Frank and Jord an [7] respectively. For more results on connectivity augmentation and its algorithmic aspects, see the survey papers by Frank [6] and Nagamochi [19] respectively. In the rest of the introduction we introduce some de nitions and our new lower bounds for the size of an ....

....new vertex s and a set of appropriately chosen edges incident to s and then obtains an optimal augmenting set by splitting o pairs of edges incident to s. This approach was initiated by Cai and Sun [1] for the k edge connectivity augmentation problem and further developed and generalized by Frank [5]. Here we adapt the method to vertexconnectivity and prove several basic properties of the extended graph as well as the splittable pairs. Given the input graph G = V; E) an extension G s = V s; E F ) of G is obtained by adding a new vertex s and a set F of new edges from s to V . In G ....

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Mathematics, Vol.5, No 1., 22-53, 1992.

....in G if G v,t still satisfies (1) It is easy to see that sv, st is not admissible if and only if some dangerous set contains both t and v. 4 The following two lemmas can be proved by standard methods using Proposition 2.1. Most of them are well known and appeared explicitely or implicitely in [7] (or later in [1] We include the proofs for completeness. Lemma 2.2 (a) A maximal dangerous set does not cross any critical set. b) If X is dangerous then d(s, V d(s, X) c) If k is even then two maximal dangerous sets X, Y which are crossing have Y ) 0. Proof: Let X be a maximal ....

.... 1, d(X X) k, d(X#Y ) k, d(X#Y ) k 2, d(X#Y,V s (X#Y ) 1, d(s, X Y ) 1, d(s, Y 1, and d(X Y ) d(X Proof: Suppose neither (0) nor (i) holds for v. Let be the collection of maximal v dangerous sets. Any two maximal v dangerous sets are crossing if d(s) is even [7]. Thus consists of (at least two) pairwise crossing sets. By Lemma 2.2(c) this shows that k is odd. Let X, Y be chosen in such a way that X # is as large as possible. Since X and Y cross, by (2) and (3) we obtain d(X) d(Y ) k 1, d(X X) k, d(X Y ) k, 5 d(X#Y ) k 2, ....

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A. Frank, Augmenting graphs to meet edge--connectivity requirements, SIAM J. Discrete Mathematics, 5 (1992) 22--53.

....related to (and generalize) splitting off. Splitting off two edges su, sv in a graph means replacing su, sv by a new edge uv. If u = v then the resulting loop is deleted. This operation is a well known and useful tool in proving theorems and designing algorithms for connectivity problems, see e.g. [3, 4, 6]. Subdividing the split edge uv by a new vertex t does not change the local edge connectivities. Thus a splitting off operation corresponds to a (2, d(s) 2) detachment of s in this sense. Also, the special case of our problem, when d(s) is even, 2, 2, 2) and r(x, y) #G (x, y) for ....

....while Nash Williams result can be deduced easily from Fleiner s result [1] As a new application, we solve a graph augmentation problem where stars of given degrees have to be attached to a given graph in order to meet given local edge connectivity requirements. This extends a theorem of A. Frank [3] on augmenting the local edge connectivities by adding a smallest set of new edges. The organization of the paper is the following. Further definitions and preliminary results are given in Section 2. The proof of the main result starts in Section 3, where we reduce the problem to the case when ....

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5, 25--53 (1992).

....most of the algorithmic details. We remark that the other three basic augmentation problems (where one wants to make G k edge connected or wants to make a digraph k edge or k vertex connected) have been shown to be polynomially solvable. These results are due to Watanabe and Nakamura [12] Frank [3], and Frank and Jord an [4] respectively. For more results on connectivity augmentation see the recent survey by Nagamochi [11] In the rest of the introduction we introduce some de nitions and our new lower bounds for the size of an augmenting set which makes G k vertex connected. We also state ....

....new vertex s and a set of appropriately chosen edges incident to s and then obtains an optimal augmenting set by splitting o pairs of edges incident to s. This approach was initiated by Cai and Sun [1] for the k edge connectivity augmentation problem and further developed and generalized by Frank [3]. In [7] we adapted the method to vertexconnectivity and proved several basic properties of the extended graph as well as the splittable pairs. Given the input graph G = V; E) an extension G s = V s; E F ) of G is obtained by adding a new vertex s and a set F of new edges from s to V . ....

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Mathematics, Vol.5, No 1., 22-53, 1992.

....connected graphs [15] and the theorem of Nash Williams for orienting 2k connected graphs to k connected digraphs (Problem 6.54 in [14] Cai and Sun [2] applied splittings for augmenting undirected edge connectivity by adding optimum number of edges to the graph. Their ideas were improved by Frank [5] to the first polynomial time algorithm for this task. Another algorithm for the same problem (Naor et al. [17] uses the cactus representation as main tool. The recent fastest augmentation algorithms of Bencz ur [1] and Gabow [8] are based on the idea of using the cactus, respectively the ....

....connectivity augmentation without using multiple edges can be found in polynomial time. One improvement we make on earlier results is that we can describe the structure of splittable pairs. The theorems of Lov asz and Mader prove only existence of such pairs. In Frank s augmentation algorithm [5] splittable pairs are found only by (quite time consuming) flow algorithms. On the other hand the algorithm of Karger and Stein [10] suggests that finding the mincuts or even all near minimum cuts might be fundamentally easier than to solve the maximum flow problem. By extending the cactus ....

Frank, A., Augmenting graphs to meet edge connectivity requirements, Proc. 31st Annual IEEE Symposium on Foundations of Comp. Sci.,

....with running time O( 2 nm) was given by Naor et al. 15] and Gabow [6] gave an O(k 2 n) time modification, where k is the desired connectivity value. These algorithms use elementary manipulations, provided some necessary structures of minimum cuts are known. On the other hand, Frank [3], based on some ideas of Cai and Sun [1] presented a different algorithm which solves the (integral) capacitated case in O(n 5 ) time. The algorithm uses maxflow algorithms and hence is not really efficient, but it can solve the capacitated problem in polynomial time. Very recently, Gabow [7] ....

....become minimal mincuts later. These sets are the so called extreme sets described by Watanabe and Nakamura [17] The first step we make in improving successive augmentation algorithms is that of eliminating the dependence of their running time on the increment. Note that by the result of Frank [3] we know that the capacitated problem can also be solved in polynomial time. We show that the situation is somewhat analogous to the Ford Fulkerson augmenting path algorithm for maxflows: in the original algorithm one finds paths that increase the flow value one by one, hence that algorithm is ....

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Frank, A., Augmenting graphs to meet edge connectivity requirements, Proc. 31st Annual Symp. on Found. of Comp. Sci. (

....can be performed while preserving certain connectivity properties of the graph. Edge splitting is an important operation for connectivity problems. For example, suppose we would like to make a graph G = V,E) k edge connected by adding the minimum number of edges. A beautiful result of Frank [2] shows that it is su#cient to add a vertex s to the graph, add the minimum even number of edges between s and V to make it k edge connected (and this is an easy task) and finally perform splitting o# while preserving k edge connectivity between the vertices in V (using Lovasz s splitting o# ....

.... s to the graph, add the minimum even number of edges between s and V to make it k edge connected (and this is an easy task) and finally perform splitting o# while preserving k edge connectivity between the vertices in V (using Lovasz s splitting o# result) For extensions of this result, see [2] and the survey [3] As another (less algorithmic) illustration of the use of edge splitting, Nagamochi, Nishimura and Ibaraki [7] have shown inductively using edge splitting that there are at most # n 2 # cuts of value strictly less than 4 3 times the minimum cut value in any undirected ....

A. Frank, "Augmenting graphs to meet edge-connectivity requirements ", SIAM J. on Disc. Math., 5, 22--53 (1992).

....k for every ; 6= X V . We use to denote proper inclusion, while means or = In this paper we focus on a di erent group of edge connectivity questions. We prove theorems on edge splittings and edge connectivity augmentation in dypergraphs, extending earlier results of Mader [8] and Frank [2] on directed graphs. 2 Preliminaries In this section we introduce further notation and prove some basic facts on dypergraphs. Let D = V; E) be a dypergraph. For X V we have already de ned the in degree (X) of X. Let (X) V X) denote the out degree of X. For a single vertex v we simply use ....

....to D 0 we obtain an admissible complete splitting at s, consisting of t splittings. The hyperedges of size at most t obtained by these splittings form the required augmenting set of size for D. If D is a directed graph and t = 2 then we get Frank s theorem as a corollary. Corollary 4.4. [2] A directed graph D = V; E) can be made k edge connected by adding at most new edges if and only if X i (k (X i ) and X i (k (X i ) hold for every sub partition fX 1 ; X t g of V . 5 Open problems Theorem 3.8 characterised when there is no admissible 1 split in a ....

A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Disc. Math. 5, 25-53, 1992.

....proof for Theorem 1.2 by using the approach we developed in the directed case. This new proof, which is based on edge splitting and edge ipping operations, seems to be simpler than the original proof [9] or the proof given in [2] We note that some parts of our proofs are similar to proofs from [1], 2] 5, 6.53] or [9] where the authors apply similar techniques. We shall use the following well known equalities for the degree function of a graph. Proposition 3.1. Let H = V; E) be a graph. For arbitrary subsets X; Y V : d(X) d(Y ) d(X Y ) d(X [ Y ) 2d(X Y; Y X) 2) d(X) ....

A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Disc. Math. 5, 25-53, 1992.

....hypotheses, there is a complete splitting at z so that the resulting (di)graph on node set V is k edge connected. An easy observation shows that the existence of a complete undirected splitting that preserves k edge connectivity is equivalent to the following degree speci ed augmentation result [16]. Theorem 2.3. We are given an undirected graph G = V; E) a degree speci cation m : V Z , and an integer k 2. There is a graph H = V; F ) so that dH (v) m(v) for every node v 2 V and G H is k edge connected if and only if m(X) k dG (X) for every nonempty subset X V: Here and ....

....E) a degree speci cation m : V Z , and an integer k 2. There is a graph H = V; F ) so that dH (v) m(v) for every node v 2 V and G H is k edge connected if and only if m(X) k dG (X) for every nonempty subset X V: Here and throughout m(X) P (m(v) v 2 X) This result was used in [16] to exhibit a short derivation of T. Watanabe and A. Nakamura s [48] earlier solution to the minimization form of the undirected edge connectivity augmentation problem: EGRES Technical Report No. 2001 11 2.2 Connectivity orientation and augmentation 7 Theorem 2.4 (Watanabe and Nakamura) An ....

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A. Frank { Augmenting graphs to meet edge-connectivity requirements, SIAM J. on Discrete Mathematics 5 No. 1. (1992 February), pp. 22-53.

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Andr'as Frank. Augmenting graphs to meet edge--connectivity requirements. SIAM J. Disc. Math., 5(1):25--53, 1992.

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A. Frank, Augmenting graphs to meet edge connectivity requirements, SIAM J. Disc. Math. Vol. 5, No. 1, (1992), pp. 25-53.

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM Journal on Discrete Mathematics, 5 (1992), pp. 25--53.

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Andr'as Frank. Augmenting graphs to meet edge--connectivity requirements. SIAM J. Disc. Math., 5(1):25--53, 1992.

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5 (1992), no. 1, 25-53.

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5 (1992), no. 1, 25--53.

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM Journal on Discrete Math., 5(1) (1992), pp. 25-53. 14

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A. Frank, \Augmenting graphs to meet edge-connectivity requirements", SIAM J. Disc. Math., 5 (1992), 25-53.

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Frank, A., Augmenting graphs to meet edge connectivity requirements, SIAM J. Discr. Math. 5(1), pp. 25-53 (

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A. Frank, \Augmenting Graphs to Meet Edge-Connectivity Requirements", SIAM Journal on Discrete Mathematics, 5 (1992), 25-53.

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A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM Journal on Discrete Math., 5(1) (1992), pp. 25--53.

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