D. Naor, D. Gus eld, and C. Martel. A fast algorithm for optimally increasing the edge-connectivity. SIAM Journal on Computing, 26(4):1139-1165, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

D. Naor, D. Gusfield, and C. Martel. A fast algorithm for optimally increasing the edge connectivity. In Proc. 31st Annual IEEE Symposium on Foundations of Computer Science, pages 698--707, 1990.

.... to Gamma n 2 Delta minimum cuts [5, 8] For many applications, it is useful to know many, or all minimum cuts of a graph, for instance, in separation algorithms for cutting plane approaches to solving integer programs [2, 6, 10] and in solving network augmentation and reliability problems [4, 3, 21]. Many other applications of minimum cuts are described in [1, 24] In 1976, Dinits, Karzanov, and Lomonosov [8] published a description of a very simple data structure called a cactus that represents all minimum cuts of a weighted, undirected graph in linear space. This is notable considering ....

D. Naor, D. Gusfield, and C. Martel. A fast algorithm for optimally increasing the edge connectivity. In Proc. 28th Annual ACM Symposium on Theory of Computing, pages 698--707, 1990.

....(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 splitting theorems. It is open whether our representation can improve on any of these algorithms. One idea could be to ....

Naor, D., D. Gusfield, Ch. Martel, A fast algorithm for optimally increasing the edge connectivity, Proc. 31st Annual IEEE Symposium on Foundations of Comp. Sci., 1990, 698--707

....a sequence G l : G; G l 1 ; G l 2 ; of graphs so that for each i l G i 1 is a supergraph of G i and G i is an i edge connected augmentation of G using a minimum number of new edges. Watanabe and Nakamura described how to compute this sequence in polynomial time. Gusfield, Naor and Martel [1990] and Gabow [1991] found improvements for the complexity. One apparent disadvantage of this approach is that the resulting algorithm is not strongly polynomial if the target edge connectivity k is very big. This is clearly so since the approach uses one by one augmentations. A. Bencz ur [1994] ....

D. Naor, D. Gusfield and Ch. Martel, A fast algorithm for optimally increasing the edge-connectivity, 31st Annual Symposium on Foundations of Computer Science, 1990, pp 698-707.

....show that we cannot optimally raise the vertex connectivity of a graph to three by first optimally raising the vertex connectivity to two and then using the special algorithm to increase the 2 vertex connectivity by one. This approach is used to optimally raise the edge connectivity of a graph [Fra92, Gab91, NGM90]. The above counter example can be extended to rule out the chance of solving our problem (for raising the vertex connectivity to four) by combining the result in [HR91] for raising the vertex connectivity to three) and the result in [Hsu92] for raising the vertex connectivity from three to ....

....of finding a smallest augmentation for a graph to reach a given edgeconnectivity, several polynomial time algorithms and efficient parallel algorithms on outerplanar graphs, hierarchically defined graphs, undirected graphs, directed graphs and mixed graphs are known. These results can be found in [Ben94, CS89, ET76, FBW89, Fra92, Gab91, Gus87, Hsu93, KU86, Kan93b, NGM90, Sor88, TW94, UKW88, Wat87, WN87, WY93]. 2.3 Augmenting a Weighted Graph Another version of the problem is to augment a graph, with a weight assigned to each edge, to meet a connectivity requirement using a set of edges with a minimum total cost. The decision version of several related problems have been proved to be NP hard. These ....

D. Naor, D. Gusfield, and C. Martel. A fast algorithm for optimally increasing the edge-connectivity. In Proc. 31th Annual IEEE Symp. on Foundations of Comp. Sci., pages 698--707, 1990.

....17 and Nakamura [WN87] gave the first polynomial time algorithms to solve the smallest augmentation problem for an arbitrary undirected graph to achieving a given edge connectivity. Cai and Sun [CS89] also gave a polynomial time algorithm for solving the same problem. Naor, Gusfield, and Martel [NGM90] gave an O(ffi 2 nm nF (n; m) time algorithm to increase the edge connectivity of an undirected graph by ffi, where n and m are the number of vertices and edges in G, respectively, and F (n; m) is the time to perform one maximum flow on G. The best known bound for F (n; m) is O(minfn 2 3 ....

....biconnectivity augmentation given in Chapter 3 to implement stage 1, since there exists a graph G such that any smallest augmentation for biconnecting G does not lead to a smallest augmentation for triconnecting G. See Section 5.3 for details. Note that for edge connectivity, it is shown in [NGM90, Wat87] that there exists a smallest augmentation A to k edge connect a graph G such that A is included in a smallest augmentation to (k 1) edge connect G, for an arbitrary k. An extended abstract of part of the work reported in this chapter appears in [HR91a] 97 98 5.2 Definitions 5.2.1 ....

[Article contains additional citation context not shown here]

D. Naor, D. Gusfield, and C. Martel. A fast algorithm for optimally increasing the edge-connectivity. In Proc. 31th Annual IEEE Symp. on Foundations of Comp. Sci., pages 698--707, 1990.

....to transform an i edge connected realization into an (i 1) edge connected realization. We introduce the notion of extreme sets and show that many extreme sets can be destroyed in one step by performing exchanges in parallel. The extreme sets can be computed from the elegant cactus representation [6, 16], which compactly represents all connectivity cuts of a graph. We give in Section 5 a brief description of this representation. Karzanov and Timofeev [13] presented an efficient sequential algorithm to compute this representation. Naor and Vazirani [17] presented an RNC algorithm to compute the ....

....that G 0 is also a cactus, and that w 0 (e) 2 if e is a cycle edge, and w 0 (e) if e is a tree edge. Dinits, Karzanov and Lomosonov [6] derived the compact and elegant cactus representation H = H(G) of a graph G = V; E) We give a brief description of H; more details can be found in [13, 16, 17]. H is an edge weighted cactus graph of O(n) nodes and edges. Every vertex in G maps to exactly one node in H and any node in H corresponds to a subset (possibly empty) of vertices from G. A cut (S; S) in H induces a cut (X; X) in G, where X consists of all vertices from G that are mapped into ....

D. Naor, D. Gusfield and C. Martel. A fast algorithm for optimally increasing the edge connectivity. Proc. 31st FOCS, pp. 698--707, IEEE Press, 1991.

....problem remains NP hard, even when all demands are also uniform. The best known approximation in this case is 2 1= when 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 ....

D. Naor, D. Gusfield, and C. Martel, A fast algorithm for optimally increasing the edge connectivity, in 31st Annual Symposium on Foundations of Computer Science, 1990, pp. 698-- 707.

....deterministic version runs in O(nm min n,# )time. For an arbitrary value of # , we can also find an element of the increasing sequence of optima ( S i # E i ) with running time not depending on # . Recently, finding efficient edge connectivity augmentation algorithms became a wellstudied area [38,7,12,32,15,18]. The algorithms use various min cut structures and are generally based on maxflow computations. Our algorithms require efficient subroutines for finding two cut data structures: the extreme sets introduced by Watanabe and Nakamura [38] and the cactus representation of Dinitz et al. 10] We ....

....historically earlier approach, when the connectivity is repeatedly increased by one unit until the target is reached. The first edge connectivity augmentation algorithm by Watanabe and Nakamura [38] has this approach. A more efficient version with running time O(# 2 nm) was given by Naor et al. [32]; Gabow [16] improved this time to O(k#m) All these running times are valid for graphs with unit edge capacities only. Algorithms of this latter approach [38,32,16] have an additional property that they can find a successive, increasing sequence of intermediate optima, for increasing values of ....

[Article contains additional citation context not shown here]

Naor, D., Gusfield, D., Martel, Ch. (1990): A fast algorithm for optimally increasing the edge connectivity. Proc. 31st Annual IEEE Symposium on Foundations of Comp. Sci. 698--707

....time (cf. 7] and set f Gamma (x) max i=1: p ffi w 2 (T i ) f (x) min i=1: p ffi w 2 (T i ) 3 Some Bottleneck Augmentation Problems In BECA1, we try to augment the (usual) edge connectivity to k under the objective to minimize the maximum increase of any edge. Several authors ([4, 10]) consider a similar problem in which the objective is to minimize the sum of increments. An efficient implementation of the strong polynomial algorithm of Frank (cf. 4] can be found in [1] The next theorem shows the connection of BECA1 to the balanced edge connectivity number. Note that for ....

D. Naor, D. Gusfield, and C. Martel. A fast algorithm for optimally increasing the edge--connectivity. In Proc. 41th Ann. Symp. on Foundations of Computer Science, pages 698--707. IEEE Computer Society, 1990. On Balanced Edge Connectivity and Applications 10

....parallel edges and edges which are parallel to edges of G as well. The edge connectivity augmentation problem and a number of its extensions can be solved e#ciently. Since Watanabe and Nakamura s first polynomial time algorithm, several other e#cient algorithms have been developed; see [4] 8] [22], 25] and also [3] 11] 21] for some important results. For a survey of this area see [9] However, there are several versions of the connectivity augmentation problem which remain open. For example, in some cases the goal is to increase the connectivity by maintaing certain properties of the ....

....In this paper we deal with another property to be preserved: the simplicity of G. As it is stated in [9] It is an important open problem to find algorithms that do not add parallel edges. Partial results in this direction have been obtained by Frank and Chou [10] Naor, Gusfield, and Martel [22], Taoka, Takafuji, and Watanabe [23] and Watanabe and Yamakado [24] the details are given below) but the complexity of the general problem was still open. Recently the second author proved that the simplicity preserving k edge connectivity augmentation problem is NP complete, even if the ....

[Article contains additional citation context not shown here]

<F3.742e+05> D. Naor, D. Gusfield, and C.<F3.881e+05> Martel,<F3.471e+05> A fast algorithm for optimally increasing the edgeconnectivity,<F3.881e+05> SIAM J. Comput., 26 (1997), pp. 1139--1165.

....[55] also developed a linear time algorithm for the augmentation to 3 nodeconnected networks. The augmentation to k edge connected graphs was studied by Watanabe and Nakamura [82] Ueno et al. 80] and Cai and Sun [14] The fastest known algorithm for this problem is the one by Naor et al. [70]. Frank [33] solved the augmentation problem completely for general edge connectivity requirements. All solution procedures allow the use of parallel edges, except those of Eswaran and Tarjan [30] and Rosenthal and Goldner [73] Again, the problem of augmentation to a node connected graph is open ....

D. Naor, D. Gusfield, and Ch. Martel. A fast algorithm for optimally increasing the edge-connectivity. In Proceedings of the Foundation of Computer Science '90, pages 698--707, St. Louis, 1990.

.... Splitting off a pair su; sv of edges means replacing su and sv by a new edge uv. Using this method Frank [6] solved several extensions of the problem. For example he showed it is tractable even if local connectivity demands or vertex costs are given. A different approach due to Naor et al. [19] resulted in a faster algorithm. The currently fastest algorithm was developed by Nagamochi et al. 17] 18] Their algorithm, also based on the splitting off method, runs in time O(n(m n log n) log n) The parameters n and m denote the number of vertices and distinct edges of the graph ....

D. Naor, D. Gusfield and Ch. Martel, A fast algorithm for optimally increasing the edgeconnectivity, 31st Annual Symposium on Foundations of Computer Science, 1990, pp 698707.

.... each other (the submodularity f(X) f(Y ) f(X Y ) f(X[Y ) does not imply the optimality of X Y or X[Y if X[Y = V ) Computing minimal and maximal optimal solutions plays a key role in many graph connectivity problems such as computing leaf vertices in a cactus structure of all minimum cuts [15] and augmenting edge connectivity by edge splitting technique [3] For the purpose of computing minimal optimal solutions, it is advantageous to work with the following function g : 2 V s 7 (where s is a new element) which is extended from f , rather than directly using the f 0 de ned in ....

D. Naor, D. Guseld and C. Martel, A fast algorithm for optimally increasing the edge connectivity, SIAM J. Computing, 26 (1997), 1139-1165.

....polynomial time, at least for the edge connectivity case. We will not survey this body of research in detail here since we are primarily interested in approximation techniques for NP hard problems. For more information on such problems see recent papers by Frank [10] and Naor, Gusfield and Martel [32]. For the vertex connectivity case, the problem appears to be significantly harder and no polynomial time algorithm is known for finding the optimal solution. In his doctoral thesis, Hsu [22] gives algorithms for vertex connectivity for small connectivity values. These algorithms are quite ....

D. Naor, D. Gusfield and C. Martel, A fast algorithm for optimally increasing the edgeconnectivity, Proc. 31st IEEE Symposium on Foundations of Computer Science, pp. 698--707, (1990).

....sequential algorithm for finding a smallest augmentation to triconnect a graph with n vertices and m edges. There is no polynomial time algorithm known for finding a smallest augmentation to k vertex connect a graph, for k 3. Results on other versions of augmentation problems can be found in [1, 2, 4, 6, 7, 12, 14, 17, 21, 22, 23, 24, 25, 27]. In this paper, we present a linear time sequential algorithm for finding a smallest augmentation to triconnect a graph. The algorithm is divided into two stages. During the first stage, we biconnect the input graph. Then in the second stage, we triconnect the resulting biconnected graph using ....

....use the algorithm in Hsu Ramachandran [10] to implement stage 1, since there exists a graph G such that any smallest augmentation for biconnecting G does not lead to a smallest augmentation for triconnecting G. An example is shown in Figure . Note that for edge connectivity, it is shown in [14, 22] that there exists a smallest augmentation to k edge connect a graph G such that it is included in a smallest augmentation to (k 1) edgeconnect G, for an arbitrary k. Owing to space limitation, many proofs are omitted in this abstract. They can be found in the full paper[11] 2 Definitions ....

D. Naor, D. Gusfield & C. Martel, "A fast algorithm for optimally increasing the edge-connectivity," Proc. 31th Annual IEEE Symp. on Foundations of Comp. Sci., 1990, pp. 698-707.

.... to Gamma n 2 Delta minimum cuts [5, 8] For many applications, it is useful to know many, or all minimum cuts of a graph, for instance, in separation algorithms for cutting plane approaches to solving integer programs [2, 6, 10] and in solving network augmentation and reliability problems [4, 3, 24]. Many other applications of minimum cuts are described in [1, 27] In 1976, Dinits, Karzanov, and Lomonosov [8] published a description of a very simple data structure called a cactus that represents all minimum cuts of a weighted, undirected graph in linear space. This is notable considering the ....

D. Naor, D. Gusfield, and C. Martel. A fast algorithm for optimally increasing the edge connectivity. In Proc. 28th Annual ACM Symposium on Theory of Computing, pages 698--707, 1990.

....will cost a lot to modify. You may still accept bad designs, but then you know that it will not cost you much to modify them later. In this respect we mention the existence of efficient algorithms for determining a minimum set of edges to be added to a graph in order to make it k connected [WN87, NGM90, Gab91, Ben95, NI96] 1.3 Testing connectivity to the rest of the graph Our algorithm for testing k edge connectivity, for k 2, uses a subroutine which may be of independent interest. To describe it, suppose that you are given as input a vertex which resides in a k connected component of the ....

D. Naor, D. Gusfield, and C. Martel. A fast algorithm for optimally increasing the edgeconnectivity. In Proceedings of the Thirty-First Annual Symposium on Foundations of Computer Science, pages 698--707, 1990.

....for a graph to reach a given edge connectivity property, several polynomial time algorithms on undirected graphs, directed graphs and mixed graphs are known. These results can be found in Cai Sun [1] Eswaran Tarjan [4] Frank [5] Gusfield [8] Kajitani Ueno [13] Naor, Gusfield Martel [15], Ueno, Kajitani Wada [24] Watanabe [25] and Watanabe Nakamura [27] Efficient parallel algorithms for finding smallest augmentations for 2 edge connectivity, strong connectivity and making a mixed graph strongly orientable can be found in Soroker [20] Another version of the problem is to ....

D. Naor, D. Gusfield, and C. Martel, A fast algorithm for optimally increasing the edge-connectivity, in Proc. 31th Annual IEEE Symp. on Foundations of Comp. Sci., 1990, pp. 698--707.

..... G 3 (k r ) of optimal solutions such that G 3 (k i 1 ) is obtained by increasing weights of some edges in G 3 (k i ) Such hierarchical structure of optimal solutions over all k is known so far only for the integer version of the edge connectivity augmentation problem in undirected graph [18, 20] and in directed graphs [3] Theorem 3 Given an edge weighted graph G = V; E; c G ) there is a set = f(C 3 i ; i 1 ; i ] j i = 1; 2; pg of p 6n 4n log n weighted cycles that can provide all optimal graphs G 3 (k) in the entire range k 2 [ G (V ) 1] Such can be ....

D. Naor, D. Guseld and C. Martel, A fast algorithm for optimally increasing the edge connectivity, Proceedings 31st Annual IEEE Symposium on Foundations of Computer Science, (1990), pp. 698-707.

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D. Naor, D. Gus eld, and C. Martel. A fast algorithm for optimally increasing the edge-connectivity. SIAM Journal on Computing, 26(4):1139-1165, 1997.

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D. Naor, D. Gusfield, and C. Martel. A fast algorithm for optimally increasing the edge--connectivity. In Proc. 41th Ann. Symp. on Foundations of Computer Science, pages 698--707. IEEE Computer Society, 1990.

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Naor, D., D. Gus eld and Ch. Martel, A fast algorithm for optimally increasing the edge connectivity, Proc. 31st Annual IEEE Symp. on Foundations of Comp. Sci. (1990), pp. 698-707. 27

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D. Naor, D. Gusfield and C. Martel (1990), "A fast algorithm for optimally increasing the edge-connectivity", Proceedings of the Foundation of Computer Science '90, 698707.

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D. Naor, D. Gusfield and C. Martel, A fast algorithm for optimally increasing the edge connectivity, 31 st Annual Symposium on Foundations of Computer Science, (1990), pp. 698-707.

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