G. Frederickson and J. Ja'Ja, "Approximation Algorithms for Several Graph Augmentation Problems," SIAM Journal of Computing, vol. 10, no. 2, pp. 270--283, May 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....By a similar argument, the outconnectivity to a subset problem is also NP hard, even for k = 2 and S = V [CJN01] It immediately follows that SNDP (which is a more general problem) is also NP hard. VCAP 0,2 is NP hard by a similar argument [ET76] and VCAP 1,2 is proved to be NP hard in [FJ81] Most previous work on approximating vertex connectivity problems concentrated on upper bounds, i.e. on designing approximation algorithms. An approximation ratio of 2k for k VCSS was obtained in [CJN01] by a straightforward application of [FT89] and the approximation ratio was later improved ....

....ratio of 1 1 k is obtained in [CT00] For a complete Euclidean graph, a polynomial time approximation scheme (i.e. factor 1 # for any fixed # 0) is devised in [CL99] The connectivity augmentation problem has also attracted a lot of attention. A 2 approximation for VCAP 1,2 is shown in [FJ81,KT93] In the case where every pair of vertices in the graph forms an augmenting edge of unit cost, VCAP k,k 1 is not known to be in P nor to be NP hard. For the latter problem, a k 2 additive approximation is presented in [Jor95] and optimal algorithms for small values of k are shown in ....

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G. N. Frederickson and J. JaJa. Approximation algorithms for several graph augmentation problems. SIAM J. Comput., 10(2):270--283, 1981.

....By a similar argument, the outconnectivity to a subset problem is also NP hard, even for k = 2 and S = V n frg [CJN01] It immediately follows that SNDP (which is a more general problem) is also NP hard. VCAP 0;2 is NP hard by a similar argument [ET76] and VCAP 1;2 is proved to be NP hard in [FJ81] Most previous work on approximating vertex connectivity problems concentrated on upper bounds, i.e. on designing approximation algorithms. An approximation ratio of 2k for k VCSS was obtained in [CJN01] by a straightforward application of [FT89] and the approximation ratio was later improved ....

....ratio of 1 1=k is obtained in [CT00] For a complete Euclidean graph, a polynomial time approximation scheme (i.e. factor 1 for any xed 0) is devised in [CL99] The connectivity augmentation problem has also attracted a lot of attention. A 2 approximation for VCAP 1;2 is shown in [FJ81, KT93] In the case where every pair of vertices in the graph forms an augmenting edge of unit cost, VCAP k;k 1 is not known to be in P nor to be NP hard. For the latter problem, a k 2 additive approximation is presented in [Jor95] and optimal algorithms for small values of k are shown in [ET76, ....

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G. N. Frederickson and J. JaJa. Approximation algorithms for several graph augmentation problems. SIAM J. Comput., 10(2):270-283, 1981. 19

....e#AUG w(e) 1) such that graph GAUG (V, E 0 AUG) is edge biconnected, i.e. CE (GAUG ) 2. In G 0 , an edge e E 0 is called a bridge if its deletion disconnects G 0 . GAUG may therefore have no bridges. Note that the E2AUG problem is also called bridge connectivity augmentation problem [4]. Besides the design of communication networks, this problem is also important to VLSI floorplanning [19] An electronic circuit can be interpreted as a graph whose vertices are the (rectangular) functional units and whose edges are the interconnections between the units. If the graph has a ....

....algorithm for the special case when the weights w(e) e E, are all equal and G is a complete graph. However, for the general case with di#erent weights, they showed that the problem is NP complete, see also [5, 8] The problem even remains if weights are chosen from set 1,2 only [4]. In general, it is computationally too expensive to solve larger problem instances to optimality using exact techniques like branch and bound. Therefore, heuristics which are able to find high quality suboptimal solutions in polynomial time are of interest. The next section gives an overview of ....

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Frederickson G. N., Jaja J.: Approximation algorithms for several graph augmentation problems, SIAM Journal on Computing 10(2), (1981), 270--283

....are drawn in Sect. 6. 2 Previous Work Eswaran and Tarjan [1] originally investigated the V2AUG problem. They showed it to be NP hard. An exact polynomial time algorithm could only be found for the special case when G is complete and each edge has unit costs [4] Frederickson and Jaja [2] provided an approximation algorithm for the general case which finds a solution within a factor 2 of the optimum. The algorithm includes a preprocessing that transforms the fixed graph G 0 into a block cut tree, see the Sect. 3. Each potential augmentation edge from E a is superimposed on the ....

....all those edges which are covered by e # A (in addition to others) e # A is obsolete and can be discarded. All such edges can be identified in O( V ) time 5 as a byproduct from a dynamic programming algorithm that computes distance values needed for the algorithm from Frederickson and Jaja [2]; see Fig. 2(b) Fixing of Edges: An edge e A EA must be included in any feasible solution to the V2AUG problem, when it represents the only possibility to connect a cut node s cut component C i to any other cut component of v c . In more detail, we process for each cut node v c its set A(v ....

G. N. Frederickson and J. Jaja. Approximation algorithms for several graph augmentation problems. SIAM Journal on Computing, 10(2):270--283, 1981.

....components. In a graph that is not edge biconnected, a critical edge whose removal would disconnect the graph is called uncovered edge or bridge. This article focuses on the edge biconnectivity augmentation (E2AUG) problem, which is sometimes also called bridge connectivity augmentation problem [4]. In it, an undirected, connected graph G = V, E) with node set V and edge set E and an additional, disjoint set A of edges between nodes in V are given. Each edge e has associated costs c(e) 0 and can be used to # This work is supported by the Austrian Science Fund (FWF) under the grant ....

....average computational e#ort is low, which allows a fast execution of the EA also on large graphs. Section 5 presents experimental results indicating the e# ciency of this approach and its superiority over two previous heuristic methods. 2. Previous work As described by Frederickson and Jaja [4], the problem of augmenting a general connected graph G can be e#ectively reduced into the problem of augmenting a spanning tree by shrinking the node sets of all already edge biconnected components into corresponding single new nodes, Fig. 1(a) shows an example. Edges in E and A between nodes of ....

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G. N. Frederickson and J. Jaja. Approximation algorithms for several graph augmentation 10(2):270--283, 1981.

....component in H 0 can be contracted into a single vertex without losing the property of the problem) Let us call the problem with a tree H 0 the minimum 2 edge connected subgraph problem containing a spanning tree (2 ECST) which is shown to be NP hard by Frederickson and J. J aJ a [6] (even if the height of a spanning tree H 0 is 2 and every edge in E E 0 connects two vertices of degree 1 in H 0 ) In the special case of H being a complete graph, 2 ECST is the problem of augmenting a tree H 0 to a 2 edge connected graph by adding a minimum number of new edges, for which ....

....graph by adding a minimum number of new edges, for which Eswaran and Tarjan [5] presented a linear time algorithm (which creates no multiple edges) If H is a general graph, we are permitted to add to H 0 only edges from E 0 E 0 . For general 2 ECST, there is a 2 approximation algorithm [6, 11], which relays on the minimum branching algorithm. However, as remarked by Khuller [10, p.263] one of the main open problems in the graph augmentation problem is to obtain a factor better than 2 for 2 ECST. In this paper, we present a (1:875 ) approximation algorithm for 2 ECST, where 0 is ....

G. N. Frederickson and J. JaJa: \Approximation algorithms for several graph augmentation problems," SIAM J. Computing, 10 (1981) 270-283.

....to find an augmentation such that the backup bandwidth reserved on edges in the augmentation is minimum. In other words, we are interested in the optimal augmentation # which minimizes the quantity Note that this is more difficult than weighted 2 edgeconnectivity (see Frederickson and Ja Ja [14] and Khuller and Thurimella [11] Indeed, in weighted 2 edge connectivity, the cost of a non tree edge # is given, while here it depends on which edges are covered by #,i.e. on####. 0 7803 7476 2 02 17.00 (c) 2002 IEEE. j i 15 15 5 5 5 5 u v x y 15 15 20 20 20 20 augmentation with ....

G. N. Frederickson and J. Ja.Ja, "Approximation algorithms for several graph augmentation problems," SIAM Journal of Computing, vol. 10-2, pp. 270--283, 1981.

....Hence, the 2 ECST Problem is equivalent to the Tree Augmentation Problem (TAP) de ned as follows. The input consists of a tree T (V; E) and a set of links E V V . The goal is to nd a smallest subset F E such that G(V; E [ F ) is 2 edge connected. Previous Results. Frederickson J aj a [7] presented a 2 approximation to the weighted version of TAP. Improving the approximation ratio below 2 was posed by Khuller [12] as one of the main open problems in graph augmentation. Nagamochi Ibaraki [11] presented a 12=7 approximation algorithm for TAP. However, the proof of the ....

....including: leaf close trees, shadows, stems, and the usage of maximum matchings that forbid certain links. Our lower bound (Claim 12) is a strengthening of [10, Lemma 4.2] Related Results. The weighted version of 2 ECST (or equivalently TAP) is called Bridge Connectivity Augmentation (BRA) in [7]. In the BRA problem, the input consists of a complete graph G(V; E) and edge weights w(e) The goal is to nd a minimum weight subset of edges F such that G 0 (V; F ) is 2 edge connected. The 2 ECST is simply the case in which the edges have f0; 1; 1g weights; the edges of the connected graph ....

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G. N. Frederickson and J. Jaja, \Approximation algorithms for several graph augmentation problems", SIAM J. Computing, 10 (1981), 270-283.

....the edge connectivity augmentation problem rst studied by Eswaran and Tarjan [2] Here we are given a graph H = U; F ) a spanning tree T in H and we wish to nd a set of edges F 0 F nT such that (U; T [F 0 ) is 2 edge connected and F 0 has minimum cardinality. Frederickson and J aJ a [3] showed that the problem is NP hard and gave a 2 approximation algorithm. The problem of covering a laminar family can be modeled as an integer program. For every edge e 2 E, associate a variable x e , which is 1 if the edge is included in E 0 and 0 otherwise. Covering a set S 2 F amounts to ....

G. N. Frederickson and J. JaJa. Approximation algorithms for several graph augmentation problems. SIAM Jounal on Computing, 10(2):270-283, 1981.

....is the function f(S) k for all subsets S ae V ; this is known as the minimum cost k strongly connected subgraph problem. For this case, a simple 2 approximation algorithm is obtained by solving two minimum cost k disjoint arborescence problems (one into and one out from) at an arbitrary node v [5]. Frank [6] showed that in the special case where the requirement function is also intersecting supermodular, i.e. the above inequality holds whenever X and Y intersect, the network design problem can be solved optimally in polynomial time. Melkonian and Tardos 1 [17] have recently shown that ....

....any strongly connected subgraph of D, possibly violating some orientation constraints, and reduces the problem of amending its violated orientation constraints to that of finding a minimum cost 2edge connected subgraph in an undirected graph. For the latter problem, a 2 approximate algorithm [5, 14] is known. We now describe our algorithm in detail: 1. Pick any node r and compute a minimum cost in branching to r, say T 1 , as well as a 9 minimum cost out branching from r, say T 2 . Consider the directed graph D 1 = V; A 1 ) induced by T 1 [T 2 . Clearly, D 1 is strongly connected and its ....

G. N. Frederickson and J. J'aJ'a, Approximation algorithms for several graph augmentation problems, SIAM Journal on Computing, 10(2) (1981), pp. 270--283.

....assuming P6=NP [4] 1.2. Other Related Work. The union of any incoming branching and any outgoing branching from the same root yields an SCSS with at most 2n Gamma 2 edges (where n is the number of vertices in the graph) This is a special case of the algorithm given by Frederickson and J aJ a [6] that uses minimumweight branchings to achieve a performance guarantee of 2 for weighted graphs. Since any SCSS has at least n edges, this yields a performance guarantee of 2 for the SCSS problem. 2 Any minimal SCSS (one from which no edge can be deleted) has at most 2n Gamma 2 edges and also ....

....characterize the various complexities of the minimum SCSS k problems. The most interesting open problem is to obtain a performance guarantee that is less than 2 for the weighted strong connectivity problem (as mentioned earlier, the performance factor of 2 is due to Frederickson and J aJ a [6]) Such an algorithm may have implications for the weighted 2 connectivity problem [15] in undirected graphs as well. The performance guarantee of k Exchange probably improves as k increases. Proving this would be interesting similar local improvement algorithms are applicable to a wide ....

G. N. Frederickson and J. J' aJ' a, Approximation algorithms for several graph augmentation problems, SIAM Journal on Computing, 10 (2), pp. 270--283, (1981). 14

....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 results can be found in [ET76, Fra92, FJ81, KT92, WHN90, WN93]. 3 Definitions We use the following notations on graphs. In this paper, G is an undirected graph with the set of vertices V and the set of edges E and is also denoted as G = V; E) The graph G is simple, i.e. one without multiple edges between a pair of vertices in G and without self loops. ....

G. N. Frederickson and J. J'aJ'a. Approximation algorithms for several graph augmentation problems. SIAM J. Comput., 10(2):270--283, May 1981.

....from the Hamiltonian circuit problem. This version of the minimum augmentation problem remains NP hard even if the input graph is connected. The problem of strongly connecting a weakly connected directed graph remains NP hard. The reductions are made from the 3 dimensional matching problem [FJ81] Watanabe and Nakamura [WN87] showed that the minimum augmentation problem for k edge connectivity or k connectivity is NP hard, for any k 2. It is also shown in [WNN89] that for 3 edge connectivity and triconnectivity, the minimum augmentation problem is NP hard even if the input graph is ....

....every pair of vertices in S, for any k 2. They showed that the problem is NP hard if we are required to have a directed cycle between every pair of vertices in S on directed graphs, even if the values of the costs are all equal to 1. 2.2. 2 Approximation Algorithms Frederickson and J aJ a [FJ81] gave approximation algorithms for directed graphs to achieve strong connectivity, and for undirected graphs to achieve 2 edge connectivity and biconnectivity. Their algorithm ran in O(n 2 ) time on an n node graph where the sum of the costs of all edges added is at most twice the minimum cost ....

G. N. Frederickson and J. J'aJ'a. Approximation algorithms for several graph augmentation problems. SIAM J. Comput., 10(2):270--283, May 1981.

....that T E 0 = V (T ) E(T ) E 0 ) is 2 edge connected; we may assume that E 0 has no multiedges. Instead of taking T to be a tree, we may take T to be a connected graph. This gives the problem CBRA which was initially studied by Eswaran Tarjan [ET 76] and by Frederickson Ja ja [FJ 81] Similarly, the problem of finding a maximum 1 packing of a capacitated laminar family H; u from among the multiedges of a k packing E may be reformulated as follows. Let T = T (H) be the tree representing H, and let the tree edges have (nonnegative) integer capacities u : E(T ) Z; the capacity ....

....that x is integral and 0. 1.3 Approximation Algorithms for NP hard Problems in Connectivity Augmentation Our results on 2 covers and 2 packings imply improved approximation algorithms for some NP hard problems in connectivity augmentation and related topics. Frederickson and Ja ja [FJ 81] showed that problem CBRA is NP hard and gave a 2 approximation algorithm. Later, Khuller and Vishkin [KV 94] gave another 2 approximation algorithm for a generalization, namely, find a minimumweight k edge connected spanning subgraph of a given weighted graph. Subsequently, Garg et al. [GVY 97, ....

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G.N.Frederickson and J.Ja'Ja', "Approximation algorithms for several graph augmentation problems," SIAM J. Comput. 10 (1981), 270--283.

....Since this problem is NP hard, we will discuss polynomial time heuristics. One heuristic for this problem works as follows: fix a root r and find a minimum weight in branching 1 B I and a minimum weight outbranching BO [7] and take the union of the two branchings. Frederickson and J aJ a [1] used this idea to obtain a 2 approximation for the SCSS problem. Observe that the weight of each branching is at most the weight of the minimum strongly connected subgraph; thus we get an approximation factor of 2. One can define similar problems in undirected graphs. One natural problem is ....

....to find a minimum weight strongly connected spanning subgraph. ffl k Augmentation: Given a graph G = V; E) a subgraph H = V; EH ) of G, and a positive integer k, the kAUG problem is to find a subset of edges of G with minimum total weight, whose addition to EH generates a k edge connected graph[1, 4]. When the initial subgraph H has no edges then this is the kECSS problem. ffl Directed edge connectivity: We also consider a natural directed analog of the k edge connectivity problem, which generalizes the SCSS problem. The input is a graph G = V; E) and a positive integer k. The problem is ....

G. N. Frederickson and J. J'aJ'a, Approximation algorithms for several graph augmentation problems, SIAM J. Comput., 10 (2), pp. 270--283, (1981).

....such that T E 0 = V (T ) E(T ) E 0 ) is 2 edge connected; we may assume that E 0 has no multiedges. Instead of taking T to be a tree, we may take T to be a connected graph. This gives the problem CBRA which was initially studied by Eswaran Tarjan [ET 76] and by Frederickson Ja ja [FJ 81] see Section 4.1. Similarly, the problem of finding a maximum 1 packing of a capacitated laminar family H; u from among the multiedges of a k packing E may be reformulated as follows. Let T = T (H) be the tree representing H, and let the tree edges have (nonnegative) integer capacities u : ....

....x e for such extreme point solutions. Approximation algorithms for NP hard problems in connectivity augmentation Our results on 2 covers and 2 packings imply improved approximation algorithms for some NP hard problems in connectivity augmentation and related topics. Frederickson and Ja ja [FJ 81] showed that problem CBRA is NP hard and gave a 2 approximation algorithm. Later, Khuller and Vishkin [KV 94] gave another 2 approximation algorithm for a generalization, namely, find a minimumweight k edge connected spanning subgraph of a given weighted graph. Subsequently, Garg et al. [GVY 97, ....

[Article contains additional citation context not shown here]

G.N.Frederickson and J.Ja'Ja', "Approximation algorithms for several graph augmentation problems, " SIAM J. Comput. 10 (1981), 270--283.

.... be pointed out that the problem of constructing a graph with n vertices, and connectivity with the least number of edges was first addressed by Harary [19] The first paper to address the issue of obtaining approximate solutions for the case when edges have weights, is by Frederickson and J aJ a [11]. They provide approximation algorithms for the cases of 2 connectivity (edge and vertex) as well as strong connectivity problems. Subsequently, their algorithm was simplified by Khuller and Thurimella[26, 27] The unweighted case was explored by Khuller and Vishkin [28] and Garg, Santosh and ....

....Given a graph G = V; E) with weights on the edges and an integer , consider the problem of finding a minimum weight spanning subgraph H = V; EH ) that is edge connected. An algorithm that achieves an approximation factor of 3 for = 2 follows by the work of Frederickson and J aJ a [11]. First find a minimum spanning tree. Now consider the problem of finding the least weight set of edges to add to the tree to obtain a 2 edge connected subgraph. Not surprisingly, this is NP hard as well [11] They give an algorithm with an approximation factor of 2 for the problem of augmenting ....

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G. N. Frederickson and J. J'aJ'a, Approximation algorithms for several graph augmentation problems, SIAM Journal on Computing, 10 (2), pp. 270--283, (1981).

....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 ....

Greg N. Frederickson & Joseph Ja'Ja', "Approximation algorithms for several graph augmentation problems, " SIAM J. Comput., Vol. 10, May 1981, pp. 270283.

.... Steiner tree r i 2 f0; 1g 2 O(n 2 ) Mehlhorn [15] 11 6 O(n 3:5 ) Zelikovsky [22] 16 9 O(n 5 ) Berman and Ramaiyer [2] Generalized Steiner tree r ij 2 f0; 1g 2 O(n 2 log n) Agrawal et al. 1] Goemans and Williamson [8] 2 edge connected subgraph r i = 2 3 O(n 2 ) Frederickson and Ja Ja [3] k edge connected subgraph r i = k 2 O(kn 3 log n) Khuller and Vishkin [12] Generalized Steiner 2 edgeconnected subgraph r ij 2 f0; 2g 3 O(n 2 log n) Klein and Ravi [13] Table 1: Previous work on special cases of SNDP listed in Table 1. Throughout this paper n and m denote the number of ....

G. N. Frederickson and J. Ja'Ja'. Approximation algorithms for several graph augmentation problems. SIAM Journal on Computing, 10:270--283, 1981.

....is due to Ravi and Williamson [25] that achieves a factor of 2H(k) where H(k) is the kth Harmonic number (H(k) 1 1 2 : 1 k ) For the special case of k = 3, Penn and Shasha Krupnik [24] have used our techniques to obtain an approximation factor of 3. Frederickson and J aJ a [9] considered the problem of computing a minimum weight biconnected spanning subgraph. They gave an approximation algorithm for a more general graph augmentation problem and used it to obtain a 3 approximation algorithm for the biconnectivity problem. For k = 2, Ravi and Williamson s algorithm also ....

G. N. Frederickson and J. J'aJ'a, Approximation algorithms for several graph augmentation problems, SIAM J. Comput., 10 (2), pp. 270--283, (1981).

....assuming P6=NP [4] 1.2. Other Related Work. The union of any incoming branching and any outgoing branching from the same root yields an SCSS with at most 2n 0 2 edges (where n is the number of vertices in the graph) This is a special case of the algorithm given by Frederickson and J aJ a [6] that uses minimum weight branchings to achieve a performance guarantee of 2 for weighted graphs. Since any SCSS has at least n edges, this yields a performance guarantee of 2 for the SCSS problem. Any minimal SCSS (one from which no edge can be deleted) has at most 2n 0 2 edges and also yields ....

....characterize the various complexities of the minimum SCSS k problems. The most interesting open problem is to obtain a performance guarantee that is less than 2 for the weighted strong connectivity problem (as mentioned earlier, the performance factor of 2 is due to Frederickson and J aJ a [6]) Such an algorithm may have implications for the weighted 2 connectivity problem [15] in undirected graphs as well. The performance guarantee of k Exchange probably improves as k increases. Proving this would be interesting similar local improvement algorithms are applicable to a wide ....

G. N. Frederickson and J. J' aJ' a, Approximation algorithms for several graph augmentation problems, SIAM Journal on Computing, 10 (2), pp. 270--283, (1981).

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G. Frederickson and J. Ja'Ja, "Approximation Algorithms for Several Graph Augmentation Problems," SIAM Journal of Computing, vol. 10, no. 2, pp. 270--283, May 1981.

No context found.

G. N. Frederickson and J. Ja'Ja', Approximation algorithms for several graph augmentation problems, SIAM Journal on Computing, 10 (1981), pp. 270--283.

No context found.

G. N. Frederickson and J. JaJa. Approximation algorithms for several graph augmentation problems. SIAM J. Comput., 10(2):270--283, 1981.

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G. N. Frederickson and J. JaJa, Approximation algorithms for several graph augmentation problems, SIAM Journal on Computing, 10(2) (1981), pp. 270-283.

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