A. Frank, T. Ibaraki, and H. Nagamochi, On sparse subgraphs preserving connectivity properties, J. Graph Theory, 17 (1993), pp. 275--281.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 0 = F 1 k , where F i (1 k) is the edge set of a maximal (but otherwise arbitrary) spanning forest of (V#En(F 1 i;1 ) see [Th 89,NI92] and a k node connected spanning subgraph (V#E 0 ) is obtained similarly, but now each F i is a maximal scan first search spanning forest, see [NI 92, FIN 93 CKT 93 ] In the approximate solution of NP hard combinatorial optimization problems, it often turns out that finding a solution within a factor of two of optimum is almost trivial, but achieving (asymptotically) better approximation guarantees needs a deeper understanding of the problem. For ....

.... second step can be improved to O(k ) as follows: we run a linear time preprocessing step to compute a sparse certificate E of G for k node connectivity, i.e. EjkjV and for all nodes v# w, V# E)hask openly disjoint v w paths iff G has k openly disjoint v w paths, see [NI 92, FIN 93 CKT93 We compute M as before, by running the first step on G.To find the set F EnM,we run the second step on M rather than on E, and for eachedgev i w i EnM,we attempt to find (k 1) openly disjoint v i i paths in the current subgraph of (V# M ) The second step runs in time ....

A. Frank, T. Ibaraki and H. Nagamochi, "On sparse subgraphs preserving connectivityproperties, " J. Graph Theory 17 (1993), 275--28

....(k 1) connected, then G is (k 1) connected. Moreover, G can be computed in linear time, NI 92] In detail, we construct a legal ordering v 1 v 2 : OE v n of V , and retain an edge v i v j , i j,inE iff igj k 1. Also, we need an extension of [CKT 93, Corollary 2. 17] and [FIN 93, Corollary 2.3] Proposition 3.4 (1) S V with jSjk is a shredder (or separator) of G iff S is a shredder (or separator) of G . 2) If G is k connected, then Q V is a tight set of G iff Q is a tight set of G . Proof: We prove part (1) for shredders. Suppose that S is a shredder of ....

....part (1) for shredders. Suppose that S is a shredder of G but not of G. Then there is an edge vw in EnE such that v and w are in different components of G nS. In the legal ordering for finding G , let v = v i and w = v j , i j, and note that igj k 1. But then the main lemma in [FIN 93] gives the desired contradiction, jS fv 1 #: #v i;1 gj jfv v j : 1#: #i; 1gj k 1. This gives an improvement on the previous theorem: By precomputing a sparse certificate ) for (k 1) connectivity, and running the algorithm for finding k shredders on G ,all the ....

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A. Frank, T. Ibaraki and H. Nagamochi, "On sparse subgraphs preserving connectivityproperties, " J. Graph Theory 17( 275--281.

....vertex connectivity, planar graphs AMS subject classifications. 68P05, 68Q20, 68R10 PII. S0097539794269072 1. Introduction. Sparse certificates, small graphs that retain some property of a larger graph, appear often in graph theory, especially in problems of edge and vertex connectivity [2, 13, 31, 35]. The main motivation for studying sparse certificates lies in the fact that they are e#ective tools for speeding up many graph algorithms. To check whether a graph G has a given property P, one can first compute a sparse certificate C for property P and then run an algorithm for P on the ....

A. Frank, T. Ibaraki, and H. Nagamochi, On sparse subgraphs preserving connectivity properties, J. Graph Theory, 17 (1993), pp. 275--281.

....hB values are integers in the interval [0; n] Thus, Step 6 can be implemented in linear time. 2 The time bound in the above theorem can be improved by precomputing a sparse certificate for (k 1) connectivity and local (k 1) node connectivities, G 0 = V; E 0 ) E 0 E, see [NI 92, CKT 93, FIN 93] The number of edges in G 0 , jE 0 j, is (k 1) n Gamma 1) O(kn) If G is (k 1) connected, then G 0 is (k 1) connected, and moreover, G 0 (v; w) min(k 1; G (v; w) for all node pairs v; w, where H (v; w) denotes the maximum number of openly disjoint v w paths in the graph H . A ....

.... 1. An ordering v 1 ; v 2 ; v n of the nodes of G is called legal if d(V i Gamma1 ; v i ) V i Gamma1 ; v j ) 81 i j n, where V = fv 1 ; v 2 ; v g, and d(Q; v) denotes the number of edges between v and Q V . Also, we need an extension of [CKT 93, Corollary 2. 17] and [FIN 93, Corollary 2.3] Proposition 3.3 (1) S ae V with jSj k is a shredder (or separator) of G iff S is a shredder (or separator) of G 0 . 2) If G is k connected, then Q ae V is a tight set of G iff Q is a tight set of G 0 . Proof: We prove part (1) for shredders. Suppose that S is a shredder ....

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A. Frank, T. Ibaraki and H. Nagamochi, "On sparse subgraphs preserving connectivity properties, " J. Graph Theory 17 (1993), 275--281.

....to know about the Gomory Hu tree to show that you can find the min cut in at most n flow computations. Find a simple way to do this yourself by a suitable choice of source destination pairs. 2 The O hides poly logarithmic factors. 10.1. A SIMPLE MINIMUM CUT ALGORITHM 147 Ibaraki and Nagamochi [8]. Since we only examining the edge version of the minimum cut problem we follow an even simpler treatment due to Stoer and Wagner [27] The paper by Frank, Ibaraki and Nagamochi remains a valuable reference for the most general result provable using this approach. 10.1.2 The Stoer Wagner ....

....node capacitated version and refer to this as the multiway cut problem. We will examine an approximation algorithm with performance ratio 2(1 Gamma 1 k ) same as for the k cut problem, for the multiway node cut problem with k terminals. This algorithm, due to Garg, Vazirani and Yannakakis [8], uses an integer programming (IP) formulation of the problem and shows that the linear programming (LP) relaxation of the formulation has an optimum solution all of whose components are half integers. Their proof also gives a method to find an approximate multiway cut from any optimal linear ....

A. Frank, T. Ibaraki, and H. Nagamochi, On Sparse Subgraphs Preserving Connectivity Properties, J. Graph Theory 17:3 (1993), pp. 275-281.

.... F i (1 i k) is the edge set of a maximal (but otherwise arbitrary) spanning forest of (V; En(F 1 [ F i Gamma1 ) see [Th 89, NI 92] and a k node connected spanning subgraph (V; E 0 ) is obtained similarly, but now each F i is a maximal scan first search spanning forest, see [NI 92, FIN 93, CKT 93] In the approximate solution of NP hard combinatorial optimization problems, it often turns out that finding a solution within a factor of two of optimum is almost trivial, but achieving (asymptotically) better approximation guarantees needs a deeper understanding of the problem. For ....

.... improved to O(k 3 jV j 2 ) as follows: we run a linear time preprocessing step to compute a sparse certificate e E of G for k node connectivity, i.e. e E E, j e Ej kjV j, and for all nodes v; w, V; e E) has k openly disjoint v w paths iff G has k openly disjoint v w paths, see [NI 92, FIN 93, CKT 93] We compute M as before, by running the first step on G. To find the set F EnM , we run the second step on e E [ M rather than on E, and for each edge v i w i 2 e EnM , we attempt to find (k 1) openly disjoint v i w i paths in the current subgraph of (V; e E [ M ) The second step ....

A. Frank, T. Ibaraki and H. Nagamochi, "On sparse subgraphs preserving connectivity properties, " J. Graph Theory 17 (1993), 275--281.

....choice of source destination pairs. 2 The O hides polylogarithmic factors. fast algorithms for constructing such subgraphs. A short proof of a generalization of the results of Nagamochi and Ibaraki to mixed cuts containing both edges and nodes was presented by Frank, Ibaraki and Nagamochi [14]. Since we only examing the edge version of the minimum cut problem we follow an even simpler treatment due to Stoer and Wagner [43] The paper by Frank, Ibaraki and Nagamochi remains a valuable reference for the most general result provable using this approach. 9.2 The Stoer Wagner Algorithm ....

A. Frank, T. Ibaraki, and H. Nagamochi, On Sparse Subgraphs Preserving Connectivity Properties, J. Graph Theory 17:3 (1993), pp. 275-281.

....was conducted in Boston University. E mail: itkis cs.technion.ac.il Part of this research was conducted while visiting the MIT Laboratory for Computer Science, and CRL Digital Equipment Corporation. Partly supported by DGAPA Projects, U.N.A.M. E mail: rajsbaum theory.lcs.mit.edu Following [FIN93], we consider a generalization of these two particular types of connectivity. Let S V . We say that a family of paths connecting vertices x; y is S independent if the paths are edge disjoint and every element of S appears as an inner vertex in at most one of these paths. The S mixed ....

....certificate of size kjV j. Moreover, if G is simple and the forests are grown according to a certain rule (see Section 4) then G k is a vertex k connectivity certificate as well. Graph G is called S simple if it has no parallel edges incident to vertices of S. Frank, Ibaraki and Nagamochi [FIN93] show that for all S V the algorithm of [NI92] applied to S simple graphs produces certificates of S mixed connectivity. Cheriyan, Kao and Thurimella [CKT93] introduce a more flexible way of constructing certificates of vertex connectivity of size kjV j, and use it in their distributed and ....

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A. Frank, T. Ibaraki, H. Nagamochi, "On sparse subgraphs preserving connectivity properties", J. of Graph Theory, 17(3), 1993, pp. 275--281.

....tuples ( B ; hB ) and the B ; hB values are integers in the interval [0; n] Thus, Step 6 can be implemented in linear time. 2 The time bound in the above theorem can be improved by precomputing a sparse certificate for (k 1) connectivity, G 0 = V; E 0 ) E 0 E, see [NI 92, CKT 93, FIN 93] G 0 has jE 0 j (k 1) n Gamma 1) O(kn) and if G is (k 1) connected, then G 0 is (k 1) connected. Moreover, G 0 can be computed in linear time, NI 92] In detail, we construct a legal ordering v 1 OE v 2 OE : OE v n of V , and retain an edge v i v j , i j, in E 0 iff ....

....(k 1) connected. Moreover, G 0 can be computed in linear time, NI 92] In detail, we construct a legal ordering v 1 OE v 2 OE : OE v n of V , and retain an edge v i v j , i j, in E 0 iff jfv : v v j 2 E; igj k 1. Also, we need an extension of [CKT 93, Corollary 2. 17] and [FIN 93, Corollary 2.3] Proposition 3.4 (1) S ae V with jSj k is a shredder (or separator) of G iff S is a shredder (or separator) of G 0 . 2) If G is k connected, then Q ae V is a tight set of G iff Q is a tight set of G 0 . Proof: We prove part (1) for shredders. Suppose that S is a shredder ....

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A. Frank, T. Ibaraki and H. Nagamochi, "On sparse subgraphs preserving connectivity properties, " J. Graph Theory 17 (1993), 275--281.

....the largest r; for each vertex y adjacent to x by an unscanned edge e = x; y) do begin r(y) r(y) c(e) q(e) r(y) Mark e scanned end; Mark x visited end; end. CAPFOREST can be executed in O(m n log n) time [8] The next property for q(e) is essential for our purposes (see [2, 7] for other properties related to graph connectivity) LEMMA 3 [8] i) The q(e) obtained by CAPFOREST satis es q(e) x; y) e = x; y) 2 E: 6) ii) Let e 3 = x 3 ; y 3 ) be the last edge scanned in CAPFOREST. Then this e 3 satis es q(e 3 ) x 3 ; y 3 ) 7) Furthermore, the ....

A. Frank, T. Ibaraki and H. Nagamochi: \On sparse subgraphs preserving connectivity properties," J. Graph Theory 17 (1993) 275-281.

....no combinatorial algorithm is known so far. Recently, Queyranne [16] found a purely combinatorial algorithm for the problem under the additional condition that f is symmetric, by extending the algorithm of Nagamochi and Ibaraki [13] for nding a minimum cut in an undirected network (see also [5, 6, 9, 14, 17] for its simpler proofs) ###o##m # [16] For a given symmetric and fully submodular function f on V , an optimal solution can be found by O(jV j 3 ) function value oracle calls. 2 In this note, we slightly generalize the above result by replacing the symmetry condition (2) with the ....

A. Frank, T. Ibaraki and H. Nagamochi, On sparse subgraphs preserving connectivity properties, J. Graph Theory, 17 (1993) 275-281.

....such a complete feasible splitting at s and by ignoring the isolated vertex s, satis es G 0 (V 0 ) k, i.e. G 0 is k edge connected. 2. 2 Maximum adjacency ordering An ordering v 1 ; v 2 ; v n of all vertices in V is called a maximum adjacency (MA) ordering (also called legal in [6]) in G if it satis es dG (fv 1 ; v 2 ; v i g; v i 1 ) dG (fv 1 ; v 2 ; v i g; v j ) 1 i j n: This plays a crucial role in this paper through the following lemmas. Lemma 2 [5, 10, 11, 15, 19] Let G = V; E; c G ) be an edge weighted graph, and let v 1 ; v 2 ; v n ....

A. Frank, T. Ibaraki and H. Nagamochi, On sparse subgraphs preserving connectivity properties, J. Graph Theory, 17, (1993), pp. 275-281.

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Frank, A., Ibaraki, T. and Nagamochi, H., \On sparse subgraphs preserving connectivity properties, " J. Graph Theory, Vol.17, 1993, pp. 275-281.

....cut algorithm is O(n(m n log n) which is currently one of the best among the existing deterministic algorithms. Given a graph G = V; E; c G ) not necessarily connected) an ordering v 1 ; v 2 ; v n of all vertices in V is called a maximum adjacency (MA) ordering (also called legal in [4]) in G if it satis es c G (fv 1 ; v 2 ; v i01 g; v i ) max fc G (fv 1 ; v 2 ; v i01 g; u) j u 2 fv i ; vngg ; 1 i n: Such an ordering can be found in O(m n log n) time [8] Lemma 1 [3, 5, 7, 8, 15] For a graph G = V; E; c G ) let v 1 ; v 2 ; v n be an ....

....CAPFOREST [8] does not a ect the correctness of the lemma. However, the proof of the lemma was rather technical and complicated, since it rst proved the case of rational valued weights, and then extended the argument to real valued weights. Since MA ordering has some other useful applications [4, 7, 11], many researchers have studied properties of MA ordering, and discovered other simpler proofs of Lemma 1 [3, 5, 15] Also [13] generalizes the lemma to a symmetric submodular function c G : 2 V R . In this paper, we rst present a new simple proof of Lemma 1. Our proof not only shows (3) ....

A. Frank, T. Ibaraki and H. Nagamochi, On sparse subgraphs preserving connectivity properties, J. Graph Theory, Vol.17, 1993, pp. 275-281.

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A. Frank, T. Ibaraki, and H. Nagamochi, On sparse subgraphs preserving connectivity properties, J. Graph Theory, 17 (1993), pp. 275--281.

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A. Frank, T. Ibaraki, H. Nagamochi, On sparse subgraphs preserving connectivity properties, Journal of Graph Theory, Vol. 17, No. 3 (1993), 275-281. References 6

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A. Frank, T. Ibaraki and H. Nagamochi, "On Sparse Subgraphs Preserving Connectivity Properties," Journal of Graph Theory, 17:275--281, 1993.

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