S. Khuller, B. Raghavachari, and N. Young, Approximating the minimum equivalent digraph, SIAM J. Comput., 24 (1995), pp. 859--872.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....# ) such that for every choice of vertices x, y # V there is a directed path from x to y in D if and only if D # has such a path. The minimum equivalent digraph problem and its generalizations to higher degrees of connectivity has practical applications and has been studied extensively, see e.g. [1, 10, 13, 14, 17]. Furthermore, for a given class of digraphs, which is closed under the operation of taking induced subdigraphs, one can find the minimum equivalent digraph in polynomial time if and only if one can solve the MSSS problem in polynomial time for that class. Hence it is of interest to find classes ....

....to find the desired strong subdigraph with few arcs, we see that the following holds: Theorem 8.5 Every strong digraph D with #(D) # n r contains a spanning strong digraph with at most (1 1 r )n arcs. Furthermore, such a subdigraph can be found in polynomial time. Khuller et at [14, 15] proved that for general digraphs, a variant of the algorithm used in the proof of Theorem 4.2 (contracting cycles which are su#ciently long and taking the arcs of the contracted cycles as a spanning subdigraph) results in a spanning strong subdigraph with no more than 1.61 times the number of ....

S. Khuller, B. Raghavachari and N. Young, Approximating the minimum equivalent digraph, SIAM J. Computing 24 (1995) 859-872.

....to acknowledge financial support from the Danish Research Council (under grant 9800435) 1 has as few arcs as possible. This problem, which generalizes the hamiltonian cycle problem and hence is NP hard, is of practical interest and has been considered several times in the literature, see e.g. [1, 12, 15, 16, 17, 18]. The MSSS problem is an essential subproblem of the so called minimum equivalent digraph problem. Here one is seeking a spanning subgraph with the minimum number of arcs in which the reachability relation is the same as in the original graph (i.e. there is a path from x to y if and only if the ....

S. Khuller, B. Raghavachari and N. Young, Approximating the minimum equivalent digraph, Siam J. Computing 24 (1995) 859-872.

....D = V; A) find a spanning subdigraph D 0 = V; A 0 ) such that for every choice of vertices x; y 2 V there is a directed path from x to y in D if and only if D 0 has such a path. The minimum equivalent digraph problem has practical applications and has been studied extensively, see e.g. [1, 13, 14, 17]. Furthermore, for a given class of digraphs, which is closed under the operation of taking induced subdigraphs, one can find the minimum equivalent digraph in polynomial time if and only if one can solve the MSSS problem in polynomial time for that class. Hence it is of interest to find classes ....

....to find the desired strong subdigraph with few arcs, we see that the following holds: Theorem 8.5 Every strong digraph D with Delta(D) n r contains a spanning strong digraph with at most (1 1 r )n arcs. Furthermore, such a subdigraph can be found in polynomial time. Khuller et at [14, 15] proved that for general digraphs, a variant of the algorithm used in the proof of Theorem 4.2 (contracting cycles which are sufficiently long and taking the arcs of the contracted cycles as a spanning subdigraph) results in a spanning strong subdigraph with no more than 1:61 times the number of ....

S. Khuller, B. Raghavachari and N. Young, Approximating the minimum equivalent digraph, SIAM J. Computing 24 (1995) 859-872. 16

....Danish Research Council (under grant 9800435) spanning subgraph D 0 of D such that D 0 has as few arcs as possible. This problem, which generalizes the hamiltonian cycle problem and hence is NP hard, is of practical interest and has been considered several times in the literature, see e.g. [1, 12, 15, 16, 17, 18]. The MSSS problem is an essential subproblem of the so called minimum equivalent digraph problem (in fact, these two problems can be reduced to each other in polynomial time) Here one is seeking a spanning subgraph with the minimum number of arcs in which the reachability relation is the same as ....

S. Khuller, B. Raghavachari and N. Young, Approximating the minimum equivalent digraph, Siam J. Computing 24 (1995) 859-872.

....by V C 7 the vertex cover problem restricted to graphs with maximum degree seven. Papadimitriou and Yannakakis [17] showed that V C 7 is MAX SNP hard. We prove that V C 7 L reduces to the minimum size 2 edge connected spanning subgraph problem, here denoted by 2MECSS. The reduction comes from [11], where a directed version of the 2 MECSS is proved to be MAX SNP hard. The rst part of the L reduction is a polynomial time algorithm f and a constant . Given any instance G of V C 7 , f produces an instance H of the 2 MECSS such that the minimum number of edges in a 2 edge connected spanning ....

....minimum size of a vertex cover in G, denoted by opt V C 7 (G) In other words, opt 2 MECSS (H) opt V C 7 (G) Let us describe algorithm f . Consider an instance G of V C 7 . G is a graph with maximum degree seven. Here is a procedure to construct an instance H of the 2 MECSS. Similarly to [11], start with a special vertex, the root. Each vertex in G will have a current vertex , initially the root. For each edge uv, add a cover testing gadget to H, as illustrated in Figure 4(a) Speci cally, add six new vertices x 1 ; x 2 ; x 3 ; y 1 ; y 2 ; y 3 . Vertex x 2 is adjacent only to ....

S. Khuller, B. Raghavachari and N. Young, \Approximating the Minimum Equivalent Digraph," SIAM Journal of Computing, 24 (4), 859-872, 1995.

....author wishes to acknowledge financial support from the Danish Research Council (under grant 9800435) Both authors wish to thank Department of Mathematics and Statistics at University of Victoria for its hospitality. interest and has been considered several times in the literature, see e.g. [1, 22, 31, 33, 34, 37]. The MSSS problem is an essential subproblem of the so called minimum equivalent digraph problem. Here one is seeking a spanning subgraph with as few arcs as possible in which the reachability relation is the same as in the original digraph (i.e. there is a path from x to y if and only if the ....

....minimum equivalent digraph problem. Here one is seeking a spanning subgraph with as few arcs as possible in which the reachability relation is the same as in the original digraph (i.e. there is a path from x to y if and only if the original digraph has such a path) Khuller, Raghavachari and Young [33] gave a 1.65 approximation algorithm for the size of a minimum strongly connected subgraph of any strongly connected digraph. This was later improved to about 1.61 using results from [34] Since the MSSS problem is NP hard, it is natural to study the problem under certain extra assumptions. For ....

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S. Khuller, B. Raghavachari and N. Young, Approximating the minimum equivalent digraph, Siam J. Computing 24 (1995) 859-872.

....Denote by V C 7 the vertex cover problem restricted to graphs with maximum degree seven. Papadimitriou and Yannakakis [PY91] showed that V C 7 is MAX SNP hard. We prove that V C 7 L reduces to the minimum k edge connected spanning subgraph problem, here denoted by k MECSS. The reduction comes from [KRY95], where a directed version of the 2 MECSS is proved to be MAX SNP hard. The first part of the L reduction is a polynomialtime algorithm f and a constant ff. Given any instance G of V C 7 , f produces an instance H of the 2MECSS such that the minimum number of edges in a 2 edge connected spanning ....

....of a vertex cover in G, denoted by opt V C7 (G) In other words, opt k GammaM ECSS (2; H) ff Delta opt V C7 (G) Let us describe algorithm f . Consider an instance G of V C 7 . G is a graph with maximum degree seven. Here is a procedure to construct an instance H of the 2 MECSS. Similarly to [KRY95], start with a special vertex, the root. Each vertex in G will have a current vertex , initially the root. For each edge uv, add a cover testing gadget to H, as illustrated in figure 2. Specifically, add six new vertices x 1 ; x 2 ; x 3 ; y 1 ; y 2 ; y 3 . Vertex x 2 is adjacent only to ....

S. Khuller, B. Raghavachari and N. Young, "Approximating the Minimum Equivalent Digraph," SIAM Journal of Computing, 24 (4), 859--872, 1995.

....NH 03755, USA. E mail: neal.young dartmouth.edu. Part of this research was done while at School of ORIE, Cornell University, Ithaca NY 14853 and supported by Eva Tardos NSF PYI grant DDM 9157199. 1 The only known c approximation algorithm for any c 2 works by repeatedly contracting cycles [6]. Each cycle contracted is either a longest cycle in the current graph, or has length at least some constant k. The set of contracted edges yields the set S. As k grows, the performance guarantee of this algorithm rapidly tends to 2 =6 1:64. A natural modification is to solve the problem ....

....guarantee of this algorithm rapidly tends to 2 =6 1:64. A natural modification is to solve the problem optimally as soon as the maximum cycle length in the current graph drops below some threshold. The problem remains NP hard even when the maximum cycle length is five, but we conjectured in [6] that it was solvable in polynomial time if the maximum cycle length is three. We use SCSS 3 to denote the SCSS problem with this restriction. In this paper we confirm the conjecture: Theorem 1.1 The SCCS 3 problem in n vertex digraphs reduces in O(n 2 ) time to Minimum Bipartite Edge Cover. ....

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S. Khuller, B. Raghavachari and N. Young, Approximating the minimum equivalent digraph, SIAM J. Comput. 24 (4), pp. 859--872, (1995).

....minimum SCSS in the special case when the input graph is guaranteed to have no cycles greater than k (a fixed constant) We call this the SCSS k problem. The SCSS 2 problem is trivial. Therefore the problem is interesting only when k 3. It was recently shown by Khuller, Raghavachari and Young [6] that the SCSS 5 problem is NP hard and that the SCSS 17 problem is MAX SNP hard (precluding the possibility of a polynomial time approximation scheme, unless P=NP) The strong dependence of the complexity on the cycle length is in marked contrast to the relation of complexity and cycle length ....

....interesting because it yields a better polynomialtime approximation algorithm for the general SCSS problem and hence for the general MEG problem. Obtaining a performance guarantee 2 for the general MEG problem is trivial any minimal solution achieves this bound. Khuller, Raghavachari and Young [6] gave the first polynomial time approximation algorithm that achieved a factor better than 2. Their algorithm finds a large cycle in G, contracts it, and recurses on the contracted graph. The set of contracted edges forms an SCSS. The cycles are chosen so that any cycle contracted either has ....

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S. Khuller, B. Raghavachari and N. Young, Approximating the minimum equivalent digraph, to appear in SIAM J. Comput..

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S. Khuller, B. Raghavachari, and N. Young, Approximating the minimum equivalent digraph, SIAM J. Comput., 24 (1995), pp. 859--872.

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S. Khuller, B. Raghavachari and N. Young, "Approximating the Minimum Equivalent Digraph," SIAM Journal of Computing, 24 (4), 859--872, 1995.

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S. Khuller, B. Raghavachari and N. Young, "Approximating the minimum equivalent digraph," SIAM J. Computing 24 (1995), 8995), Also in Proc. 5th Annual ACM-SIAM Symposium on Discrete Algorithms 1994,177-18

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S. Khuller, B. Raghavachari and N. Young, Approximating the minimum equivalent digraph, Siam J. Computing 24 (1995) 859-872.

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