S. Khuller, B. Raghavachari, and N. Young, On strongly connected digraphs with bounded cycle length, Discrete Appl. Math., 69 (1996), pp. 281--289.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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, On strongly connected digraphs with bounded cycle length, Disc. Applied Math. 69 (1996) 281-289.

....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 ....

....of digraphs for which we can solve the MSSS problem in polynomial time, we must consider classes of digraphs for which we can solve the hamiltonian cycle problem in polynomial time. This follows from the fact that the hamiltonian cycle problem can be solved if we can solve the MSSS problem. In [17] the MSSS problem was considered for digraphs whose longest cycle has length r for some r. It was shown that if r # 3, then the problem is polynomial and that it is NP hard already when r = 5. In this paper we study the MSSS problem for quasi transitive digraphs. These digraphs have a nice, ....

S. Khuller, B. Raghavachari and N. Young, On strongly connected digraphs with bounded cycle length, Disc. Applied Math. 69 (1996) 281-289.

....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, On strongly connected digraphs with bounded cycle length, Disc. Applied Math. 69 (1996) 281-289.

....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 ....

....of digraphs for which we can solve the MSSS problem in polynomial time, we must consider classes of digraphs for which we can solve the hamiltonian cycle problem in polynomial time. This follows from the fact that the hamiltonian cycle problem can be solved if we can solve the MSSS problem. In [17] the MSSS problem was considered for digraphs whose longest cycle has length r for some r. It was shown that if r 3, then the problem is polynomial and that it is NP hard already when r = 5. In this paper we study the MSSS problem for quasi transitive digraphs. These digraphs have a nice, ....

S. Khuller, B. Raghavachari and N. Young, On strongly connected digraphs with bounded cycle length, Disc. Applied Math. 69 (1996) 281-289.

....a solution whose value is always within the factor ff of the optimum value. The quantity ff is called the approximation guarantee of the algorithm. Previous results Results in this paper Undirected Graphs Digraphs Undirected Graphs Digraphs k ECSS 2 Gamma [1=k] for k 2 [K 96] 1. 61 for k = 1 [KRY 96] 1 [2= k 1) 1 [4= p k] 1.85 for k 2 [KR 96] 2 for k 2 improves for k 3 improves for k 17 1 p O(log n) k [Ka 94] k NCSS 1.5 for k = 2 [GSS 93] 1.61 for k = 1 [KRY 96] 1 [1=k] 1 [1=k] 2 for k 3 2 for k 2 improves for k 3 improves for k 2 Table 1: A summary of previous new ....

....paper Undirected Graphs Digraphs Undirected Graphs Digraphs k ECSS 2 Gamma [1=k] for k 2 [K 96] 1.61 for k = 1 [KRY 96] 1 [2= k 1) 1 [4= p k] 1.85 for k 2 [KR 96] 2 for k 2 improves for k 3 improves for k 17 1 p O(log n) k [Ka 94] k NCSS 1.5 for k = 2 [GSS 93] 1. 61 for k = 1 [KRY 96] 1 [1=k] 1 [1=k] 2 for k 3 2 for k 2 improves for k 3 improves for k 2 Table 1: A summary of previous new approximation guarantees for minimum size k edge connected spanning subgraphs (k ECSS) and minimum size k node connected spanning subgraphs (k NCSS) 1.1 Previous work Results of ....

[Article contains additional citation context not shown here]

S. Khuller, B. Raghavachari and N. Young, "On strongly connected digraphs with bounded cycle length," Discrete Applied Mathematics 69 (1996), 281--289.

....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 ....

....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 example to consider special classes of digraphs. One restriction is to study digraphs with longest cycle at most r. Then the problem is known under the name SCCS r [33] In [34] it is shown ....

[Article contains additional citation context not shown here]

S. Khuller, B. Raghavachari and N. Young, On strongly connected digraphs with bounded cycle length, Disc. Applied Math. 69 (1996) 281-289.

....performance guarantee to 2 =6 Gamma 1=36 1:617. We use SCSS c to denote the minimum SCSS problem restricted to digraphs with no cycle longer than c. The minimumSCSS 2 problem is trivial. The minimum SCSS 3 problem can be solved in polynomial time, as shown by Khuller, Raghavachari and Young [14]. However, further improvement in this direction is limited: we show that the minimum SCSS 5 problem is NP hard. In fact, we show that the minimum SCSS 17 problem is MAX SNP hard. This precludes the possibility of a polynomial time approximation scheme, assuming P6=NP [4] 1.2. Other Related ....

....k to c k Gamma 1=36 (for k 4) matching the lower bound in Table 1. The lower bound given holds for the modified algorithm. This leads us to consider the minimum SCSS c problem the minimum SCSS problem restricted to graphs with cycle length bounded by c. The following theorem is shown in [14]. Theorem 7.1. There is a polynomial time algorithm for the SCSS 3 problem. We make no conjecture concerning the SCCS 4 problem. However, we next show that the SCCS 5 problem is NP hard, and that the SCSS 17 problem is MAX SNP hard. 7.1. NP hardness of SCSS 5 . We prove the following theorem. ....

S. Khuller, B. Raghavachari and N. Young, On strongly connected digraphs with bounded cycle length, UMIACS-TR-94-10/CS-TR-3212, January (1994).

.... case was explored by Khuller and Vishkin [28] and Garg, Santosh and Singla [16] For any k, fast algorithms for finding sparse certificates were given by Nagamochi and Ibaraki [33] and Cheriyan, Kao and Thurimella [5] The strong connectivity case is addressed by Khuller, Raghavachari and Young [24, 25]. When parallel edges are allowed, Goemans and Bertsimas provide an approximation algorithm [17] 1.1 Outline of Chapter The problems we deal with are divided broadly into four categories: edge connectivity, vertex connectivity, strong connectivity and connectivity augmentation. In each case, ....

....is obtained by taking the union of a minimum weight in branching and a minimum weight out branching, rooted at an arbitrary vertex. For the unweighted case, Khuller, Raghavachari and Young [24] obtained an approximation algorithm with a performance ratio of about 1:64, which was improved to 1:61 [25]. The algorithms have a relatively high running time, albeit polynomial. An almost linear time algorithm that achieves a ratio of 1:75 is also described. The MEG (minimum equivalent graph) problem is the following: Given a directed graph, find a smallest subset of the edges that maintains all ....

[Article contains additional citation context not shown here]

S. Khuller, B. Raghavachari and N. Young, On strongly connected digraphs with bounded cycle length, UMIACS-TR-94-10/CS-TR-3212, (1994).

....performance guarantee to 2 =6 0 1=36 1:617. We use SCSS c to denote the minimum SCSS problem restricted to digraphs with no cycle longer than c. The minimum SCSS 2 problem is trivial. The minimum SCSS 3 problem can be solved in polynomial time, as shown by Khuller, Raghavachari and Young [14]. However, further improvement in this direction is limited: we show that the minimum SCSS 5 problem is NP hard. In fact, we show that the minimum SCSS 17 problem is MAX SNP hard. This precludes the possibility of a polynomial time approximation scheme, assuming P6=NP [4] 1.2. Other Related ....

....k to c k 0 1=36 (for k 4) matching the lower bound in Table 1. The lower bound given holds for the modified algorithm. This leads us to consider the minimum SCSS c problem the minimum SCSS problem restricted to graphs with cycle length bounded by c. The following theorem is shown in [14]. Theorem 7.1. There is a polynomial time algorithm for the SCSS 3 problem. We make no conjecture concerning the SCCS 4 problem. However, we next show that the SCCS 5 problem is NP hard, and that the SCSS 17 problem is MAX SNP hard. 7.1. NP hardness of SCSS 5 . We prove the following ....

S. Khuller, B. Raghavachari and N. Young, On strongly connected digraphs with bounded cycle length, UMIACS-TR-94-10/CS-TR-3212, (1994).

No context found.

S. Khuller, B. Raghavachari, and N. Young, On strongly connected digraphs with bounded cycle length, Discrete Appl. Math., 69 (1996), pp. 281--289.

No context found.

S. Khuller, B. Raghavachari and N. Young, "On strongly connected digraphs with bounded cycle length," Discrete Applied Mathematics69 (1996), 2896),

No context found.

S. Khuller, B. Raghavachari and N. Young, On strongly connected digraphs with bounded cycle length, Disc. Applied Math. 69 (1996) 281-289. 14

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC