C.C. Wang, E.L. Lloyd, and M.L. So#a. Feedback vertex sets and cyclically reducible graphs. Journal of the Association for Computing Machinery, 32(2):296-- 313, 1985.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....was at INRIA for a summer internship of the Indian Institute of Technology Delhi, New Delhi 110016, India. the graph. Shamir [11] proposed a linear time algorithm for reducible ow graphs. Rosen [2] modi ed this algorithm in an approximate algorithm for general graphs. Wang, Lloyd and So a [14] found an O(jEj jV j ) algorithm for an unrelated class of cyclically reducible graphs. Smith and Walford [13] proposed an exponential time algorithm for general graphs that behaves in O(jEj jV j ) in certain classes of graphs. The comparison of these di erent reducibility properties was ....

C.C. Wang, E.L. Lloyd, and M.L. Soa. Feedback vertex sets and cyclically reducible graphs. Journal of the Association for Computing Machinery, 32(2):296-313, 1985.

....also Miyano [263, 266] A.2.17 Minimum Feedback Vertex Set (MFVS) Given: A directed graph G = V, E) that is cyclically reducible (defined below) and a designated vertex v. Problem: Is v contained in the minimum feedback set of G that is computed by the algorithm given by Wang, Lloyd, and So#a [371] Reference: Bovet, De Agostino, and Petreschi [44] Hint: We review some terminology from [371] A vertex z of G is deadlocked if there is a directed path in G from z to a vertex y that lies on a directed cycle. The associated graph of vertex x with respect to G, A(G, x) consists of vertex x ....

....E) that is cyclically reducible (defined below) and a designated vertex v. Problem: Is v contained in the minimum feedback set of G that is computed by the algorithm given by Wang, Lloyd, and So#a [371] Reference: Bovet, De Agostino, and Petreschi [44] Hint: We review some terminology from [371]. A vertex z of G is deadlocked if there is a directed path in G from z to a vertex y that lies on a directed cycle. The associated graph of vertex x with respect to G, A(G, x) consists of vertex x and all vertices of G that are not deadlocked if x is removed from G. A directed graph is ....

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C.-C. Wang, E. L. Lloyd, and M. L. So#a. Feedback vertex sets and cyclically reducible graphs. Journal of the ACM, 32(2):296--313, April 1985. (139, 140)

....membership problem is NP complete [Gre89] A.2.14 Minimum Feedback Vertex Set (MFVS) Given: A directed graph G = V; E) that is cyclically reducible (defined below) and a designated vertex v. Problem: Is v contained in the minimum feedback set of G that is computed by the algorithm given in [WLS85] Reference: BDAP88] Hint: We review some terminology [WLS85] A node z of G is deadlocked if there is a directed path in G from z to a node y that lies on a directed cycle. The associated graph of node x with respect to G, A(G,x) consists of node x and all nodes of G that are not deadlocked ....

....Vertex Set (MFVS) Given: A directed graph G = V; E) that is cyclically reducible (defined below) and a designated vertex v. Problem: Is v contained in the minimum feedback set of G that is computed by the algorithm given in [WLS85] Reference: BDAP88] Hint: We review some terminology [WLS85]. A node z of G is deadlocked if there is a directed path in G from z to a node y that lies on a directed cycle. The associated graph of node x with respect to G, A(G,x) consists of node x and all nodes of G that are not deadlocked if x is removed from G. A directed graph is cyclically reducible ....

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C. C. Wang, E. L. Lloyd, and M. L. Soffa. Feedback vertex sets and cyclically reducible graphs. Journal of the ACM, 32(2):296--313, 1985. 112 ffl A Compendium of Problems Complete for P (Preliminary: RCS Revision: 1.46)

....hypercubes. Given a graph, the minimum feedback vertex set problem consists of nding a subset of vertices of minimum size whose removal induces an acyclic subgraph. The problem is NP hard for general networks [2] but interesting polynomial solutions have been found for particular graphs [5, 6, 7, 8, 9, 12, 14]. keywords: feedback vertex set, bounds, hypercube, combinatorial problems, algorithms. 1 Introduction The feedback vertex set of a graph G = V; E) is a subset of vertices V V whose removal from G, induces an acyclic subgraph G 0 = V 0 ; E 0 ) where V 0 = V n V and E 0 = f(x; y) 2 ....

....the set is called a minimum feedback vertex set for G. The problem is NP hard for general networks [2] although there exist polynomial time algorithms for particular topologies, e.g. reducible ow graphs [12] cocomparability graphs and convex bipartite graphs [6] cyclically reducible graphs [14], and other [5, 7, 8, 9, 13] The problem has been widely studied since it has interesting applications. As an example, consider the typical operating system problem of allocating resources to processors while preventing deadlocks. It can be solved by considering the graph where vertices represent ....

[Article contains additional citation context not shown here]

C. Wang and E.L. Lloyd and M.L. Soa. Feedback Vertex Sets and Cyclically Reducible Graphs. JACM 32, (2), 296-313, 1985. 6

....hypercubes. Given a graph, the minimum feedback vertex set problem consists of finding a subset of vertices of minimum size whose removal induces an acyclic subgraph. The problem is NP hard for general networks [2] but interesting polynomial solutions have been found for particular graphs [5, 6, 7, 8, 9, 12, 14]. keywords: feedback vertex set, bounds, hypercube, combinatorial problems, algorithms. 1 Introduction The feedback vertex set of a graph G = V; E) is a subset of vertices V V whose removal from G, induces an acyclic subgraph G 0 = V 0 ; E 0 ) where V 0 = V n V and E 0 = f(x; y) 2 ....

....the set is called a minimum feedback vertex set for G. The problem is NP hard for general networks [2] although there exist polynomial time algorithms for particular topologies, e.g. reducible flow graphs [12] cocomparability graphs and convex bipartite graphs [6] cyclically reducible graphs [14], and other [5, 7, 8, 9, 13] The problem has been widely studied since it has interesting applications. As an example, consider the typical operating system problem of allocating resources to processors while preventing deadlocks. It can be solved by considering the graph where vertices represent ....

[Article contains additional citation context not shown here]

C. Wang and E.L. Lloyd and M.L. Soffa. Feedback Vertex Sets and Cyclically Reducible Graphs. JACM 32, (2), 296-313, 1985. 6

....is to identify those specially structured problems which can be solved in polynomial time. Research along this line started with the pioneering work of Shamir [76] in which a linear time algorithm was proposed to find a feedback vertex set for a reducible flow graph. Wang, Lloyd, and Soffa [87] developed an O E G ) V G 2 algorithm for finding a feedback vertex set. The class of graphs known as cyclically reducible graphs, which is shown to be unrelated to the class of quasi reducible graphs. Although the exact algorithm proposed by Smith and Walford [80] has ....

....a n with a 1 C a 2 CQ RC a n . The algorithm is based on dynamic programming techniques and the special structure of the graph. 3.3. Approximation algorithms and provable bounds. The feedback vertex (arc) set problem has found applications in many fields, including deadlock prevention [87], program verification [76] and Bayesian inference [2] Therefore, it is natural that in the past few years there have been intensive efforts on approximation algorithms for these kinds of problems, for the cases that are not known to be polynomially solvable. To quantify the quality of an ....

C. Wang, E. Lloyd, and M. Soffa, Feedback vertex sets and cyclically reducible graphs, Journal of the Association for Computing Machinery Vol.32 No.2 (1985) pp. 296-313.

....proved NP complete [12] There are weighted versions of both of these problems as well. In the weighted feedback arc (vertex) set problem arcs (vertices) are assigned non negative real weights. These types of problems have applications in the study of feedback systems and in deadlock detection [20]. Problems modeled in this domain can be decomposed into an acyclic component and a set of edges. A solution to the acyclic part can usually be generated efficiently. Edges forming cycles are reintroduced and a new solution incorporating all links is developed through iteration. Ramachandran ....

..... Thus, the decision problem Is e contained in the subgraph computed by Algorithm B is also P complete. The reduction also holds for graphs that are restricted to having indegree and outdegree of at most three. Additionally, since the instances we construct are cyclically reducible flow graphs [20], Problem B remains P complete for the restricted class of cyclically reducible flow graphs of indegree at most 3. To see the graphs are cyclically reducible, we construct a complete D sequence. Such a sequence can be built if we proceed in reverse topological order through the graph. Each time ....

C.C. Wang, E.L. Llyod, and M.L. Soffa. Feedback vertex sets and cyclically reducible graphs. Journal of the ACM, 32(2):296--313, April 1985.

....membership problem is NP complete [Gre89] A.2.14 Minimum Feedback Vertex Set (MFVS) Given: A directed graph G = V; E) that is cyclically reducible (defined below) and a designated vertex v. Problem: Is v contained in the minimum feedback set of G that is computed by the algorithm given in [WLS85] Reference: BDAP88] Hint: We review some terminology [WLS85] A node z of G is deadlocked if there is a directed path in G from z to a node y that lies on a directed cycle. The associated graph of node x with respect to G, A(G,x) consists of node x and all nodes of G that are not deadlocked ....

....Feedback Vertex Set (MFVS) Given: A directed graph G = V; E) that is cyclically reducible (defined below) and a designated vertex v. Problem: Is v contained in the minimum feedback set of G that is computed by the algorithm given in [WLS85] Reference: BDAP88] Hint: We review some terminology [WLS85]. A node z of G is deadlocked if there is a directed path in G from z to a node y that lies on a directed cycle. The associated graph of node x with respect to G, A(G,x) consists of node x and all nodes of G that are not deadlocked if x is removed from G. A directed graph is cyclically reducible ....

[Article contains additional citation context not shown here]

C. C. Wang, E. L. Lloyd, and M. L. Soffa. Feedback vertex sets and cyclically reducible graphs. Journal of the ACM, 32(2):296--313, 1985. 112 ffl A Compendium of Problems Complete for P (Preliminary: RCS Revision: 1.46)

....from the original graph such that the new graph has no directed cycle Theoretically, FVS is a well known combinatorial problem among other classical problems like satisfiability, vertex cover, etc. Practically, FVS enjoys the direct applications in partial scan design [3] deadlock prevention [17], program verification [15] constraint satisfaction problems [20] Bayesian inference [20] The exact version of FVS is NP complete [5] and the approximation literature can be divided into two groups. On the undirected graph side, approximating FVS can be approximated within some constant ratio. ....

Wang, C. Lloyd, E. and Soffa, M. Feedback Vertex Sets And Cyclically Reducible Graphs, Journal Of The Association For Computing Machinery, Vol. 32, No. 2, pp. 296-313, (1985).

....whose removal from the graph eliminates all directed cycles of the graph. The minimum cardinality cutset is called the feedback vertex set. The FVS problem is known to be NP complete [8, 19] and has found applications in many diverse areas, including program verification [17] deadlock prevention [18], and Bayesian inference [20] The FVS problem can be formulated as a set covering problem. Let CG denote the set of all cycles in the graph G and define x j = ae 1 if vertex j is in the feedback vertex set 0 otherwise. The minimum number of vertices that need to be removed so that the ....

....tqian dollar.biz.uiowa.edu x Information Sciences Research Center, AT T Research, Murray Hill, NJ 07974 USA. e mail: mgcr research.att.com 2 x j 2 f0; 1g; j = 1; n: Among the classical NP complete problems, the FVS problem has been regarded as one of the least understood problems [18]. Solvable special cases are studied in [17, 18] From the point of view of approximation algorithms, Erdos and P osa [3] proposed an algorithm with a ratio of approximation of 2 log n, which was later improved by Monien and Schultz [11] to a ratio of p log n. Recently, Bafna et al. 1] proposed ....

[Article contains additional citation context not shown here]

C. Wang, E. Lloyd, and M. Soffa, Feedback vertex sets and cyclically reducible graphs, Journal of the ACM, 32 (1985), pp. 296--313.

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C.C. Wang, E.L. Lloyd, and M.L. So#a. Feedback vertex sets and cyclically reducible graphs. Journal of the Association for Computing Machinery, 32(2):296-- 313, 1985.

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