The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Steiner Tree Gilbert and Pollak [GP68] first conjectured that the ratio of the length of the minimum Steiner tree over the minimum spanning tree is always . After twenty years of active research, the Gilbert- Pollak ratio was finally proven by Du and Hwang [DH92]. The Euclidean minimum spanning tree for n points in the plane can be constructed in time [PS85]. Expositions on the proof that the Steiner tree problem for graphs is hard [Kar72] include [Eve79a]. Expositions on exact algorithms for Steiner trees in graphs include [Law76]. The hardness of Steiner tree for Euclidean and rectilinear metrics was established in [GGJ77, GJ77]. Euclidean Steiner tree is not known to be in NP, because of numerical issues in representing distances. Analogies can be drawn between minimum Steiner trees and minimum energy configurations in certain physical systems. The case that such analog systems, including the behavior of soap films over wire frames, ``solve'' the Steiner tree problem is discussed in [Mie58]. Related Problems: Minimum spanning tree (see page ), shortest path (see page ). Next: Feedback Edge/Vertex Set Up: Graph Problems: Hard Problems Previous: Graph Isomorphism Algorithms Mon Jun 2 23:33:50 EDT 1997 file:///E|/BOOK/BOOK4/NODE181.HTM (4 of 4) [19/1/2003 1:31:31]
Feedback Edge/Vertex Set Next: Computational Geometry Up: Graph Problems: Hard Problems Previous: Steiner Tree Feedback Edge/Vertex Set Input description: A (directed) graph G=(V,E). Problem description: What is the smallest set of edges E' or vertices V' whose deletion leaves an acyclic graph? Discussion: Feedback set problems arise because many algorithmic problems are much easier or much better defined on directed acyclic graphs than on arbitrary digraphs. Topological sorting (see Section ) can be used to test whether a graph is a DAG, and if so, to order the vertices so as to respect the edges as precedence scheduling constraints. But how can you design a schedule if there are cyclic constraints, such as A must be done before B, which must be done before C, which must be done before A? By identifying a feedback set, we identify the smallest number of constraints that must be dropped so as to permit a valid schedule. In the feedback edge (or arc) set problem, we drop precedence constraints (job A must come before job B). In the feedback vertex set problem, we drop entire jobs and any associated constraints. It is also referred to in the literature as the maximum acyclic subgraph problem. ● Do any constraints have to be dropped? - Not if the graph is a DAG, which can be tested via file:///E|/BOOK/BOOK4/NODE182.HTM (1 of 3) [19/1/2003 1:31:32]
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Feedback Edge/Vertex Set<br />
Next: Computational Geometry Up: Graph Problems: Hard Problems Previous: Steiner Tree<br />
Feedback Edge/Vertex Set<br />
Input description: A (directed) graph G=(V,E).<br />
Problem description: What is the smallest set of edges E' or vertices V' whose deletion leaves an acyclic<br />
graph?<br />
Discussion: Feedback set problems arise because many algorithmic problems are much easier or much<br />
better defined on directed acyclic graphs than on arbitrary digraphs. Topological sorting (see Section<br />
) can be used to test whether a graph is a DAG, and if so, to order the vertices so as to respect the<br />
edges as precedence scheduling constraints. But how can you design a schedule if there are cyclic<br />
constraints, such as A must be done before B, which must be done before C, which must be done before<br />
A?<br />
By identifying a feedback set, we identify the smallest number of constraints that must be dropped so as<br />
to permit a valid schedule. In the feedback edge (or arc) set problem, we drop precedence constraints<br />
(job A must come before job B). In the feedback vertex set problem, we drop entire jobs and any<br />
associated constraints. It is also referred to in the literature as the maximum acyclic subgraph problem.<br />
● Do any constraints have to be dropped? - Not if the graph is a DAG, which can be tested via<br />
file:///E|/BOOK/BOOK4/NODE182.HTM (1 of 3) [19/1/2003 1:31:32]
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