Minimum equivalent precedence relation systems
Kin Cheong Sou
- 01 Dec 2015
- pp 3299-3304
2
TL;DR: The first problem is shown to be solvable in polynomial-time, with a full parameterization of all solutions described, and a decomposition of the first problem into independent tractable and intractable subproblems is derived.
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Abstract: In this paper two related simplification problems for systems of linear inequalities describing precedence relation systems are considered. Given a precedence relation system, the first problem seeks a minimum equivalent subset of the precedence relations (i.e., inequalities) which has the same solution set as that of the original system. The second problem is similar to the first one, but the minimum equivalent system need not be a subset of the original system. This paper shows that the first problem is NP-hard. However, a sufficient condition is derived under which the first problem is solvable in polynomial-time. In addition, a decomposition of the first problem into independent tractable and intractable subproblems is derived. The second problem is shown to be solvable in polynomial-time, with a full parameterization of all solutions described. The results in this paper generalize those in [Moyles and Thompson 1969, Aho, Garey, and Ullman 1972] for the minimum equivalent graph problem and transitive reduction problem, which are applicable to unweighted directed graphs.
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Citations
Minimum equivalent precedence relation systems
Kin Cheong Sou
- 01 Dec 2015
TL;DR: The first problem is shown to be solvable in polynomial-time, with a full parameterization of all solutions described, and a decomposition of the first problem into independent tractable and intractable subproblems is derived.
2
•Posted Content
Minimum Equivalent Precedence Relation Systems
TL;DR: In this article, two related simplification problems for systems of linear inequalities describing precedence relation systems are considered, and it is shown that the first problem is NP-hard and the second problem is solvable in polynomial time.
2
References
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Michael Randolph Garey,David S. Johnson +1 more
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The Transitive Reduction of a Directed Graph
TL;DR: It is shown that the time complexity of the best algorithm for finding the transitive reduction of a graph is the same as the time to compute the transitives closure of agraph or to perform Boolean matrix multiplication.
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An Algorithm for Finding a Minimum Equivalent Graph of a Digraph
TL;DR: The present paper gives an algorithm for finding the maximum number of edges that can be removed from a digraph without affecting its reachability properties.
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Approximating the Minimum Equivalent Digraph
TL;DR: In this article, the authors give an approximation algorithm for the minimum SCSS problem with a performance guarantee of 1.64 in polynomial time, which is the best known algorithm for this problem.