Maximal and Maximum Transitive Relation Contained in a Given Binary Relation
We study the problem of finding a maximal transitive relation contained in a given binary relation. Given a binary relation of size m defined on a set of size n, we present a polynomial time algorithm that finds a maximal transitive sub-relation in time \(O(n^2 + nm)\).
We also study the problem of finding a maximum transitive relation contained in a binary relation. For the class of triangle-free relations (directed graphs), we present a 0.874-approximation via the problem of maximum directed cut.
KeywordsDirected Graph Binary Relation Transitive Closure Transitive Relation Transitive Property
Unable to display preview. Download preview PDF.
- 2.Alvarado, F.L., Pothen, A., Schreiber, R.: Highly parallel sparse triangular solution. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds.) Graph Theory and Sparse Matrix Computation. The IMA Volumes in Mathematics and its Applications, vol. 56, pp. 141–157. Springer, New York (1993)CrossRefGoogle Scholar
- 3.Chakraborty, S., Jha, N.: On the size of maximum directed cuts in triangle free graphs (2015) (unpublished)Google Scholar
- 8.Nuutila, E.: Efficient transitive closure computation in large digraphs. Acta Polytechnica Scandinavia: Math. Comput. Eng. 74, 1–124 (1995)Google Scholar
- 10.Williams, V.V.: Multiplying matrices faster than coppersmith-winograd. In: Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19–22, 2012, pp. 887–898 (2012)Google Scholar
- 11.Yannakakis, M.: Node- and edge-deletion np-complete problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1–3, 1978, San Diego, California, USA, pp. 253–264 (1978)Google Scholar