Chandran, Sunil L (1999) A high girth graph construction and a lower bound for hitting set size for combinatorial rectangles. In: Proceedings 19th FST & TCS 1999: Nineteenth Conference on the Foundations of Software Technology and Theoretical Computer Science, 13-15 Dec. 1999, Chennai, India, pp. 283-290.
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We give the following two results. First, we give a deterministic algorithm which constructs a graph of girth $log_k(n)+O(1)$ and minimum degree k-1, taking number of nodes n and the number of edges e=[nk/2] as input. The graphs constructed by our algorithm are expanders of sub-linear sized subsets, that is subsets of size at most $n^\delta$, where $\delta$<¼. Although methods which construct high girth graphs are known, the proof of our construction uses only very simple counting arguments in comparison. Also our algorithm works for all values of n or k. We also give a lower bound of m/8ϵ for the size of hitting sets for combinatorial rectangles of volume ϵ. This result is an improvement of the previously known lower bound, namely Ω(m+1/ϵ+log(d)). The known upper bound for the size of the hitting set is m poly(log(d)/ϵ) (N. Linial et al., 1997)
| Item Type: | Conference Paper |
|---|---|
| Publisher: | Springer-Verlag |
| Additional Information: | Copyright of this article belongs to Springer-Verlag |
| Keywords: | computational complexity:graph theory;set theory;theorem proving |
| Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |
| Date Deposited: | 25 Jul 2007 |
| Last Modified: | 27 Aug 2008 12:42 |
| URI: | http://eprints.iisc.ac.in/id/eprint/10273 |
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