Randomized algorithms
Information Processing Letters
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
An Approximation Algorithm for Feedback Vertex Sets in Tournaments
SIAM Journal on Computing
An Approximation Algorithm for Feedback Vertex Sets in Tournaments
SIAM Journal on Computing
The Approximation of Maximum Subgraph Problems
ICALP '93 Proceedings of the 20th International Colloquium on Automata, Languages and Programming
On Feedback Problems in Diagraphs
WG '89 Proceedings of the 15th International Workshop on Graph-Theoretic Concepts in Computer Science
Parameterized algorithms for feedback set problems and their duals in tournaments
Theoretical Computer Science - Parameterized and exact computation
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Given a simple directed graph D=(V,A), let the size of the largest induced acyclic tournament be denoted by mat(D). Let $D \in \mathcal{D}(n,p)$ be a random instance, obtained by choosing each of the ${{n}\choose{2}}$ possible undirected edges independently with probability 2p and then orienting each chosen edge in one of two possible directions with probability 1/2. We show that for such a random instance, mat(D) is asymptotically almost surely one of only 2 possible values, namely either b* or b*+1, where $b^* = \lfloor 2(\log_{p^{-1}} n)+0.5 \rfloor$. It is then shown that almost surely any maximal induced acyclic tournament is of a size which is at least nearly half of any optimal solution. We also analyze a polynomial time heuristic and show that almost surely it produces a solution whose size is at least $\log_{p^{-1}} n + \Theta(\sqrt{\log_{p^{-1}} n})$. Our results also carry over to a related model in which each possible directed arc is chosen independently with probability p. An immediate corollary is that (the size of a) minimum feedback vertex set can be approximated within a ratio of 1+o(1) for random tournaments.