## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, November 08, 2016, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Daniel Graf

## 2-Reachability is Almost as Easy as Transitive Closure

In the 2-reachability problem we are given a directed graph G and we wish to determine if there are two (edge or vertex) disjoint paths from u to v, for every pair of vertices u and v. In this talk, we present an algorithm that computes 2-reachability information for all pairs of vertices in $O(n^{\omega}\log n)$ time, where n is the number of vertices and $\omega$ the matrix multiplication exponent. Hence, rather surprisingly, we show that the running time of 2-reachability is within a log-factor of transitive closure.

Moreover, our algorithm produces a witness (separating edge or separating vertex) whenever 2-reachability does not hold. By processing these witnesses, we can compute all the edge- and vertex-dominator trees of G in $O(n^2)$ additional time, which in turn enables us to answer various connectivity queries in O(1) time. For instance, we can test in constant time if there is a path from u to v avoiding an edge e, for any pair of query vertices u and v, and any query edge e, or if there is a path from u to v avoiding a vertex w, for any query vertices u, v and w.

This is joint work with Loukas Georgiadis, Giuseppe F. Italiano, Nikos Parotsidis and Przemysław Uznański.

Information for students and suggested topics for student talks

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