All-Subsets Important Separators with Applications to Sample Sets, Balanced Separators and Vertex Sparsifiers in Directed Graphs

Given a directed graph $G$ with $n$ vertices and $m$ edges, a parameter $k$ and two disjoint subsets $S,T \subseteq V(G)$, we show that the number of all-subsets important separators, which is the number of $A$-$B$ important vertex separators of size at most $k$ over all $A \subseteq S$ and $B \subseteq T$, is at most $β(|S|, |T|, k) = 4^k {|S| \choose \leq k} {|T| \choose \leq 2k}$, where ${x \choose \leq c} = \sum_{i = 1}^c {x \choose i}$, and that they can be enumerated in time $O(β(|S|,|T|,k)k^2(m+n))$. This is a generalization of the folklore result stating that the number of $A$-$B$ important separators for two fixed sets $A$ and $B$ is at most $4^k$ (first implicitly shown by Chen, Liu and Lu Algorithmica '09). From this result, we obtain the following applications: We give a construction for detection sets and sample sets in directed graphs, generalizing the results of Kleinberg (Internet Mathematics' 03) and Feige and Mahdian (STOC' 06) to directed graphs. Via our new sample sets, we give the first FPT algorithm for finding balanced separators in directed graphs parameterized by $k$, the size of the separator. Our algorithm runs in time $2^{O(k)} (m + n)$. We also give a $O({\sqrt{\log k}})$ approximation algorithm for the same problem. Finally, we present new results on vertex sparsifiers for preserving small cuts.