Listing Small Minimal $s,t$-separators in FPT-Delay

TL;DR

This paper introduces fixed-parameter tractable algorithms for listing minimal $s,t$-separators of bounded size in graphs, with applications in database query evaluation and probabilistic inference.

Contribution

It presents the first FPT-delay algorithm for enumerating minimal $s,t$-separators of size at most $k$, and a simple ranking algorithm for all $s,t$-separators.

Findings

Efficient enumeration of minimal $s,t$-separators with FPT-delay.

Ranking of all $s,t$-separators by size.

Applicability to algorithms parameterized by treewidth.

Abstract

Let $G$ be an undirected graph, and $s,t$ distinguished vertices of $G$. A minimal $s,t$-separator is an inclusion-wise minimal vertex-set whose removal places $s$ and $t$ in distinct connected components. We present an algorithm for listing the minimal $s,t$-separators of a graph, whose cardinality is at most $k$, with FPT-delay, where the parameter depends only on $k$. This problem finds applications in various algorithms parameterized by treewidth, which include query evaluation in relational databases, probabilistic inference, and many more. We also present a simple algorithm that enumerates all of the (not necessarily minimal) $s,t$-separators of a graph in ranked order by size.