On the Complexity of Connected $(s,t)$-Vertex Separator

TL;DR

This paper investigates the computational complexity of the connected $(s,t)$-vertex separator problem, establishing hardness results, special cases where it is polynomial-time solvable, approximation algorithms, and parameterized complexity insights.

Contribution

It provides new hardness bounds, polynomial algorithms for specific graph classes, approximation strategies, and parameterized complexity results for the connected $(s,t)$-vertex separator problem.

Findings

$(s,t)$-CVS is $ ext{Omega}( ext{log}^{2- ext{epsilon}} n)$-hard to approximate.

NP-complete on graphs with chordality at least 5.

Polynomial-time algorithm for bipartite chordality 4 graphs.

Abstract

We show that minimum connected $(s,t)$-vertex separator ($(s,t)$-CVS) is $\Omega(log^{2-\epsilon}n)$-hard for any $\epsilon >0$ unless NP has quasi-polynomial Las-Vegas algorithms. i.e., for any $\epsilon >0$ and for some $\delta >0$, $(s,t)$-CVS is unlikely to have $\delta.log^{2-\epsilon}n$-approximation algorithm. We show that $(s,t)$-CVS is NP-complete on graphs with chordality at least 5 and present a polynomial-time algorithm for $(s,t)$-CVS on bipartite chordality 4 graphs. We also present a $\lceil\frac{c}{2}\rceil$-approximation algorithm for $(s,t)$-CVS on graphs with chordality $c$. Finally, from the parameterized setting, we show that $(s,t)$-CVS parameterized above the $(s,t)$-vertex connectivity is $W[2]$-hard.