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# On the Parameterized Complexity of Multiway Near-Separator

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LIPIcs.IPEC.2023.28.pdf
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## Cite As

Bart M. P. Jansen and Shivesh K. Roy. On the Parameterized Complexity of Multiway Near-Separator. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.28

## Abstract

We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph G, integer k, and terminal set T ⊆ V(G), it asks whether there is a vertex set S ⊆ V(G) ⧵ T of size at most k such that in graph G-S, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in G-S by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of G-S. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time 2^{𝒪(k log k)} ⋅ n^{𝒪(1)}. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size k plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph G and terminal set T ⊆ V(G) along with a single vertex x ∈ V(G) that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing x.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
• Theory of computation → Parameterized complexity and exact algorithms
• Theory of computation → Approximation algorithms analysis
##### Keywords
• fixed-parameter tractability
• multiway cut
• near-separator

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