# Quick separation in chordal and split graphs

Misra, P. and Panolan, Fahad and Rai, A. and et al, . (2020) Quick separation in chordal and split graphs. In: 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, 25-26 August 2020, Prague.

## Abstract

In this paper we study two classical cut problems, namely Multicut and Multiway Cut on chordal graphs and split graphs. In the Multicut problem, the input is a graph G, a collection of ℓ vertex pairs (si, ti), i ∈ [ℓ], and a positive integer k and the goal is to decide if there exists a vertex subset S ⊆ V (G) \ {si, ti: i ∈ [ℓ]} of size at most k such that for every vertex pair (si, ti), si and ti are in two different connected components of G − S. In Unrestricted Multicut, the solution S can possibly pick the vertices in the vertex pairs {(si, ti): i ∈ [ℓ]}. An important special case of the Multicut problem is the Multiway Cut problem, where instead of vertex pairs, we are given a set T of terminal vertices, and the goal is to separate every pair of distinct vertices in T × T. The fixed parameter tractability (FPT) of these problems was a long-standing open problem and has been resolved fairly recently. Multicut and Multiway Cut now admit algorithms with running times 2O(k3)nO(1) and 2knO(1), respectively. However, the kernelization complexity of both these problems is not fully resolved: while Multicut cannot admit a polynomial kernel under reasonable complexity assumptions, it is a well known open problem to construct a polynomial kernel for Multiway Cut. Towards designing faster FPT algorithms and polynomial kernels for the above mentioned problems, we study them on chordal and split graphs. In particular we obtain the following results. 1. Multicut on chordal graphs admits a polynomial kernel with O(k3ℓ7) vertices. Multiway Cut on chordal graphs admits a polynomial kernel with O(k13) vertices. 2. Multicut on chordal graphs can be solved in time min{O(2k · (k3 + ℓ) · (n + m)), 2O(ℓ log k) · (n + m) + ℓ(n + m)}. Hence Multicut on chordal graphs parameterized by the number of terminals is in XP. 3. Multicut on split graphs can be solved in time min{O(1.2738k+kn+ℓ(n+m), O(2ℓ·`·(n+m))}. Unrestricted Multicut on split graphs can be solved in time O(4ℓ · ℓ · (n + m)). © Nathalie Bertrand; licensed under Creative Commons License CC-BY 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020).

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Item Type: Conference or Workshop Item (Paper)
Additional Information: Funding Ashutosh Rai: Supported by Center for Foundations of Modern Computer Science (Charles University projectUNCE/SCI/004). Saket Saurabh: European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416), and Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.
Uncontrolled Keywords: Chordal graphs, FPT, Kernel, Multicut, Multiway cut
Subjects: Computer science
Divisions: Department of Computer Science & Engineering
Depositing User: . LibTrainee 2021
Date Deposited: 18 Nov 2022 15:10