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Fixed-Parameter Tractable Algorithm and Polynomial Kernel for Max-Cut Above Spanning Tree

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Abstract

Every connected graph on n vertices has a cut of size at least n − 1. We call this bound the spanning tree bound. In the Max-Cut Above Spanning Tree (Max-Cut-AST) problem, we are given a connected n-vertex graph G and a non-negative integer k, and the task is to decide whether G has a cut of size at least n − 1 + k. We show that Max-Cut-AST admits an algorithm that runs in time \(\mathcal {O}(8^{k}n^{\mathcal {O}(1)})\), and hence it is fixed parameter tractable with respect to k. Furthermore, we show that Max-Cut-AST has a polynomial kernel of size \(\mathcal {O}(k^{5})\).

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Correspondence to Jayakrishnan Madathil.

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This article is part of the Topical Collection on Computer Science Symposium in Russia (2018)

Appendix: Index for Section 4

Appendix: Index for Section 4

\(\mathcal {B}_{= 2}\), 36

bd(⋅), 20

Cext, 20

\({{C}_{\text {int}}}, {{C}_{\text {int}}^1}, {{C}_{\text {int}}^0}\), 20

J, J0, J1, 20

Δ-block, 21

-block, 21

Bad component, 29

Block-degree, 20

Clique-block, 19

Component-block, 21

Cycle-block, 19

Exterior vertex, 20

Good component, 29

Group-D, Group-E, Group-F, 37

Interior vertex, 20

Leaf block, 20

Superblock, 39

Type-A block, 19

Type-B block, 19

Type-I component, 21

Type-II component, 21

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Madathil, J., Saurabh, S. & Zehavi, M. Fixed-Parameter Tractable Algorithm and Polynomial Kernel for Max-Cut Above Spanning Tree. Theory Comput Syst 64, 62–100 (2020). https://doi.org/10.1007/s00224-018-09909-5

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  • DOI: https://doi.org/10.1007/s00224-018-09909-5

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