## Abstract

Max-Cut is a well-known classical NP-hard problem. This problem asks whether the vertex-set of a given graph G= (V, E) can be partitioned into two disjoint subsets, A and B, such that there exist at least p edges with one endpoint in A and the other endpoint in B. It is well known that if p≤ | E| / 2 , the answer is necessarily positive. A widely-studied variant of particular interest to parameterized complexity, called (k, n- k) -Max-Cut, restricts the size of the subset A to be exactly k. For the (k, n- k) -Max-Cut problem, we obtain an O^{∗}(2 ^{p}) -time algorithm, improving upon the previous best O^{∗}(4 ^{p} ^{+} ^{o} ^{(} ^{p} ^{)}) -time algorithm, as well as the first polynomial kernel. Our algorithm relies on a delicate combination of methods and notions, including independent sets, depth-search trees, bounded search trees, dynamic programming and treewidth, while our kernel relies on examination of the closed neighborhood of the neighborhood of a certain independent set of the graph G.

Original language | English |
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Pages (from-to) | 3844-3860 |

Number of pages | 17 |

Journal | Algorithmica |

Volume | 80 |

Issue number | 12 |

DOIs | |

State | Published - 1 Dec 2018 |

Externally published | Yes |

## Keywords

- Bounded search tree
- Kernel
- Max-Cut
- Parameterized algorithm