(k,n-k)-Max-Cut: An O∗(2p)-Time Algorithm and a Polynomial Kernel

• Published in: Algorithmica Vol. 80; no. 12; pp. 3844 - 3860
• Main Authors: Saurabh, Saket, Zehavi, Meirav
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2018; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Dynamic programming; Mathematics of Computing; Polynomials; Set theory; Theory of Computation; Trees; Bounded search tree; Parameterized algorithm; Kernel; Max-Cut
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-018-0418-5

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 .