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 \(p\le |E|/2\), 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 \({\mathcal O}^*(2^p)\)-time algorithm, improving upon the previous best \({\mathcal O}^*(4^{p+o(p)})\)-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.
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Appendix
Appendix
Treewidth: A tree decomposition of a graph G is a pair \((D,\beta )\), where D is a rooted tree and \(\beta : V(D)\rightarrow 2^{V(G)}\) is a mapping that satisfies the following conditions.
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For each vertex \(v\in V(G)\), the set \(\{d\in V(D): v\in \beta (d)\}\) induces a nonempty and connected subtree of D.
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For each edge \(\{v,u\}\in E(G)\), there exists \(d\in V(D)\) such that \(\{v,u\}\subseteq \beta (d)\).
The set \(\beta (d)\) is called the bag at d, and the width of \((D,\beta )\) is the size of the largest bag minus one (i.e., \(\max _{d\in V(D)}|\beta (d)|-1\)). The treewidth of G, \(tw_G\), is the minimum width among all possible tree decompositions of G.
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Saurabh, S., Zehavi, M. (2016). \((k,n-k)\)-Max-Cut: An \({\mathcal O}^*(2^p)\)-Time Algorithm and a Polynomial Kernel. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_51
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DOI: https://doi.org/10.1007/978-3-662-49529-2_51
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