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

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.