Parameterized Algorithms for the 2-Clustering Problem with Minimum Sum and Minimum Sum of Squares Objective Functions

Abstract

In the Min-Sum 2-Clustering problem, we are given a graph and a parameter k, and the goal is to determine if there exists a 2-partition of the vertex set such that the total conflict number is at most k, where the conflict number of a vertex is the number of its non-neighbors in the same cluster and neighbors in the different cluster. The problem is equivalent to 2-Cluster Editing and 2-Correlation Clustering with an additional multiplicative factor two in the cost function. In this paper we show an algorithm for Min-Sum 2-Clustering with time complexity O(n⋅2.619^r/(1−4r/n)+n ³), where n is the number of vertices and r=k/n. Particularly, the time complexity is O ^∗(2.619^k/n) for k∈o(n ²) and polynomial for k∈O(nlogn), which implies that the problem can be solved in subexponential time for k∈o(n ²). We also design a parameterized algorithm for a variant in which the cost is the sum of the squared conflict-numbers. For k∈o(n ³), the algorithm runs in subexponential O(n ³⋅5.171^θ) time, where $\theta=\sqrt{k/n}$.