The Correlation Clustering Problem (CorCP) is a significant clustering problem based on the similarity of data. It has significant applications in different fields, such as machine learning, biology, and data mining, and many different problems in other areas. In this paper, the Balanced $\mathrm{2}$-CorCP (B $\mathrm{2}$-CorCP) is introduced and examined, and a new interesting variant of the CorCP is described. The goal of this clustering problem is to partition the vertex set into two clusters with equal size, such that the number of disagreements is minimized. We first present a polynomial time algorithm for the B $\mathrm{2}$-CorCP on $M$-positive edge dominant graphs $\mathrm{(}M\mathrm{⩾}\mathrm{3}\mathrm{)}$. Then, we provide a series of numerical experiments, and the results show the effectiveness of our algorithm.

In this paper, we study the prize-collecting $k$-Steiner tree (PC $k$ST) problem. We are given a graph $G\mathrm{=}\mathrm{(}V\mathrm{,}E\mathrm{)}$ and an integer $k$. The graph is connected and undirected. A vertex $r\mathrm{\in }V$ called root and a subset $R\mathrm{\subseteq }V$ called terminals are also given. A feasible solution for the PC $k$ST is a tree $F$ rooted at $r$ and connecting at least $k$ vertices in $R$. Excluding a vertex from the tree incurs a penalty cost, and including an edge in the tree incurs an edge cost. We wish to find a feasible solution with minimum total cost. The total cost of a tree $F$ is the sum of the edge costs of the edges in $F$ and the penalty costs of the vertices not in $F$. We present a simple approximation algorithm with the ratio of $\mathrm{5.9672}$ for the PC $k$ST. This algorithm uses the approximation algorithms for the prize-collecting Steiner tree (PCST) problem and the $k$-Steiner tree ( $k$ST) problem as subroutines. Then we propose a primal-dual based approximation algorithm and improve the approximation ratio to $\mathrm{5}$.