Parameterized algorithms for min–max 2-cluster editing

Abstract

For a given graph and an integer t, the Min–Max 2-Clustering problem asks if there exists a modification of a given graph into two maximal disjoint cliques by inserting or deleting edges such that the number of the editing edges incident to each vertex is at most t. It has been shown that the problem can be solved in polynomial time for $t<n/4$, where n is the number of vertices. In this paper, we design parameterized algorithms for different ranges of t. Let $k=t-n/4$. We show that the problem is polynomial-time solvable when roughly $k<\sqrt{n/32}$. When $k\in o(n)$, we design a randomized and a deterministic algorithm with sub-exponential time parameterized complexity, i.e., the problem is in SUBEPT. We also show that the problem can be solved in $O({2}^{n/r}\cdot n^2)$ time for $k<n/12$ and in $O(n^2\cdot 2^{3n/4+k})$ time for $n/12\le k< n/4$, where $r=2+\lfloor (n/4-3k-2)/(2k+1) \rfloor \ge 2$.

Appendix

We give a proof of that $(1-x)^{(1/x)}< 1/e$ for all $1>x>0$.

Let $j=(1-x)^{(1/x)}$, i.e., $j^x=1-x$. Taking derivatives on both sides, we have that $\frac{dj^x}{dx}=\frac{d(1-x)}{dx}=-1$. Since

$$\begin{aligned} \frac{dj^x}{dx}=\frac{dj^x}{dj}\cdot \frac{dj}{dx}= x\cdot j^{x-1}\cdot \frac{dj}{dx} \end{aligned}$$

and $j>0$, we have that

$$\begin{aligned} \frac{dj}{dx}=\frac{d(1-x)^{(1/x)}}{dx}<0 \end{aligned}$$

for all $0<x<1$. The proof is completed by the following well-known identity Finney et al. (2001)

$$\begin{aligned} \lim _{x \rightarrow 0^+} (1-x)^{(1/x)}=1/e. \end{aligned}$$