Finding All Leftmost Separators of Size $\leq k$

We define a notion called leftmost separator of size at most $k$. A leftmost separator of size $k$ is a minimal separator $S$ that separates two given sets of vertices $X$ and $Y$ such that we "cannot move $S$ more towards $X$" such that $|S|$ remains smaller than the threshold. One of the incentives is that by using leftmost separators we can improve the time complexity of treewidth approximation. Treewidth approximation is a problem which is known to have a linear time FPT algorithm in terms of input size, and only single exponential in terms of the parameter, treewidth. It is not known whether this result can be improved theoretically. However, the coefficient of the parameter $k$ (the treewidth) in the exponent is large. Hence, our goal is to decrease the coefficient of $k$ in the exponent, in order to achieve a more practical algorithm. Hereby, we trade a linear-time algorithm for an $\mathcal{O}(n \log n)$-time algorithm. The previous known $\mathcal{O}(f(k) n \log n)$-time algorithms have dependences of $2^{24k}k!$, $2^{8.766k}k^2$ (a better analysis shows that it is $2^{7.671k}k^2$), and higher. In this paper, we present an algorithm for treewidth approximation which runs in time $\mathcal{O}(2^{6.755k}\ n \log n)$, Furthermore, we count the number of leftmost separators and give a tight upper bound for them. We show that the number of leftmost separators of size $\leq k$ is at most $C_{k-1}$ (Catalan number). Then, we present an algorithm which outputs all leftmost separators in time $\mathcal{O}(\frac{4^k}{\sqrt{k}}n)$.