A Faster Algorithm for Fewer Vertex-Disjoint Paths Parameterized by Treewidth
The $k$ vertex-disjoint paths problem asks whether, given a graph $G$ and $k$ pairs of vertices $(s_1,t_1)$, \ldots, $(s_k,t_k)$, $G$ has $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all $1\leq i\leq k$. If $G$ is undirected, then this problem is NP-complete, but there exist FPT algorithms parameterized by $k$.Since these algorithms involve an extremely large function on $k$, algorithms for restricted graphs have also been investigated. In particular, a $2^{2tw\log tw+O(tw)}\cdot n$ time algorithm for undirected graphs with $n$ vertices and treewidth $tw$ is proposed by Scheffler (Technical Report 396, TU Berlin, '94), and it is proved by Lokshtanov, Marx, and Saurabh (SIAM J. Comput. '18) that, under the ETH, there exists no $2^{o(pw\log pw)}\cdot n^{O(1)}$ time algorithm for either directed or undirected graphs with pathwidth $pw$ and for $k=Ω(pw^4)$. It has not been known whether the lower bound also holds for a smaller $k$. In this paper, we prove that, for both the directed and undirected cases, there is an algorithm faster than Lokshtanov et al.'s lower bound for $k=tw^{o(1)}$ by proposing a $2^{O((tw+k)\log k)}\cdot n$ time algorithm. Besides, we prove a lower bound that, under the SETH, there exists no $(2-ε)^{pw\log pw}\cdot n^{O(1)}$ time algorithm for directed graphs and for a general $k$. This lower bound is tight because, with slight modifications, Scheffler's algorithm runs in $2^{pw\log pw+O(pw)}\cdot n$ time also for directed graphs.