一种由树宽参数化的更少顶点不相交路径的更快算法
A Faster Algorithm for Fewer Vertex-Disjoint Paths Parameterized by Treewidth
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中文总结 AI 辅助
本文针对顶点不相交路径问题，提出一种由树宽参数化的更快算法，在 $k=tw^{o(1)}$ 时突破已知下界，并证明有向图在 SETH 下的紧下界。
中文摘要 AI 辅助
$k$ 顶点不相交路径问题询问：给定一个图 $G$ 和 $k$ 对顶点 $(s_1,t_1), \ldots, (s_k,t_k)$，$G$ 是否存在 $k$ 条两两顶点不相交的路径，分别连接 $s_i$ 和 $t_i$，对所有 $1\leq i\leq k$ 成立。若 $G$ 是无向图，则此问题是 NP 完全的，但存在以 $k$ 为参数的 FPT 算法。由于这些算法涉及关于 $k$ 的极大函数，受限图上的算法也已被研究。特别地，Scheffler（技术报告 396，柏林工业大学，'94）提出了一个针对具有 $n$ 个顶点和树宽 $tw$ 的无向图的 $2^{2tw\log tw+O(tw)}\cdot n$ 时间算法，并且 Lokshtanov、Marx 和 Saurabh（SIAM J. Comput. '18）证明了在 ETH 假设下，对于路径宽为 $pw$ 的有向图或无向图，当 $k=\Omega(pw^4)$ 时，不存在 $2^{o(pw\log pw)}\cdot n^{O(1)}$ 时间算法。尚不清楚该下界是否也适用于更小的 $k$。在本文中，我们证明对于有向和无向两种情况，当 $k=tw^{o(1)}$ 时，存在一个比 Lokshtanov 等人的下界更快的算法，通过提出一个 $2^{O((tw+k)\log k)}\cdot n$ 时间算法。此外，我们证明了一个下界：在 SETH 假设下，对于有向图且一般 $k$，不存在 $(2-\epsilon)^{pw\log pw}\cdot n^{O(1)}$ 时间算法。该下界是紧的，因为经过轻微修改，Scheffler 的算法对于有向图也能在 $2^{pw\log pw+O(pw)}\cdot n$ 时间内运行。
英文摘要
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.
发表机构
- Division of Electrical, Information and Communication Engineering, Kanazawa University（金泽大学电气信息通信工程分部）
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