1 Introduction
In the Disjoint Paths problem, the input is an undirected graph $G$ on $n$ vertices and a set of $k$ pairwise disjoint vertex pairs, ${s_i, t_i}_{i=1}^k$, and the task is to find $k$ pairwise vertex-disjoint paths connecting $s_i$ to $t_i$ for each $i \in {1, \dots, k}$. The Disjoint Paths problem is a fundamental routing problem that finds applications in VLSI layout and virtual circuit routing, and has a central role in Robertson and Seymour's Graph Minors series. We refer to surveys such as [21, 43] for a detailed overview. The Disjoint Paths problem was shown to be NP-complete by Karp (who attributed it to Knuth) in a followup paper [25] to his initial list of 21 NP-complete problems [24]. It remains NP-complete even if $G$ is restricted to be a grid [33, 30]. On directed graphs, the problem remains NP-hard even for $k = 2$ [20]. For undirected graphs, Perl and Shiloach [35] designed a polynomial time algorithm for the case where $k = 2$. Then, the seminal work of Robertson and Seymour [37] showed that the problem is polynomial time solvable for every fixed $k$. In fact, they showed that it is fixed parameter tractable (FPT) by designing an algorithm with running time $f(k)n^3$. The currently fastest parameterized algorithm for Disjoint Paths has running time $h(k)n^2$ [26]. However, all we know about $h$ and $f$ is that they are computable functions. That is, we still have no idea about what the running time dependence on $k$ really is. Similarly, the problem appears difficult in the realm of approximation, where one considers the optimization variant of the problem where the aim is to find disjoint paths connecting as many of the ${s_i, t_i}$ pairs as possible. Despite substantial efforts, the currently best known approximation algorithm remains a simple greedy algorithm that achieves approximation ratio $O(\sqrt{n})$.
The Disjoint Paths problem has received particular attention when the input graph is re- stricted to be planar [2, 17, 42, 14]. Adler et al. [2] gave an algorithm for Disjoint Paths on planar graphs (Planar Disjoint Paths) with running time $2^{2^{\mathcal{O}(k)}} n^2$, giving at least a concrete form for the dependence of the running time on $k$ for planar graphs. Schrijver [42] gave an algo- rithm for Disjoint Paths on directed planar graphs with running time $n^{\mathcal{O}(k)}$, in contrast to the NP-hardness for $k=2$ on general directed graphs. Almost 20 years later, Cygan et al. [14] im- proved over the algorithm of Schrijver and showed that Disjoint Paths on directed planar graphs is FPT by giving an algorithm with running time $2^{2^{\mathcal{O}(k^2)}} n^{\mathcal{O}(1)}$. The Planar Disjoint Paths prob- lem is well-studied also from the perspective of approximation algorithms, with a recent burst of activity [7, 8, 9, 10, 11]. Highlights of this work include an approximation algorithm with factor $\mathcal{O}(n^{9/19} \log^{\mathcal{O}(1)} n)$ [8] and, under reasonable complexity-theoretic assumptions, hardness of approximating the problem within a factor of $2^{\Omega(\frac{1}{(\log \log n)^2})}$ [10].
In this paper, we consider the parameterized complexity of Planar Disjoint Paths. Prior to our work, the fastest known algorithm was the $2^{2^{\mathcal{O}(k)}} n^2$ time algorithm of Adler et al. [2]. Double exponential dependence on k for a natural problem on planar graphs is something of an outlier–the majority of problems that are FPT on planar graphs enjoy running times of the form $2^{\mathcal{O}(\sqrt{k} \text{ polylog } k)} n^{\mathcal{O}(1)}$ (see, e.g., [15, 18, 19, 29, 36]). This, among other reasons (discussed below), led Adler [1] to pose as an open problem in GROW 2013¹ whether Planar Disjoint Paths admits an algorithm with running time $2^{k^{\mathcal{O}(1)}} n^{\mathcal{O}(1)}$. By integrating tools with origins in algebra and topology, we resolve this problem in the affirmative. In particular, we prove the following.
Theorem 1.1. The Planar Disjoint Paths problem is solvable in time 2$^{{\mathcal{O}(k^2)}}$n^{{\mathcal{O}}(1)}.²*
In addition to its value as a stand-alone result, our algorithm should be viewed as a piece of an on-going effort of many researchers to make the Graph Minor Theory of Robertson and Seymour algorithmically efficient. The graph minors project is abound with powerful algorithmic and
¹The conference version of [2] appeared in 2011, before [1]. The document [1] erroneously states the open problem for Disjoint Paths instead of for Planar Disjoint Paths—that Planar Disjoint Paths is meant is evident from the statement that a $2^{2^{\mathcal{O}(k)}} n^{\mathcal{O}(1)}$ time algorithm is known.
²In fact, towards this we implicitly design a w*$^{\mathcal{O}(k)}$-time algorithm, where w is the treewidth of the input graph.