Almost Polynomial Hardness of Node-Disjoint Paths in Grids

In the classical Node-Disjoint Paths (NDP) problem, we are given an $n$-vertex graph $G=(V,E)$, and a collection $M=\{(s_1,t_1),\ldots,(s_k,t_k)\}$ of pairs of its vertices, called source-destination, or demand pairs. The goal is to route as many of the demand pairs as possible, where to route a pair we need to select a path connecting it, so that all selected paths are disjoint in their vertices. The best current algorithm for NDP achieves an $O(\sqrt{n})$-approximation, while, until recently, the best negative result was a factor $\Omega(\log^{1/2-\epsilon}n)$-hardness of approximation, for any constant $\epsilon$, unless $NP \subseteq ZPTIME(n^{poly \log n})$. In a recent work, the authors have shown an improved $2^{\Omega(\sqrt{\log n})}$-hardness of approximation for NDP, unless $NP\subseteq DTIME(n^{O(\log n)})$, even if the underlying graph is a subgraph of a grid graph, and all source vertices lie on the boundary of the grid. Unfortunately, this result does not extend to grid graphs. The approximability of the NDP problem on grid graphs has remained a tantalizing open question, with the best current upper bound of $\tilde{O}(n^{1/4})$, and the best current lower bound of APX-hardness. In this paper we come close to resolving the approximability of NDP in general, and NDP in grids in particular. Our main result is that NDP is $2^{\Omega(\log^{1-\epsilon} n)}$-hard to approximate for any constant $\epsilon$, assuming that $NP\not\subseteq RTIME(n^{poly\log n})$, and that it is $n^{\Omega (1/(\log \log n)^2)}$-hard to approximate, assuming that for some constant $\delta>0$, $NP \not \subseteq RTIME(2^{n^{\delta}})$. These results hold even for grid graphs and wall graphs, and extend to the closely related Edge-Disjoint Paths problem, even in wall graphs.