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We study the classical Node-Disjoint Paths (NDP) problem: given an undirected $n$-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy $O(\sqrt{n})$-approximation algorithm. A special case of the problem called NDP-Grid, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for NDP-Grid achieves an $\tilde{O}(n^{1/4})$-approximation factor. On the negative side, a recent result by the authors shows that NDP is hard to approximate to within factor $2^{\Omega(\sqrt{\log n})}$, even if the underlying graph is a sub-graph of a grid, and all source vertices lie on the grid boundary. In a follow-up work, the authors further show that NDP-Grid is hard to approximate to within factor $\Omega(2^{\log^{1-\epsilon}n})$ for any constant $\epsilon$ under standard complexity assumptions, and to within factor $n^{\Omega(1/(\log\log n)^2)}$ under randomized ETH. In this paper we study NDP-Grid, where all source vertices {s_1,...,s_k} appear on the grid boundary. Our main result is an efficient randomized $2^{O(\sqrt{\log n} \cdot \log\log n)}$-approximation algorithm for this problem. We generalize this result to instances where the source vertices lie within a prescribed distance from the grid boundary. Much of the work on approximation algorithms for NDP relies on the multicommodity flow relaxation of the problem, which is known to have an $\Omega(\sqrt n)$ integrality gap, even in grid graphs. Our work departs from this paradigm, and uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods., Comment: To appear in the proceedings of ICALP 2018

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.

In the classical Node-Disjoint Paths (NDP) problem, the input consists of an undirected $n$-vertex graph $G$, and a collection $\mathcal{M}=\{(s_1,t_1),\ldots,(s_k,t_k)\}$ of pairs of its vertices, called source-destination, or demand, pairs. The goal is to route the largest possible number of the demand pairs via node-disjoint paths. The best current approximation for the problem is achieved by a simple greedy algorithm, whose approximation factor is $O(\sqrt n)$, while the best current negative result is an $\Omega(\log^{1/2-\delta}n)$-hardness of approximation for any constant $\delta$, under standard complexity assumptions. Even seemingly simple special cases of the problem are still poorly understood: when the input graph is a grid, the best current algorithm achieves an $\tilde O(n^{1/4})$-approximation, and when it is a general planar graph, the best current approximation ratio of an efficient algorithm is $\tilde O(n^{9/19})$. The best currently known lower bound on the approximability of both these versions of the problem is APX-hardness. In this paper we prove that NDP is $2^{\Omega(\sqrt{\log n})}$-hard to approximate, unless all problems in NP have algorithms with running time $n^{O(\log n)}$. Our result holds even when the underlying graph is a planar graph with maximum vertex degree $3$, and all source vertices lie on the boundary of a single face (but the destination vertices may lie anywhere in the graph). We extend this result to the closely related Edge-Disjoint Paths problem, showing the same hardness of approximation ratio even for sub-cubic planar graphs with all sources lying on the boundary of a single face.

We study the classical Node-Disjoint Paths (NDP) problem: given an $n$-vertex graph $G$ and a collection $M=\{(s_1,t_1),\ldots,(s_k,t_k)\}$ of pairs of vertices of $G$ called demand pairs, find a maximum-cardinality set of node-disjoint paths connecting the demand pairs. NDP is one of the most basic routing problems, that has been studied extensively. Despite this, there are still wide gaps in our understanding of its approximability: the best currently known upper bound of $O(\sqrt n)$ on its approximation ratio is achieved via a simple greedy algorithm, while the best current negative result shows that the problem does not have a better than $\Omega(\log^{1/2-\delta}n)$-approximation for any constant $\delta$, under standard complexity assumptions. Even for planar graphs no better approximation algorithms are known, and to the best of our knowledge, the best negative bound is APX-hardness. Perhaps the biggest obstacle to obtaining better approximation algorithms for NDP is that most currently known approximation algorithms for this type of problems rely on the standard multicommodity flow relaxation, whose integrality gap is $\Omega(\sqrt n)$ for NDP, even in planar graphs. In this paper, we break the barrier of $O(\sqrt n)$ on the approximability of the NDP problem in planar graphs and obtain an $\tilde O(n^{9/19})$-approximation. We introduce a new linear programming relaxation of the problem, and a number of new techniques, that we hope will be helpful in designing more powerful algorithms for this and related problems.