Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths

We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) $n$-vertex graph $G$ along with $k$ terminal pairs $(s_1,t_1),(s_2,t_2),\ldots,(s_k,t_k)$. The task is to connect as many terminal pairs as possible by pairwise vertex-disjoint paths such that each path is a shortest path between the respective terminals. Our work is anchored in the recent breakthrough by Lochet [SODA '21], which demonstrates the polynomial-time solvability of the problem for a fixed value of $k$. Lochet's result implies the existence of a polynomial-time $ck$-approximation for Maximum Vertex-Disjoint Shortest Paths, where $c \leq 1$ is a constant. Our first result suggests that this approximation algorithm is, in a sense, the best we can hope for. More precisely, assuming the gap-ETH, we exclude the existence of an $o(k)$-approximations within $f(k) \cdot$poly($n$) time for any function $f$ that only depends on $k$. Our second result demonstrates the infeasibility of achieving an approximation ratio of $n^{\frac{1}{2}-\varepsilon}$ in polynomial time, unless P = NP. It is not difficult to show that a greedy algorithm selecting a path with the minimum number of arcs results in a $\lceil\sqrt{\ell}\rceil$-approximation, where $\ell$ is the number of edges in all the paths of an optimal solution. Since $\ell \leq n$, this underscores the tightness of the $n^{\frac{1}{2}-\varepsilon}$-inapproximability bound. Additionally, we establish that Maximum Vertex-Disjoint Shortest Paths is fixed-parameter tractable when parameterized by $\ell$ but does not admit a polynomial kernel. Our hardness results hold for undirected graphs with unit weights, while our positive results extend to scenarios where the input graph is directed and features arbitrary (non-negative) edge weights

Approximating maximum integral multiflows on bounded genus graphs

We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances

An Approximation Algorithm for Fully Planar Edge-Disjoint Paths

We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem

Shortest disjoint paths on a grid

The well-known k-disjoint paths problem involves finding pairwise vertex-disjoint paths between k specified pairs of vertices within a given graph if they exist. In the shortest k-disjoint paths problem one looks for such paths of minimum total length. Despite nearly 50 years of active research on the k-disjoint paths problem, many open problems and complexity gaps still persist. A particularly well-defined scenario, inspired by VLSI design, focuses on infinite rectangular grids where the terminals are placed at arbitrary grid points. While the decision problem in this context remains NP-hard, no prior research has provided any positive results for the optimization version. The main result of this paper is a fixed-parameter tractable (FPT) algorithm for this scenario. It is important to stress that this is the first result achieving the FPT complexity of the shortest disjoint paths problem in any, even very restricted classes of graphs where we do not put any restriction on the placements of the terminals

Almost Polynomial Hardness of Node-Disjoint Paths in Grids

We study the hardness of the Node-Disjoint Paths (NDP) problem. In this problem, we are given a graph and a collection of source-sink pairs of vertices, called demand pairs. The goal is to route as many of the demand pairs as possible, along paths which are disjoint in their vertices. The best current algorithm for NDP achieves a factor O(\sqrt{n})-approximation. In a recent work, we showed a 2^{\Omega(\sqrt{log n})}-hardness of approximation for NDP, improving the best previous hardness bound of roughly \Omega(\sqrt{\log n}) by Andrews et al. This new hardness result holds even for subgraphs of grid graphs, but unfortunately it does not extend to grids themselves. The question of approximability of NDP on grid graphs has remained widely open, with the best current upper bound of O(n^{1/4}), and the best current lower bound of APX-hardness. In this talk I will present a new result that comes close to resolving the approximability of NDP in general, and of NDP grids in particular. We show that NDP is hard to approximate to within near-polynomial factors, even in grid graphs. We also obtain similar hardness of approximation results for the closely related Edge-Disjoint Paths (EDP) problem, even in wall graphs. Joint work with Julia Chuzhoy and David H. K. Kim.Non UBCUnreviewedAuthor affiliation: Toyota Technological Institute at ChicagoGraduat

Parameterized Algorithm for the Disjoint Path Problem on Planar Graphs: Exponential in k2k^2 and Linear in nn

In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. We present a $2^{O(k^2)}n$-time algorithm for the \textsf{Planar Disjoint Paths} problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].Comment: SODA 202