New algorithms for maximum disjoint paths based on tree-likeness

Abstract

We study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is 2 Ω( √ log n) , assuming NP DTIME (n O(log n) (or nodes) may be used by O (log n/ log log n) paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:-For MaxEDP, we give an O( √ r log(kr))-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio O( √ n) due to Chekuri et al., as r ≤ n. -Further, we show how to route Ω(OPT * ) pairs with congestion bounded by O(log(kr)/ log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson. -For MaxNDP, we give an algorithm that gives the optimal answer in time

Walking Through Waypoints

“…For capacitated graphs, researchers have explored the admission control variant: the problem of admitting a maximal number of routing requests such that capacity constraints are met. Chekuri et al [17], Ene et al [22], and Fleszar et al [32] presented approximation algorithms for maximizing the benefit of admitting disjoint paths in graphs admitting treelike structures with both edge and vertex capacities. Even et al [25,26] and Rost et al [47,58] initiated the study of approximation algorithms for admitting a maximal number of routing walks through waypoints.…”

On Structural Parameterizations of the Edge Disjoint Paths Problem

“…The EdgE disjoint Paths (EdP) and nodE disjoint Paths (ndP) are fundamental routing graph problems. In the EdP (ndP) problem the input is a graph G, and a set P containing k pairs of vertices and the objective is to decide whether there is a set of k pairwise edge disjoint (respectively vertex disjoint) paths connecting each pair in P. These problems and their optimization versions-MaxEdP and MaxndP-have been at the center of numerous results in structural graph theory, approximation algorithms, and parameterized algorithms [4,10,12,17,21,23,26,27,29].…”

“…While the aforementioned research considered the number of paths to be the parameter, another line of research investigates the effect of structural parameters of the input graphs on the complexity of these problems. Fleszar, Mnich, and Spoerhase [12] initiated the study of NDP and EDP parameterized by the feedback vertex set number (the size of the smallest feedback vertex set) of the input graph and showed that EDP remains NP-hard even on graphs with feedback vertex set number two. Since EDP is known to be polynomial time solvable on forests [17], this left only the case of feedback vertex set number one open, which they conjectured to be polynomial time solvable.…”

The Power of Cut-Based Parameters for Computing Edge Disjoint Paths

“…Edge Disjoint Paths (EDP) is a fundamental routing graph problem: we are given a graph G and a set P containing pairs of vertices (terminals), and are asked to decide whether there is a set of |P | pairwise edge disjoint paths in G connecting each pair in P . Similarly to its counterpart, the Vertex Disjoint Paths (VDP) problem, EDP has been at the center of numerous results in structural graph theory, approximation algorithms, and parameterized algorithms [2,8,9,15,17,19,21,22,26]. Both EDP and VDP are NP-complete in general [16], and a significant amount of research has focused on identifying structural properties which make these problems tractable.…”

“…Here, we find a stark contrast in the difficulty between these two, otherwise closely related, problems. Indeed, while VDP is known to be FPT with respect to the well-established structural parameter treewidth [24], EDP is NP-hard even on graphs of treewidth 3 [9]. What's worse, the same reduction shows that EDP remains NP-hard even on graphs with a vertex cover of size 3 [9], which rules out fixed-parameter and XP algorithms for the vast majority of studied graph parameters (including, e.g., treedepth and the size of a minimum feedback vertex set).…”