On Routing Disjoint Paths in Bounded Treewidth Graphs

We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph $G$ and a collection of $k$ source-destination pairs $\mathcal{M} = \{(s_1, t_1), \dots, (s_k, t_k)\}$. The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset $\mathcal{M}'$ of the pairs is a collection $\mathcal{P}$ of paths such that, for each pair $(s_i, t_i) \in \mathcal{M}'$, there is a path in $\mathcal{P}$ connecting $s_i$ to $t_i$. In the Maximum Edge Disjoint Paths (MaxEDP) problem, the graph $G$ has capacities $\mathrm{cap}(e)$ on the edges and a routing $\mathcal{P}$ is feasible if each edge $e$ is in at most $\mathrm{cap}(e)$ of the paths of $\mathcal{P}$. The Maximum Node Disjoint Paths (MaxNDP) problem is the node-capacitated counterpart of MaxEDP. In this paper we obtain an $O(r^3)$ approximation for MaxEDP on graphs of treewidth at most $r$ and a matching approximation for MaxNDP on graphs of pathwidth at most $r$. Our results build on and significantly improve the work by Chekuri et al. [ICALP 2013] who obtained an $O(r \cdot 3^r)$ approximation for MaxEDP

Distances and cuts in planar graphs

AbstractWe prove the following theorem. Let G = (V, E) be a planar bipartite graph, embedded in the euclidean plane. Let O and I be two of its faces. Then there exist pairwise edge-disjoint cuts C1, …, Ct so that for each two vertices v, w with v, w ϵ O of v, w ϵ I, the distance from v to w in G is equal to the number of cuts Cj separating v and w. This theorem is dual to a theorem of Okamura on plane multicommodity flows, in the same way as a theorem of Karzanov is dual to one of Lomonosov

Shortest path and maximum flow problems in planar flow networks with additive gains and losses

In contrast to traditional flow networks, in additive flow networks, to every edge e is assigned a gain factor g(e) which represents the loss or gain of the flow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) is less than the designated capacity of e, then f(e) + g(e) = 0 units of flow reach the end point of e, provided e is used, i.e., provided f(e) != 0. In this report we study the maximum flow problem in additive flow networks, which we prove to be NP-hard even when the underlying graphs of additive flow networks are planar. We also investigate the shortest path problem, when to every edge e is assigned a cost value for every unit flow entering edge e, which we show to be NP-hard in the strong sense even when the additive flow networks are planar

Algorithms for Multicommodity Flows in Planar Graphs

This paper gives efficient algorithms for the multicommodity flow problem for two classes Ct2 and Co~ of planar undirected graphs. Every graph in Ct2 has two face boundaries B t and B 2 such that each of the source-sink pairs lies on B 1 or B 2. On the other hand, every graph in Cot has a face boundary B t such that some of the source-sink pairs lie on B 1 and all the other pairs share a common sink lying on B t. The algorithms run in O(kn + nT(n)) time if a graph has n vertices and k source-sink pairs and T(n) is the time required for finding the single-source shortest paths in a planar graph of n vertices