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

Approximation Algorithms and Hardness of Integral Concurrent Flow

We study an integral counterpart of the classical Maximum Concurrent Flow problem, that we call Integral Concurrent Flow (ICF). In the basic version of this problem (basic-ICF), we are given an undirected n-vertex graph G with edge capacities c(e), a subset T of vertices called terminals, and a demand D(t, t ′ ) for every pair (t, t ′ ) of the terminals. The goal is to find the maximum value λ, and a collection P of paths, such that every pair (t, t ′ ) of terminals is connected by ⌊λD(t, t ′) ⌋ paths in P, and the number of paths containing any edge e is at most c(e). We show an algorithm that achieves a poly log n-approximation for basic-ICF, while violating the edge capacities by only a constant factor. We complement this result by proving that no efficient algorithm can achieve a factor α-approximation with congestion c for any values α, c satisfying α · c = O(log log n / log log log n), unless NP ⊆ ZPTIME(n poly log n). We then turn to study the more general group version of the problem (group-ICF), in which we are given a collection {(S1, T1),..., (Sk, Tk)} of pairs of vertex subsets, and for each 1 ≤ i ≤ k, a demand Di is specified. The goal is to find the maximum value λ and a collection P of paths, such that for each i, at least ⌊λDi ⌋ paths connect the vertices of Si to the vertices of Ti, while respecting the( edge capacities. We sho