Maximum Flows on Disjoint Paths

Abstract. We consider the question: What is the maximum flow achievable in a network if the flow must be decomposable into a collection of edgedisjoint paths? Equivalently, we wish to find a maximum weighted packing of disjoint paths, where the weight of a path is the minimum capacity of an edge on the path. Our main result is an Ω(log n) lower bound on the approximability of the problem. We also show this bound is tight to within a constant factor. Surprisingly, the lower bound applies even for the simple case of undirected, planar graphs. Our results extend to the case in which the flow must decompose into at most k disjoint paths. There we obtain Θ(log k) upper and lower approximability bounds. 1

Flow Allocation for Maximum Throughput and Bounded Delay on Multiple Disjoint Paths for Random Access Wireless Multihop Networks

In this paper, we consider random access, wireless, multi-hop networks, with multi-packet reception capabilities, where multiple flows are forwarded to the gateways through node disjoint paths. We explore the issue of allocating flow on multiple paths, exhibiting both intra- and inter-path interference, in order to maximize average aggregate flow throughput (AAT) and also provide bounded packet delay. A distributed flow allocation scheme is proposed where allocation of flow on paths is formulated as an optimization problem. Through an illustrative topology it is shown that the corresponding problem is non-convex. Furthermore, a simple, but accurate model is employed for the average aggregate throughput achieved by all flows, that captures both intra- and inter-path interference through the SINR model. The proposed scheme is evaluated through Ns2 simulations of several random wireless scenarios. Simulation results reveal that, the model employed, accurately captures the AAT observed in the simulated scenarios, even when the assumption of saturated queues is removed. Simulation results also show that the proposed scheme achieves significantly higher AAT, for the vast majority of the wireless scenarios explored, than the following flow allocation schemes: one that assigns flows on paths on a round-robin fashion, one that optimally utilizes the best path only, and another one that assigns the maximum possible flow on each path. Finally, a variant of the proposed scheme is explored, where interference for each link is approximated by considering its dominant interfering nodes only.Comment: IEEE Transactions on Vehicular Technolog

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

Optimization of Free Space Optical Wireless Network for Cellular Backhauling

With densification of nodes in cellular networks, free space optic (FSO) connections are becoming an appealing low cost and high rate alternative to copper and fiber as the backhaul solution for wireless communication systems. To ensure a reliable cellular backhaul, provisions for redundant, disjoint paths between the nodes must be made in the design phase. This paper aims at finding a cost-effective solution to upgrade the cellular backhaul with pre-deployed optical fibers using FSO links and mirror components. Since the quality of the FSO links depends on several factors, such as transmission distance, power, and weather conditions, we adopt an elaborate formulation to calculate link reliability. We present a novel integer linear programming model to approach optimal FSO backhaul design, guaranteeing $K$-disjoint paths connecting each node pair. Next, we derive a column generation method to a path-oriented mathematical formulation. Applying the method in a sequential manner enables high computational scalability. We use realistic scenarios to demonstrate our approaches efficiently provide optimal or near-optimal solutions, and thereby allow for accurately dealing with the trade-off between cost and reliability

Maximum Skew-Symmetric Flows and Matchings

The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor stylistic corrections and shortenings to the original versio

Finding flows in the one-way measurement model

The one-way measurement model is a framework for universal quantum computation, in which algorithms are partially described by a graph G of entanglement relations on a collection of qubits. A sufficient condition for an algorithm to perform a unitary embedding between two Hilbert spaces is for the graph G, together with input/output vertices I, O \subset V(G), to have a flow in the sense introduced by Danos and Kashefi [quant-ph/0506062]. For the special case of |I| = |O|, using a graph-theoretic characterization, I show that such flows are unique when they exist. This leads to an efficient algorithm for finding flows, by a reduction to solved problems in graph theory.Comment: 8 pages, 3 figures: somewhat condensed and updated version, to appear in PR