Abstract
We establish a tight max-flow min-cut theorem for multi-commodity routing in random geometric graphs. We show that, as the number of nodes in the network n tends to infinity, the maximum concurrent flow (MCF) and the minimum cut-sparsity scale as ¿(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> (n)/k), for a random choice of k = ¿(n) source-destination pairs, where n and r(n) are the number of nodes and the communication range in the network respectively. The MCF equals the interference-free capacity of an ad-hoc network. We exploit this fact to develop novel graph theoretic techniques that can be used to deduce tight order bounds on the capacity of ad-hoc networks. We generalize all existing capacity results reported to date by showing that the per-commodity capacity of the network scales as ¿(1/r(n)k) for the single-packet reception model suggested by Gupta and Kumar, and as ¿(nr(n)/k) for the multiple-packet reception model suggested by others. More importantly, we show that, if the nodes in the network are capable of (perfect) multiple-packet transmission (MPT) and reception (MPR), then it is feasible to achieve the optimal scaling of ¿(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> (n)/k), despite the presence of interference. In comparison to the Gupta-Kumar model, the realization of MPT and MPR may require the deployment of a large number of antennas at each node or bandwidth expansion. Nevertheless, in stark contrast to the existing literature, our analysis presents the possibility of actually increasing the capacity of ad-hoc networks with n even while the communication range tends to zero!
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