Multiflow Transmission in Delay Constrained Cooperative Wireless   Networks
  This paper considers the problem of energy-efficient transmission in multi-flow multihop cooperative wireless networks. Although the performance gains of cooperative approaches are well known, the combinatorial nature of these schemes makes it difficult to design efficient polynomial-time algorithms for joint routing, scheduling and power control. This becomes more so when there is more than one flow in the network. It has been conjectured by many authors, in the literature, that the multiflow problem in cooperative networks is an NP-hard problem. In this paper, we formulate the problem, as a combinatorial optimization problem, for a general setting of $k$-flows, and formally prove that the problem is not only NP-hard but it is $o(n^{1/7-\epsilon})$ inapproxmiable. To our knowledge*, these results provide the first such inapproxmiablity proof in the context of multiflow cooperative wireless networks. We further prove that for a special case of k = 1 the solution is a simple path, and devise a polynomial time algorithm for jointly optimizing routing, scheduling and power control. We then use this algorithm to establish analytical upper and lower bounds for the optimal performance for the general case of $k$ flows. Furthermore, we propose a polynomial time heuristic for calculating the solution for the general case and evaluate the performance of this heuristic under different channel conditions and against the analytical upper and lower bounds. 
Marjan Baghaie, Dorit S. Hochbaum, Bhaskar Krishnamachari
cs.NI cs.CC