In a multihop wireless network, wireless interference is a crucial factor in the maximum multiflow (MMF) problem, which studies the maximum throughput between multiple pairs of sources and sinks with a link schedule to support it. In this paper, we observe that network coding could help to decrease the impact of wireless interference, and thus propose a framework to study the MMF problem for multihop wireless networks with network coding. Firstly, a network model is established to describe the new conflict relations and schedulability modified by network coding. Next, we formulate the MMF problem to compute the maximum throughput of multiple unicast flows supported by the multihop wireless network with network coding, and show that its capacity region could be enlarged by performing network coding. Finally, we show that determining the capacity region of a multihop wireless network with network coding is an NP-hard problem, and thus propose a greedy heuristic algorithm, called coding-first collecting (CFC), to determine a capacity subregion of the network. We also show that finding an optimal hyperarc schedule to meet a given link demand function is NP-hard, and propose a polynomial algorithm, called coding-first scheduling (CFS), to find an approximate fractional hyperarc schedule in the multihop wireless network with network coding. A numerical analysis of a grid wireless network and a random wireless network is presented to demonstrate the efficiencies of the CFC algorithm and the CFS algorithm based on the framework.

Most large-scale Peer-to-Peer (P2P) live streaming systems are constructed as a mesh structure, which can provide robustness in the dynamic P2P environment. The pull scheduling algorithm is widely used in this mesh structure, which degrades the performance of the entire system. Recently, network coding was introduced in mesh P2P streaming systems to improve the performance, which makes the push strategy feasible. One of the most famous scheduling algorithms based on network coding is R2, with a random push strategy. Although R2 has achieved some success, the push scheduling strategy still lacks a theoretical model and optimal solution. In this paper, we propose a novel optimal pull-push scheduling algorithm based on network coding, which consists of two stages: the initial pull stage and the push stage. The main contributions of this paper are: 1) we put forward a theoretical analysis model that considers the scarcity and timeliness of segments; 2) we formulate the push scheduling problem to be a global optimization problem and decompose it into local optimization problems on individual peers; 3) we introduce some rules to transform the local optimization problem into a classical min-cost optimization problem for solving it; 4) We combine the pull strategy with the push strategy and systematically realize our scheduling algorithm. Simulation results demonstrate that decode delay, decode ratio and redundant fraction of the P2P streaming system with our algorithm can be significantly improved, without losing throughput and increasing overhead.