1)Effect of Network Coding in Graphs Undirecting the edges is roughly as strong as allowing network coding simplicity is the main benefit 2)Effect of Network Coding in Wireless Networks Mainly From: Erasures (unlimited) Interference (unlimited) Duplex constraints On the Multiple Unicast Network Conjecture – Langberg, Medard Existing approaches: the crux of previous proofs include a reduction in which the multicast instance undergoes several splitting modifications, until it is turned into an instance of a broadcast problem Our approach: we consider a flow- based approach and we consider k-multicast coding rate on one hand and k-unicast routing rate on the other Towards characterizing the fundamental contributions of coding over routing MAIN ACHIEVEMENT: We have shown that, in undirected graphs that are r -strongly connected, the use of network coding for k- multicast is comparable (within a factor of 3) to the routing rate of an arbitrary set of k unicast connections. HOW IT WORKS: Create flow of decomposition of the graph Complete them to multicast ASSUMPTIONS AND LIMITATIONS: Does not take into account the simplicity of coding over routing (see picture below) While wireless networks are undirected, interference and duplex constraints do not allow us to operate them as undirected graphs Capacity of general relay channel unknown For directed links, the gap between routing and coding is known to be arbitrarily large For undirected networks, the gap for broadcast and point-to-point is nil, the gap for multicast is bounded by 2 and the general case is not understood, although it has been conjectured there is no gap The question is thus – is most of the gain from network coding coming from the fact that it allows quasi undirected operation? Taxonomy of Benefits of Network Coding: How much of the benefit is present in an undiredted setting and to what extent does traditional routing, by acting as a directed graph, negate the theoretical gap? IMPACT NEXT-PHASE GOALS ACHIEVEMENT DESCRIPTION STATUS QUO NEW INSIGHTS Example from Li, Li, Lau: With network coding: 2 symbols Without network coding: symbols This comes at a cost of optimizing over Steiner trees

The advantage of network coding It was shown that for unicast and broadcast there is no advantage in the use of network coding over traditional routing For the case of multicast, the coding advantage was shown to be at most 2, and this advantage may be at least 8/7 [Agarwal, Charikar 2004] Little is known regarding the coding advantage for the more general k-unicast setting To this day, the possibility that the advantage be unbounded (i.e., a function of the size of the network) has not been ruled out

The k-unicast network coding conjecture It has been conjectured by Li and Li that, for undirected graphs, there is no coding advantage at all This fact was verified on several special cases such as bipartite graphs and planar graphs Loosely speaking, the Li and Li conjecture states that an undirected graph allowing a k-unicast connection using network coding also allows the same connection using routing We address a relaxed version of this conjecture

k-unicast and k-multicast In the k-unicast problem, there are k sources, k terminals, and one is required to design an information flow allowing each source to transmit information to its corresponding terminal In the k-multicast problem, one is required to design an information flow allowing each source to transmit information to all the terminals Requiring that a network allows a k-multicast connection implies the corresponding k-unicast connection. We show that an undirected graph allowing a k-multicast connection at rate r using network coding will allow the corresponding k-unicast connection at rate r/3.

Interpretation in terms of undirecting graphs Given a directed graph G which allows k-multicast communication at rate r on k source/terminal pairs, by undirecting the edges of G one can obtain a feasible k- unicast routing solution of rate at least r/3 In the setting in which one is guaranteed k-multicast communication, but requires only k-unicast:undirecting the edges of G is as strong as allowing network coding (up to a factor of 3) Informally, undirecting the edges of G is as strong (within a small multiplicative factor) as allowing network coding

Need for new techniques The approach of Li and Li operates by reducing to the broadcast problem (in which the terminal set includes the entire vertex set of G) This reduction does not adapt to the k-multicast scenario addressed in this work because of the lack of a single source governing the multicast connection We adopt a multicommodity flow approach

Main lemma Proof outline:

Proof outline Proof outline continued

Conclusions This may have interesting consequences for wireless networks, since they are generally undirected While it may at first blush seem that our results imply a bound of a factor of 3 for the advantage of k-multicast coding versus k-unicast non-coding in wireless networks,such a conclusion would misinterpret our results –broadcast conditions –half-duplex constraints.