On the Multiple Unicast Network Conjecture – Langberg, Medard

1. ACHIEVEMENT DESCRIPTION STATUS QUO IMPACT NEXT-PHASE GOALS NEW INSIGHTS On the Multiple Unicast Network Conjecture – Langberg, Medard 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? • 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 • Effect of Network Coding in Graphs • Undirecting the edges is roughly as strong as allowing network coding simplicity is the main benefit • Effect of Network Coding in Wireless Networks Mainly From: • Erasures (unlimited) • Interference (unlimited) • Duplex constraints 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 Example from Li, Li, Lau: With network coding: 2 symbols Without network coding: 1.786 symbols This comes at a cost of optimizing over 119104 Steiner trees 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? Towards characterizing the fundamental contributions of coding over routing

2. 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

3. 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

4. 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.

5. 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

6. 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

7. Main lemma Proof outline:

8. Proof outline • Proof outline continued

9. 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.