On the Multiple Unicast Network Conjecture – Langberg,
Medard
• 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?
ACHIEVEMENT DESCRIPTION
MAIN ACHIEVEMENT:
We have shown that, in undirected
graphs that are r -strongly connected,
the use of network coding for kmulticast is comparable (within a factor
of 3) to the routing rate of an arbitrary
set of k unicast connections.
1)Effect of Network Coding in
Graphs
IMPACT
STATUS QUO
Capacity of general relay channel
unknown
2)Effect of Network Coding in
Wireless Networks Mainly
From:
HOW IT WORKS:
• Erasures (unlimited)
• Create flow of decomposition of the graph
• Interference (unlimited)
Our approach: we consider a flowbased approach and we consider
k-multicast coding rate on one
hand and k-unicast routing rate on
the other
• Duplex constraints
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
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
NEXT-PHASE GOALS
NEW INSIGHTS
• Complete them to multicast
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
Undirecting the edges is roughly
as strong as allowing network
coding simplicity is the main
benefit
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
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 kunicast 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.