241 research outputs found
The Approximate Optimality of Simple Schedules for Half-Duplex Multi-Relay Networks
In ISIT'12 Brahma, \"{O}zg\"{u}r and Fragouli conjectured that in a
half-duplex diamond relay network (a Gaussian noise network without a direct
source-destination link and with non-interfering relays) an approximately
optimal relay scheduling (achieving the cut-set upper bound to within a
constant gap uniformly over all channel gains) exists with at most active
states (only out of the possible relay listen-transmit
configurations have a strictly positive probability). Such relay scheduling
policies are said to be simple. In ITW'13 we conjectured that simple relay
policies are optimal for any half-duplex Gaussian multi-relay network, that is,
simple schedules are not a consequence of the diamond network's sparse
topology. In this paper we formally prove the conjecture beyond Gaussian
networks. In particular, for any memoryless half-duplex -relay network with
independent noises and for which independent inputs are approximately optimal
in the cut-set upper bound, an optimal schedule exists with at most
active states. The key step of our proof is to write the minimum of a
submodular function by means of its Lov\'{a}sz extension and use the greedy
algorithm for submodular polyhedra to highlight structural properties of the
optimal solution. This, together with the saddle-point property of min-max
problems and the existence of optimal basic feasible solutions in linear
programs, proves the claim.Comment: Submitted to IEEE Information Theory Workshop (ITW) 201
Efficiently Finding Simple Schedules in Gaussian Half-Duplex Relay Line Networks
The problem of operating a Gaussian Half-Duplex (HD) relay network optimally
is challenging due to the exponential number of listen/transmit network states
that need to be considered. Recent results have shown that, for the class of
Gaussian HD networks with N relays, there always exists a simple schedule,
i.e., with at most N +1 active states, that is sufficient for approximate
(i.e., up to a constant gap) capacity characterization. This paper investigates
how to efficiently find such a simple schedule over line networks. Towards this
end, a polynomial-time algorithm is designed and proved to output a simple
schedule that achieves the approximate capacity. The key ingredient of the
algorithm is to leverage similarities between network states in HD and edge
coloring in a graph. It is also shown that the algorithm allows to derive a
closed-form expression for the approximate capacity of the Gaussian line
network that can be evaluated distributively and in linear time. Additionally,
it is shown using this closed-form that the problem of Half-Duplex routing is
NP-Hard.Comment: A short version of this paper was submitted to ISIT 201
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