Minimum-Cost Network-Wide Broadcast over Reliable MAC-Layer Multicast

We consider the network-wide broadcast problem in multihop wireless networks with reliable multicast at the Medium Access Control (MAC) layer, where the cost of transmitting to downstream nodes at each branch point in the broadcast tree depends on the number
$k$
of recipients, specifically
$1+A k^b$
in our model (for some
$b \geq 0$
,
$A \geq 0$
). This allows us to capture a wide array of MAC-layer approaches and their costs, simply by varying the value of
$b$
(relative to
$A$
), in a problem formulation subsuming the Connected Dominating Set and Spanning Tree problems. We give a systematic analysis of this problem, including positive and negative results. In particular, we show the problem is: approximable by a factor varying from
$2H_{\Delta}+2$
down to 2 as
$b$
varies from 0 to 1 (where
$\Delta$
is the maximum degree of the network graph and
$H_\Delta$
is the
$\Delta$
th harmonic number); approximable by a factor varying from 2 to 1 (i.e., optimal) as
$b$
varies from 1 to
$\log _2 (\frac{1}{A}+2)$
; and optimally solvable thereafter. Finally, we present numerical results comparing the two algorithms above with other natural heuristics. We find there is an advantage in algorithms taking into consideration the value
$b$
, even if
$b$
can only be roughly estimated.