  In this paper we deal with a network of computing agents with local
processing and neighboring communication capabilities that aim at solving
(without any central unit) a submodular optimization problem. The cost function
is the sum of many local submodular functions and each agent in the network has
access to one function in the sum only. In this \emph{distributed} set-up, in
order to preserve their own privacy, agents communicate with neighbors but do
not share their local cost functions. We propose a distributed algorithm in
which agents resort to the Lov\`{a}sz extension of their local submodular
functions and perform local updates and communications in terms of single
blocks of the entire optimization variable. Updates are performed by means of a
greedy algorithm which is run only until the selected block is computed, thus
resulting in a reduced computational burden. The proposed algorithm is shown to
converge in expected value to the optimal cost of the problem, and an
approximate solution to the submodular problem is retrieved by a thresholding
operation. As an application, we consider a distributed image segmentation
problem in which each agent has access only to a portion of the entire image.
While agents cannot segment the entire image on their own, they correctly
complete the task by cooperating through the proposed distributed algorithm.
