The paper deals with the distributed minimum sharing problem, in which a
network of decision-makers - exchanging information through a communication
network - computes the minimum of some local quantities of interest in a
distributed and decentralized way. The problem is equivalently cast into a
cost-coupled distributed optimization problem, and an adjustable approximate
(or sub-optimal) solution is presented which enjoys several properties of
crucial importance in applications. In particular, the proposed solution is
scalable in that the dimension of the state space does not grow with the size
or topology of the communication network. Moreover, a global and uniform (both
in the initial time and in the initial condition) asymptotic stability result
is provided, as well as an attractiveness property towards a steady state which
can be made arbitrarily close to the sought minimum. Exact asymptotic
convergence is also recovered at the price, however, of loosing uniformity of
the convergence with respect to the initial time.