Problem :

Given a directed acyclic graph and capacity of every edge. Let S be set of all Max flows ( sum of flows of outgoing edges is maximum ), find the flow among S with the minimum positive-flow edges OR maximum zero flow edges.

The problem is NP Hard.

I was trying to think of heuristics to approximate the sparsest max flow with a good bound but couldn't find any. The topic seems interesting but less explored. I would appreciate any kind of insights from others or already done research in this area.