A polymatroid on a finite ground set 
is the polytope defined by a normalized nondecreasing
submodular function 
:
 |
Its base polytope imposes the additional equality 
. For integer-valued
, its integer bases are the nonnegative
integer vectors in that base polytope. A matroid
rank function is a special case; its integer bases
are the incidence vectors of matroid
bases. For 
and 
, the integer bases are 
. Counting common
integer polymatroid bases counts vectors once, rather
than their representations by labeled copies.
The integer bases also support an activity generating function, the polymatroid Tutte polynomial. For connected hypergraphs, the corresponding bases yield the hypergraph Tutte polynomial.
