  We describe the twisted $K$-polynomial of multiplicity-free varieties in a
multiprojective setting. More precisely, for multiplicity-free varieties, we
show that the support of the twisted $K$-polynomial is a generalized
polymatroid. As applications, we show that the support of the M\"obius function
of a linear polymatroid is a generalized polymatroid, and we settle a
conjecture of Monical, Tokcan and Yong regarding Grothendieck polynomials for
the case of zero-one Schubert polynomials.
