# Covering a crossing supermodular function with pairwise non-parallel arcs

Given a crossing supermodular function $p:2^V\to \mathbb{Z}$ satisfying $p(\emptyset)=p(V)=0$, what is the minimum number of pairwise non-parallel arcs (i.e. only one copy of the arc uv is allowed for any pair $u,v\in V$, but opposite pairs are allowed) covering p, where a digraph D=(V,A) is said to cover p if $\varrho_D(X)\ge p(X)$ holds for every $X\subseteq V$?
Theorem: A crossing supermodular function $p:2^V\to \mathbb{Z}$ satisfying $p(\emptyset)=p(V)=0$ can be covered by $\gamma$ arcs if and only if $\sum_{i=1}^t p(V_i)\le \gamma$ and $\sum_{i=1}^t p(V-V_i)\le \gamma$ holds for every partition $V_1,V_2,\dots,V_t$ of V.