Consider the following linear programming problem
max $w^\top x$ s.t.
$Ax\leq b, \,\,0\leq x\leq 1$,
where $x\in R^E, w \in\{0,1\}^E$, $A$ is a $\{0,\pm 1\}$ matrix and $b$ is a vector. Further assume that $A$ is totally unimodular, i.e., every nonsingular square submatrix has determinant -1 or 1.
Define a set function $f: 2^E\to R$ as follows: for every $U\subseteq E$,
$f(U):=max \{\chi^U x | Ax\leq b, 0\leq x\leq 1\}$.
where $\chi^U$ is the incidence vector of $U$.
My question is what combinatorial properties (i.e., submodularity) would function $f$ have? Is there any literature that discusses this problem? Thanks.