\b[No.] Take the <hypergraph> with $V=\{a,b\}$ and its single <hyperedge> $\{a,b\}$ of weight $1$. Then $\{a\}\subseteq V$ but $f(\{a\})=1>0=f(V)$, violating the <monotone set function> property. This example already has strictly positive weights.

Solution

Past exam of the computer science course of the University of Oxford / 2023 / Combinatorial optimisation / 1 / d / ii / Solution

 Solution (source code)