For my project, I try to use minimum fractional vertex cover number (MFVC). (Please find below definition details) MFVC can be formulated as optimal solution of a linear program relaxation.
However I find that every linear program relaxation is either infeasible, unbounded, or has an optimum solution.
Does is mean that MFVC may not exist, or may not be a finite number, but MFVC is less or equal to minimum vertex cover number? Very confusing! :(
Many thanks!
Definitions:
Minimum fractional vertex cover number (MFVC): link (on page 8)
MFVC as linear programming relaxation:
The minimum fractionl vertex cover $\tau^* (G)$ in a k-uniform hypergraph $G=(V,E)$ is $\min \sum_{v\in V}w (v)$
such that $0 \le w(v) \le 1$,
and for each $e \in E$, we have $\sum_{v\in e} w(v) \ge 1$.