A minimal complex is a CW complex whose only cells are the homology cells. 


Is there some sort of criterion on CW complexes about existence of minimal complexes?


Actually I am working on a problem of understanding homotopy type of certain spaces 
(see: [How to show that a space has the homotopy type of wedge of spheres ?](https://mathoverflow.net/questions/43711/how-to-show-that-a-space-has-the-homotopy-type-of-wedge-of-spheres))


My hope was to use discrete Morse theory (acyclic matching of face poset to be precise) and find the minimal complex. But then I don't know if the existence of the minimal complex is always guaranteed. 



Discrete Morse theory and existence of minimal complex
This is not exactly what you asked, but it's certainly not the case that every CW complex has a discrete vector field where the Morse complex has trivial differential. In particular this would imply that chain complex is simple-homotopy equivalent to a chain complex with no differential. However, simple-homotopy equivalence is well-known not to generate homotopy equivalence. In particular, the Whitehead torsion is an obstruction which lives in the Whitehead group of the fundamental group. Marshall Cohen's book on Whitehead torsion is the canonical place to learn about this.


