# Time complexity of Vertex Cover vs Clique for fixed k

I have 2 ways of solving Independent Set problem of fixed size $$k$$ for graph $$G = (V, E)$$:
- Vertex Cover algorithm running in $$O^*(1.47^{V - k})$$ (optimized recursive algorithm)
- Clique algorithm running in $$O({V\choose k})$$ (simple enumerate subsets of $$V$$ and check algorithm)

How can I determine which one has a lower time complexity? I'm not very familiar with algorithms for NP-complete problems and $$O^*$$ notation. Would plotting those functions suffice? I think that VC algorithm can have any polynomial $$n^{O(1)}$$ as a multiplication because of the $$O^*$$ notation and this could affect the running times, but I'm not sure.

For any fixed $$k$$, $$O(\binom{V}{k}) = O(V^k)$$ is polynomial, whereas $$O^*(1.47^{V-k}) = O^*(1.47^V)$$ is exponential. Exponentials grow much faster than polynomials.
That said, if you're interested in particular $$V$$ and $$k$$, then your best option is to empirically check which of these algorithms is faster. Asymptotic notation is not helpful here, since it hides constant factors, and these could make a big difference for concrete values of $$V$$ and $$k$$.
Vertex Cover is fixed-parameter tractable. There is a simple $$2^k n$$ algorithm to find a VC of size $$k$$. This should beat the naive algorithm. The current state of the art is something like $$1.24^k n$$.
Under some assumptions there is no algorithm for k clique with running time $$f(k) n^c$$.