Greedy Best First Search
There are a few ways to modulate the heuristic function to different effect such as in weighted A*. Another way is to totally discard the g-cost in the computation of the f-value and use only the h-value to guide the search.
This algorithm is complete, but sub-optimal. It could find an optimal solution but it is not guaranteed to.
\[
f(n) = h(n)
\]