Hardness of Approximation of MAX-3SAT
I believe that Håstad gave such an algorithm in 2000, see On bounded occurrence constraint satisfaction. His algorithm is stated for instances where each variable occurs at most $B$ times, and the advantage over 7/8 depends on $B$. The running time, however, seems not to depend on $B$.

In order to prove that P=NP, you need to find an algorithm which, given a satisfiable instance, satisfies at least $7/8+\epsilon$ clauses, for some constant $\epsilon&gt;0$. This will enable you to distinguish between satisfiable instances and instances which are at most $7/8+\epsilon/2$ satisfiable, which is NP-hard as my answer to the question you mention shows.