Fastest algorithm for finding the longest palindrome subsequence
There's an algorithm named after Manacher's algorithm, which is really fast, a linear time algorithm.

See Wikipedia's reference



Postscript: If you're really familiar with Z Algorithm, you will find that they're alike.



Edit

I've just misunderstood the OP's meaning (but I don't want to delete the proceding information. It's somewhat useful). He means the longest palindrome subsequence of a string, so dynamic programming seems good:
\begin{align*}
f_{j,k}&amp;=\max(f_{j,k+1},f_{j+1,k},2[S_j=S_k]+f_{j+1,k-1}),\qquad j&lt;k\\
f_{k,k}&amp;=1\\
f_{j,k}&amp;=0,\qquad j&gt;k\\
\end{align*}
where $f_{j,k}$ denotes the length of the longgest palindrome subsequence of $S_{j..k}$, and $[P]$ is Iverson bracket I think it's just like LCS.