The study of Succinct representation of graphs was initiated by Galperin and Wigderson in a paper from 1983, where they prove that for many simple problems like finding a triangle in a graph, the corresponding succinct version in $\mathsf{NP}$-complete. Papadimitriou and Yanakkakis further this line of research, and prove that for a problem $\Pi$ which is $\mathsf{NP}$-complete/$\mathsf{P}$-complete, the corresponding Succinct version, namely Succinct $\Pi$ is respectively, $\mathsf{NEXP}$-complete and $\mathsf{EXP}$-complete. (They also show that if $\Pi$ is $\mathsf{NL}$-complete, then Succinct $\Pi$ is $\mathsf{PSPACE}$-complete. Now my question is, are there any problems $\Pi$ known for which, the corresponding Succinct version is in $\mathsf{P}$? I'd be interested in knowing about any other related results(both positive and impossibility results, if any) which I might have missed above. (I couldn't locate anything of interest by a google search, since search words like succinct, representation, problems, graphslead to just almost any complexity result! :))
Succinct Problems in $\mathsf{P}$
cc.complexity theory;reference request;complexity classes
Here is an interesting problem whose succinct version has interesting properties. Define Circuit-Size-$2^{n/2}$ to be the problem: given a Boolean function as a $2^n$-bit string, does this function have a circuit of size at most $2^{n/2}$?  Note this problem is in $NP$.One way to define Succinct-Circuit-Size-$2^{n/2}$ would be: for a constant $k$, given an $n$-input, $n^k$-size circuit $C$, we want to know if its truth table is an instance of Circuit-Size-$2^{n/2}$. But this is a trivial problem: all inputs which are actual circuits are yes-instances. So this problem is in $P$.A more general way to define Succinct-Circuit-Size-$2^{n/2}$ would be: we are given an arbitrary circuit $C$ and want to know if its truth table encodes an instance of Circuit-Size-$2^{n/2}$. But if $n$ is the number of inputs to $C$, $m$ is the size of $C$, and $m \leq 2^{n/2}$, we can automatically accept: the input itself is a witness for the language. Otherwise, we have $m \geq 2^{n/2}$. In that case, the input length $m$ is already huge, so we can try all possible $2^n$ assignments in $m^{O(1)}$ time, obtain the truth table of the function, and now we are back to the original $NP$ problem again. So this is a problem in $NP$ whose succinct version is also in $NP$. This problem is believed to be not $NP$-hard; see the paper by Kabanets and Cai (http://www.cs.sfu.ca/~kabanets/Research/circuit.html)