## The program-enumeration bottleneck in average-case complexity theory

by Luca Trevisan

These three theorems of Levin are among my favorites. I note that, while they are often cricitized for the overhead incurred by the enumeration, this is a secondary consideration that arises only one understand their proofs; that is, beforehand, one would probably consider their statement to be bluntly wrong (or at least very challenging). (Indeed, clever researchers tend to forget the state of mind they were in before understanding a simple proof, and tell themselves that they gained nothing from the theorem -- but this comment belongs to my opinion page on What's wrong with STOC/FOCS?)

Anyhow, I always had a feeling that the effect of enumeration is unavoidable in these results, but I guess I was not brave enough to try to articulate this feeling (let alone instantiate it). Note that the same may be true also with respect to the Cook-Levin Theorem (i.e., NP-completeness of SAT or of Bounded Halting), but it is not clear how to articulate this in that content. Indeed, the reduction to pathological sub-cases that goes "unpunished" in worst-case complexity, has a clear cost in the context of average-case complexity, which is indeed a satisfying feature of that theory.

#### The original abstract

Three fundamental results of Levin involve algorithms or reductions whose running time is exponential in the length of certain programs. We study the question of whether such dependency can be made polynomial.

1. Levin's ``optimal search algorithm'' performs at most a constant factor more slowly than any other fixed algorithm. The constant, however, is exponential in the length of the competing algorithm.
We note that the running time of a universal search cannot be made ``fully polynomial'' (that is, the relation between slowdown and program length cannot be made polynomial), unless P=NP.
2. Levin's ``universal one-way function'' result has the following structure: there is a polynomial time computable function fLevin such that if there is a polynomial time computable adversary A that inverts fLevin on an inverse polynomial fraction of inputs, then for every polynomial time computable function g there also is a polynomial time adversary Ag that inverts g on an inverse polynomial fraction of inputs. Unfortunately, again the running time of Ag depends exponentially on the bit length of the program that computes g in polynomial time.
We show that a fully polynomial uniform reduction from an arbitrary one-way function to a specific one-way function is not possible relative to an oracle that we construct, and so no ``universal one-way function'' can have a fully polynomial security analysis via relativizing techniques.
3. Levin's completeness result for distributional NP problems implies that if a specific problem in NP is easy on average under the uniform distribution, then every language L in NP is also easy on average under any polynomial time computable distribution. The running time of the implied algorithm for L, however, depends exponentially on the bit length of the non-deterministic polynomial time Turing machine that decides L.
We show that if a completeness result for distributional NP can be proved via a ``fully uniform'' and ``fully polynomial'' time reduction, then there is a worst-case to average-case reduction for NP-complete problems. In particular, this means that a fully polynomial completeness result for distributional NP is impossible, even via randomized truth-table reductions, unless the polynomial hierarchy collapses.

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