Derandomizing from Random Strings

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings

Derandomizing Arthur-Merlin Games using Hitting Sets

We prove that AM (and hence Graph Nonisomorphism) is in NPif for some epsilon > 0, some language in NE intersection coNE requires nondeterministiccircuits of size 2^(epsilon n). This improves recent results of Arvindand K¨obler and of Klivans and Van Melkebeek who proved the sameconclusion, but under stronger hardness assumptions, namely, eitherthe existence of a language in NE intersection coNE which cannot be approximatedby nondeterministic circuits of size less than 2^(epsilon n) or the existenceof a language in NE intersection coNE which requires oracle circuits of size 2^(epsilon n)with oracle gates for SAT (satisfiability).The previous results on derandomizing AM were based on pseudorandomgenerators. In contrast, our approach is based on a strengtheningof Andreev, Clementi and Rolim's hitting set approach to derandomization.As a spin-off, we show that this approach is strong enoughto give an easy (if the existence of explicit dispersers can be assumedknown) proof of the following implication: For some epsilon > 0, if there isa language in E which requires nondeterministic circuits of size 2^(epsilon n),then P=BPP. This differs from Impagliazzo and Wigderson's theorem"only" by replacing deterministic circuits with nondeterministicones

Graph Isomorphism and the Lasserre Hierarchy

In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to expf O(n^(1/5) [5]). It has since been an open question to remove the requirement that the graph be strongly regular. Recent algorithmic results show that for many problems the Lasserre hierarchy works surprisingly well when the underlying graph has expansion properties. Moreover, recent work of Atserias and Maneva [3] shows that k rounds of the Lasserre hierarchy is a generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph Isomorphism. These two facts combined make the Lasserre hierarchy a good candidate for solving graph isomorphism on expander graphs. Our main result rules out this promising direction by showing that even Omega(n) rounds of the Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism problem even on expander graphs.Comment: 22 pages, 3 figures, submitted to CC

On Transformations of Interactive Proofs that Preserve the Prover's Complexity

Goldwasser and Sipser [GS89] proved that every interactive proof system can be transformed into a public-coin one (a.k.a., an Arthur-Merlin game). Their transformation has the drawback that the computational complexity of the prover's strategy is not preserved. We show that this is inherent, by proving that the same must be true of any transformation which only uses the original prover and verifier strategies as "black boxes". Our negative result holds even if the original proof system is restricted to be honest-verifier perfect zero knowledge and the transformation can also use the simulator as a black box. We also examine a similar deficiency in a transformation of Fürer et al. [FGM+89] from interactive proofs to ones with perfect completeness. We argue that the increase in prover complexity incurred by their transformation is necessary, given that their construction is a black-box transformation which works regardless of the verifier's computational complexity.Engineering and Applied Science

On optimal language compression for sets in PSPACE/poly

We show that if DTIME[2^O(n)] is not included in DSPACE[2^o(n)], then, for every set B in PSPACE/poly, all strings x in B of length n can be represented by a string compressed(x) of length at most log(|B^{=n}|)+O(log n), such that a polynomial-time algorithm, given compressed(x), can distinguish x from all the other strings in B^{=n}. Modulo the O(log n) additive term, this achieves the information-theoretic optimum for string compression. We also observe that optimal compression is not possible for sets more complex than PSPACE/poly because for any time-constructible superpolynomial function t, there is a set A computable in space t(n) such that at least one string x of length n requires compressed(x) to be of length 2 log(|A^=n|).Comment: submitted to Theory of Computing System

Nonuniform Reductions and NP-Completeness

Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP0completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP. Under various hypotheses, we obtain the following separations: 1. There is a set complete for NP under nonuniform many-one reductions, but not under uniform many-one reductions. This is true even with a single bit of nonuniform advice. 2. There is a set complete for NP under nonuniform many-one reductions with polynomial-size advice, but not under uniform Turing reductions. That is, polynomial nonuniformity is stronger than a polynomial number of queries. 3. For any fixed polynomial p(n), there is a set complete for NP under uniform 2-truth-table reductions, but not under nonuniform many-one reductions that use p(n) advice. That is, giving a uniform reduction a second query makes it more powerful than a nonuniform reduction with fixed polynomial advice. 4. There is a set complete for NP under nonuniform many-one reductions with polynomial ad- vice, but not under nonuniform many-one reductions with logarithmic advice. This hierarchy theorem also holds for other reducibilities, such as truth-table and Turing. We also consider uniform upper bounds on nonuniform completeness. Hirahara (2015) showed that unconditionally every set that is complete for NP under nonuniform truth-table reductions that use logarithmic advice is also uniformly Turing-complete. We show that under a derandomization hypothesis, the same statement for truth-table reductions and truth-table completeness also holds

Homotopy equivalences between p-subgroup categories

Let p be a prime number and G a finite group of order divisible by p. Quillen showed that the Brown poset of nonidentity p-subgroups of G is homotopy equivalent to its subposet of nonidentity elementary abelian subgroups. We show here that a similar statement holds for the fusion category of nonidentity p-subgroups of G. Other categories of p-subgroups of G are also considered.Comment: 19 pages. Second versio