Rahul Santhanam

Infeasibility of instance compression and succinct PCPs for NP

The OR-SAT problem asks, given Boolean formulae Φ1,...,Φm each of size at most n, whether at least one of the Φis is satisfiable. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the Polynomial-Time Hierarchy collapses. This result settles an open problem proposed by Bodlaender et. al. [4] and Harnik and Naor [15] and has a number of implications. A number of parametric

Hierarchy theorems for probabilistic polynomial time

\NP

Circuit lower bounds for Merlin-Arthur classes

problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless NP ⊆ coNP/poly. Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP/poly. An approach of Harnik and Naor to constructing collision-resistant hash functions from one-way functions is unlikely to be viable in its present form. (Buhrman-Hitchcock) There are no subexponential-size hard sets for NP unless NP is in co-NP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions.

Graph model selection using maximum likelihood

We show a hierarchy for probabilistic time with one bit of advice, specifically we show that for all real numbers 1 /spl les/ /spl alpha/ /spl les/ /spl beta/, BPTIME(n/sup /spl alpha//)/l /spl sube/ BPTIME(n/sup /spl beta//)/l. This result builds on and improves an earlier hierarchy of Barak using O(log log n) bits of advice. We also show that for any constant d > 0, there is a language L computable on average in BPP but not on average in BPTIME (n/sup d/). We build on Baraks techniques by using a different translation argument and by a careful application of the fact that there is a PSPACE-complete problem L such that worst-case probabilistic algorithms for L take only slightly more time than average-case algorithms.

Hierarchies for semantic classes

We show that for each k> 0, MA/ 1( MA with 1 bit of advice) does not have circuits of size n k . This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM ,a ndZPP NP . We extend our main result in several ways. For each k ,w e give an explicit language in (MA ∩ coMA)/1 which does not have circuits of size nk. We also adapt our lower bound to the average-case setting; i.e., we show that MA/1 cannot be solved on more than 1/ 2+1 /n k fraction of inputs of length n by circuits of size n k . Furthermore, we prove that MA does not have arithmetic circuits of size n k for any k. As a corollary to our main result, we obtain that derandomization of MA/O(1) implies the existence of pseudorandom generators computable using O(1) bits of advice.

Circuit Lower Bounds for Merlin-Arthur Classes

In recent years, there has been a proliferation of theoretical graph models, e.g., preferential attachment and small-world models, motivated by real-world graphs such as the Internet topology. To address the natural question of which model is best for a particular data set, we propose a model selection criterion for graph models. Since each model is in fact a probability distribution over graphs, we suggest using Maximum Likelihood to compare graph models and select their parameters. Interestingly, for the case of graph models, computing likelihoods is a difficult algorithmic task. However, we design and implement MCMC algorithms for computing the maximum likelihood for four popular models: a power-law random graph model, a preferential attachment model, a small-world model, and a uniform random graph model. We hope that this novel use of ML will objectify comparisons between graph models.

Pebbles and Branching Programs for Tree Evaluation

We show that for any constant a, ZPP/b(n) strictly contains ZPTIME(na)/b(n) for some b(n) = O(log n log log n). Our techniques are very general and give the same hierarchy for all common semantic time classes including RTIME, NTIME ∩ coNTIME, UTIME, MATIME, AMTIME and BQTIME.We show a stronger hierarchy for RTIME: For every constant c, RP/1 is not contained in RTIME(nc)/(log n)1/2c. To prove this result we first prove a similar statement for NP by building on Záks proof of the nondeterministic time hierarchy.

On the limits of sparsification

We show that for each

Holographic Proofs and Derandomization

k>0

Improved Algorithms for Sparse MAX-SAT and MAX-k-CSP

,