Alexander E. Andreev

A new general derandomization method

We show that <italic>quick hitting set generators</italic> can replace <italic>quick pseudorandom generators</italic> to derandomize any probabilistic <italic>two-sided error</italic> algorithms. Up to now <italic>quick hitting set generators</italic> have been known as the general and uniform derandomization method for probabilistic <italic>one-sided error</italic> algorithms, while <italic>quick pseudorandom generators</italic> as the generators as the general and uniform method to derandomize probabilistic <italic>two-sided error</italic> algorithms. Our method is based on a deterministic algorithm that, given a Boolean circuit <italic>C</italic> and given access to a hitting set generator, constructs a <italic>discrepancy set</italic> for <italic>C</italic>. The main novelty is that the discrepancy set depends on <italic>C</italic>, so the new derandomization method is not uniform (i.e., not <italic>oblivious</italic>). The algorithm works in time exponential in <italic>k(p(n))</italic> where <italic>k</italic>(*) is the <italic>price</italic> of the hitting set generator and <italic>p</italic>(*) is a polynomial function in the size of <italic>C</italic>. We thus prove that if a logarithmic price quick hitting set generator exists then BPP = P.

Hitting Sets Derandomize BPP

We show that hitting sets can derandomize any probabilistic, two-sided error algorithm. This gives a positive answer to a fundamental open question in probabilistic algorithms. More precisely, we present a polynomial time deterministic algorithm which uses any given hitting set to approximate the fractions of 1s in the output of any boolean circuit of polynomial size. This new algorithm implies that if a quick hitting set generator with logarithmic price exists then BPP = P. Furthermore, we generalize this result by showing that the existence of a quick hitting set generator with price k implies that BPTIME(t) \(\subseteq DTIME(2^{O(k(t^{O(1)} ))} )\). The existence of quick hitting set generators is thus a new weaker sufficient condition to obtain BPP = P; this can be considered as another strong indication that the gap between probabilistic and deterministic computational power is not large.

Worst-case hardness suffices for derandomization: a new method for hardness-randomness trade-offs

Abstract Up to now, the known derandomization methods for BPP have been derived assuming the existence of an EXP function that has a “hard” average-case circuit complexity. In this paper we instead present the first construction of a de-randomization method for BPP that relies on the existence of an EXP function that is hard only in the worst-case. The construction is based on a new method that departs significantly from the usual known methods based on pseudo-random generators. Indeed, we prove new particular bounds on the circuit complexity of partial Boolean functions which are then used to derive efficient constructions of hitting set generators. As recently proved, such generators can derandomize any BPP-algorithm. Our method is efficiently parallelizable and hence yields also a hardness condition on the worst-case circuit complexity of Boolean operators running in NC which is sufficient to obtain NC = BPNC.

Weak random sources, hitting sets, and BPP simulations

We show how to simulate any BPP algorithm in polynomial time using a weak random source of min-entropy r/sup /spl gamma// for any /spl gamma/>0. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP (Saks et al., 1995) and a n(log/sup (k)/n)-time simulation of BPP for fixed k (Ta-Shma, 1996). Departing significantly from previous related works, we do not use extractors; instead we use the OR-disperser of (Saks et al., 1995) in combination with a tricky use of hitting sets borrowed from Andreev et al. (1996). Of independent interest is our new (simplified) proof of the main result of Andreev et al., (1996). Our proof also gives some new hardness/randomness trade-offs for parallel classes.

A Deciding Algorithm for Linear Isomorphism of Types with Complexity O (n log2(n))

It is known, that ordinary isomorphisms (associativity and commutativity of “times”, isomorphisms for “times” unit and currying) provide a complete axiomatisation of isomorphism of types in multiplicative linear lambda calculus (isomorphism of objects in a free symmetric monoidal closed category). One of the reasons to consider linear isomorphism of types instead of ordinary isomorphism was that better complexity could be expected. Meanwhile, no upper bounds reasonnably close to linear were obtained. We describe an algorithm deciding if two types are linearly isomorphic with complexity O(nlog 2(n)).

The parallel complexity of approximating the high degree subgraph problem

Abstract The high degree subgraph problem is to find a subgraph H of a graph G such that the minimum degree of H is as large as possible. This problem is known to be P-hard so that parallel approximation algorithms are very important for it. Our first goal is to determine how effectively the approximation algorithm based on a well-known extremal graph result parallelizes. In particular, we show that two natural decision problems associated with this algorithm are P-complete: these results suggest that the parallel implementation of the algorithm itself requires more sophisticated techniques. Successively, we study the high degree subgraph problem for random graphs with any edge probability function and we provide different parallel approximation algorithms depending on the type of this function.

Efficient Construction of Hitting Sets for Systems of Linear Functions

Given a positive number δ ∈ (0,1), a subset H ⊑ {0,1}n is a δ-Hitting Set for a class R of boolean functions with n inputs if, for any function f ∈ R such that Pr (f=1)≥δ, there exists an element h ∈ H such that f(h)=1. Our paper presents a new deterministic method to efficiently construct δ-Hitting Set for the class of systems (i.e. logical conjunctions) of boolean linear functions. Systems of boolean linear functions can be considered as the algebraic generalization of boolean combinatorial rectangular functions, the only significative example for which an efficient deterministic construction of Hitting Sets were previously known. In the restricted case of boolean rectangular functions, our method (even though completely different) achieves equivalent results to those obtained in [11]. Our results also gives an upper bound on the minimum cardinality of solution covers for the class of systems of linear equations defined over a finite field. Furthermore, as preliminary result, we show a new upper bound on the circuit complexity of integer monotone functions generalizing previous results obtained in [12].

Constructing the highest degree subgraph for dense graphs in NCAL

Abstract In this paper, we first show that the Highest Degree Subgraph problem remains P -complete for dense graphs (i.e. when m = Ω( n 2 poly log n ) ). This hardness result gives a clear motivation in studying the approximability of the Highest Degree Problem even for this restricted case. We then provide an N b -approximation scheme computing approximate solutions for dense graphs, thus proving that, in this case, the problem belongs to the N b A S class.

Note: Very large cliques are easy to detect

It is known that, for every constant

Memory Organization Schemes for Large Shared Data: A Randomized Solution for Distributed Memory Machines

kgeq 3