21 research outputs found
PPSZ is better than you think
PPSZ, for long time the fastest known algorithm for -SAT, works by going
through the variables of the input formula in random order; each variable is
then set randomly to or , unless the correct value can be inferred by an
efficiently implementable rule (like small-width resolution; or being implied
by a small set of clauses).
We show that PPSZ performs exponentially better than previously known, for
all . For Unique--SAT we bound its running time by
, which is somewhat better than the algorithm of Hansen,
Kaplan, Zamir, and Zwick, which runs in time . Before that, the
best known upper bound for Unique--SAT was .
All improvements are achieved without changing the original PPSZ. The core
idea is to pretend that PPSZ does not process the variables in uniformly random
order, but according to a carefully designed distribution. We write "pretend"
since this can be done without any actual change to the algorithm
An approximation algorithm for #k-SAT
"Vegeu el resum a l'inici del document del fitxer adjunt"
An Approximation Algorithm for #k-SAT
We present a simple randomized algorithm that approximates the number of
satisfying assignments of Boolean formulas in conjunctive normal form. To the
best of our knowledge this is the first algorithm which approximates #k-SAT for
any k >= 3 within a running time that is not only non-trivial, but also
significantly better than that of the currently fastest exact algorithms for
the problem. More precisely, our algorithm is a randomized approximation scheme
whose running time depends polynomially on the error tolerance and is mildly
exponential in the number n of variables of the input formula. For example,
even stipulating sub-exponentially small error tolerance, the number of
solutions to 3-CNF input formulas can be approximated in time O(1.5366^n). For
4-CNF input the bound increases to O(1.6155^n).
We further show how to obtain upper and lower bounds on the number of
solutions to a CNF formula in a controllable way. Relaxing the requirements on
the quality of the approximation, on k-CNF input we obtain significantly
reduced running times in comparison to the above bounds
Quantum Speed-ups for Boolean Satisfiability and Derivative-Free Optimization
In this thesis, we have considered two important problems, Boolean satisfiability (SAT) and derivative free optimization in the context of large scale quantum computers. In the first part, we survey well known classical techniques for solving satisfiability. We compute the approximate time it would take to solve SAT instances using quantum techniques and compare it with state-of-the heart classical heuristics employed annually in SAT competitions. In the second part of the thesis, we consider a few classically well known algorithms for derivative free optimization which are
ubiquitously employed in engineering problems. We propose a quantum speedup to this classical algorithm by using techniques of the quantum minimum finding algorithm. In the third part of the thesis, we consider practical applications in the fields of bio-informatics, petroleum refineries and civil engineering which involve solving either satisfiability or derivative free optimization. We investigate if using known quantum techniques to speedup these algorithms directly translate to
the benefit of industries which invest in technology to solve these problems. In the last section, we propose a few open problems which we feel are immediate hurdles, either from an algorithmic or architecture perspective to getting a convincing speedup for the practical problems considered
Short PCPs with projection queries
We construct a PCP for NTIME(2 n) with constant soundness, 2 n poly(n) proof length, and poly(n) queries where the verifier’s computation is simple: the queries are a projection of the input randomness, and the computation on the prover’s answers is a 3CNF. The previous upper bound for these two computations was polynomial-size circuits. Composing this verifier with a proof oracle increases the circuit-depth of the latter by 2. Our PCP is a simple variant of the PCP by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (CCC 2005). We also give a more modular exposition of the latter, separating the combinatorial from the algebraic arguments. If our PCP is taken as a black box, we obtain a more direct proof of the result by Williams, later with Santhanam (CCC 2013) that derandomizing circuits on n bits from a class C in time 2 n /n ω(1) yields that NEXP is not in a related circuit class C ′. Our proof yields a tighter connection: C is an And-Or of circuits from C ′. Along the way we show that the same lower bound follows if the satisfiability of the And of any 3 circuits from C ′ can be solved in time 2 n /n ω(1). ∗The research leading to these results has received funding from the European Community’