20 research outputs found
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
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure
Improved Bounds for Sampling Solutions of Random CNF Formulas
Let be a random -CNF formula on variables and clauses,
where each clause is a disjunction of literals chosen independently and
uniformly. Our goal is to sample an approximately uniform solution of
(or equivalently, approximate the partition function of ).
Let be the density. The previous best algorithm runs in time
for any [Galanis,
Goldberg, Guo, and Yang, SIAM J. Comput.'21]. Our result significantly improves
both bounds by providing an almost-linear time sampler for any
.
The density captures the \emph{average degree} in the random
formula. In the worst-case model with bounded \emph{maximum degree}, current
best efficient sampler works up to degree bound [He, Wang, and Yin,
FOCS'22 and SODA'23], which is, for the first time, superseded by its
average-case counterpart due to our bound. Our result is the first
progress towards establishing the intuition that the solvability of the
average-case model (random -CNF formula with bounded average degree) is
better than the worst-case model (standard -CNF formula with bounded maximal
degree) in terms of sampling solutions.Comment: 51 pages, all proofs added, and bounds slightly improve
Counting Solutions to Random CNF Formulas
We give the first efficient algorithm to approximately count the number of
solutions in the random -SAT model when the density of the formula scales
exponentially with . The best previous counting algorithm was due to
Montanari and Shah and was based on the correlation decay method, which works
up to densities , the Gibbs uniqueness threshold
for the model. Instead, our algorithm harnesses a recent technique by Moitra to
work for random formulas. The main challenge in our setting is to account for
the presence of high-degree variables whose marginal distributions are hard to
control and which cause significant correlations within the formula