20,767 research outputs found
Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover
Stochastic Boolean Function Evaluation is the problem of determining the
value of a given Boolean function f on an unknown input x, when each bit of x_i
of x can only be determined by paying an associated cost c_i. The assumption is
that x is drawn from a given product distribution, and the goal is to minimize
the expected cost. This problem has been studied in Operations Research, where
it is known as "sequential testing" of Boolean functions. It has also been
studied in learning theory in the context of learning with attribute costs. We
consider the general problem of developing approximation algorithms for
Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for
evaluating Boolean linear threshold formulas. We also present an approximation
algorithm for evaluating CDNF formulas (and decision trees) achieving a factor
of O(log kd), where k is the number of terms in the DNF formula, and d is the
number of clauses in the CNF formula. In addition, we present approximation
algorithms for simultaneous evaluation of linear threshold functions, and for
ranking of linear functions.
Our function evaluation algorithms are based on reductions to the Stochastic
Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and
Krause. They presented an approximation algorithm for the problem, called
Adaptive Greedy. Our main technical contribution is a new approximation
algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an
extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito,
which is a generalization of Hochbaum's algorithm for the classical Set Cover
Problem. We also give a new bound on the approximation achieved by the Adaptive
Greedy algorithm of Golovin and Krause
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits
For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. In 2006,
Gopalan et al. studied connectivity properties of the solution graph and
related complexity issues for CSPs, motivated mainly by research on
satisfiability algorithms and the satisfiability threshold. They proved
dichotomies for the diameter of connected components and for the complexity of
the st-connectivity question, and conjectured a trichotomy for the connectivity
question. Recently, we were able to establish the trichotomy [arXiv:1312.4524].
Here, we consider connectivity issues of satisfiability problems defined by
Boolean circuits and propositional formulas that use gates, resp. connectives,
from a fixed set of Boolean functions. We obtain dichotomies for the diameter
and the two connectivity problems: on one side, the diameter is linear in the
number of variables, and both problems are in P, while on the other side, the
diameter can be exponential, and the problems are PSPACE-complete. For
partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement
Spin Glasses, Boolean Satisfiability, and Survey Propagation
In recent years statistical physics and computational complexity have found mutually interesting subjects of research. The theory of spin glasses from statistical physics has been successfully applied to the boolean satisfiability problem, which is the canonical topic of computational complexity.
The study of spin glasses originated from experimental studies of the magnetic properties of impure metallic alloys, but soon the study of the theoretical models outshone the interest in the experimental systems. The model studied in this thesis is that of Ising spins with random interactions. In this thesis we discuss two analytical derivations on spin glasses: the famous replica trick on the Sherrington-Kirkpatrick model and the cavity method on a Bethe lattice spin glass.
Computational complexity theory is a branch of theoretical computer science that studies how the running time of algorithms scales with the size of the input. Two important classes of algorithms or problems are P and NP, or colloquially easy and hard problems. The first problem to be proven to belong to the class of NP-complete problems is that of boolean satisfiability, i.e., the study of whether there is an assignment of variables for a random boolean formula so that the formula is satisfied. The boolean satisfiability problem can be tackled with spin glass theory; the cavity method can be applied to it.
Boolean satisfiability exhibits a phase transition. As one increases the ratio of constraints to variables the probability of a random formula being satisfiable drops from unity to zero. This transition of random formulas from satisfiable to unsatisfiable is continuous for small formulas. It grows sharper with increasing problem size and becomes discrete at the limit of an infinite number of variables. The cavity method gives a value for the location of the phase transition that is in agreement with the numerical value.
The cavity method is an analytical tool for studying average values over a distribution, but it introduces so called surveys that can also be calculated numerically for a single instance. These surveys inspire the survey propagation algorithm that is implemented as a numerical program to efficiently solve large instances of random boolean satisfiability problems.
In this thesis I present a parallel version of survey propagation that achieves a speedup by a factor of 3 with 4 processors. With the improved version we are able to gain further knowledge on the detailed workings of survey propagation. It is found, firstly, that the number of iterations needed for one convergence of survey propagation depends on the number of variables, seemingly as ln(N). Secondly, it is found that the constraint to variable ratio for which survey propagation succeeds is dependent on the number of variables
Understanding the complexity of #SAT using knowledge compilation
Two main techniques have been used so far to solve the #P-hard problem #SAT.
The first one, used in practice, is based on an extension of DPLL for model
counting called exhaustive DPLL. The second approach, more theoretical,
exploits the structure of the input to compute the number of satisfying
assignments by usually using a dynamic programming scheme on a decomposition of
the formula. In this paper, we make a first step toward the separation of these
two techniques by exhibiting a family of formulas that can be solved in
polynomial time with the first technique but needs an exponential time with the
second one. We show this by observing that both techniques implicitely
construct a very specific boolean circuit equivalent to the input formula. We
then show that every beta-acyclic formula can be represented by a polynomial
size circuit corresponding to the first method and exhibit a family of
beta-acyclic formulas which cannot be represented by polynomial size circuits
corresponding to the second method. This result shed a new light on the
complexity of #SAT and related problems on beta-acyclic formulas. As a
byproduct, we give new handy tools to design algorithms on beta-acyclic
hypergraphs
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