149 research outputs found
Unsatisfiable Linear CNF Formulas Are Large and Complex
We call a CNF formula linear if any two clauses have at most one variable in
common. We show that there exist unsatisfiable linear k-CNF formulas with at
most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at
most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic
means, and we have no explicit construction coming even close to it. One reason
for this is that unsatisfiable linear formulas exhibit a more complex structure
than general (non-linear) formulas: First, any treelike resolution refutation
of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$.
This implies that small unsatisfiable linear k-CNF formulas are hard instances
for Davis-Putnam style splitting algorithms. Second, if we require that the
formula F have a strict resolution tree, i.e. every clause of F is used only
once in the resolution tree, then we need at least a^a^...^a clauses, where a
is approximately 2 and the height of this tower is roughly k.Comment: 12 pages plus a two-page appendix; corrected an inconsistency between
title of the paper and title of the arxiv submissio
Strictly monotone and smooth nonparametric regression for two or more variables
In this article a new monotone nonparametric estimate for a regression function of two or more variables is proposed. The method starts with an unconstrained nonparametric regression estimate and uses successively one-dimensional isotonization procedures. In the case of a strictly monotone regression function, it is shown that the new estimate is first order asymptotic equivalent to the unconstrained estimate, and asymptotic normality of an appropriate standardization of the estimate is established. Moreover, if the regression function is not monotone in one of its arguments, the constructed estimate has approximately the same Lp-norm as the initial unconstrained estimate. The methodology is also illustrated by means of a simulation study, and two data examples are analyzed. --multivariate nonparametric regression,isotonic regression,order restricted inference,nondecreasing rearrangement
A finite sample comparison of nonparametric estimates of the effective dose in quantal bioassay
To estimate the effective dose level ED a in the common binary response model, several parametric and nonparametric estimators have been proposed in the literature. In the present paper, we focus on nonparametric methods and present a detailed numerical comparison of four different approaches to estimate the ED a nonparametrically. The methods are briefly reviewed and their finite sample properties are studied by means of a detailed simulation study. Moreover, a data example is presented to illustrate the different concepts. --Binary response model,effective dose level,nonparametric regression,isotonic regression,order restricted inference,local linear regression
Satisfiability of Almost Disjoint CNF Formulas
We call a CNF formula linear if any two clauses have at most one variable in
common. Let m(k) be the largest integer m such that any linear k-CNF formula
with <= m clauses is satisfiable. We show that 4^k / (4e^2k^3) <= m(k) < ln(2)
k^4 4^k. More generally, a (k,d)-CSP is a constraint satisfaction problem in
conjunctive normal form where each variable can take on one of d values, and
each constraint contains k variables and forbids exacty one of the d^k possible
assignments to these variables. Call a (k,d)-CSP l-disjoint if no two distinct
constraints have l or more variables in common. Let m_l(k,d) denote the largest
integer m such that any l-disjoint (k,d)-CSP with at most m constraints is
satisfiable. We show that 1/k (d^k/(ed^(l-1)k))^(1+1/(l-1))<= m_l(k,d) < c
(k^2/l ln(d) d^k)^(1+1/(l-1)). for some constant c. This means for constant l,
upper and lower bound differ only in a polynomial factor in d and k
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
A finite sample comparison of nonparametric estimates of the effective dose in quantal bioassay
To estimate the effective dose level ED in the common binary response model, several
parametric and nonparametric estimators have been proposed in the literature. In the present paper, we focus on nonparametric methods and present a detailed numerical comparison of four different approaches to estimate the ED nonparametrically. The methods are briefly reviewed and their finite sample properties are studied by means of a detailed simulation study. Moreover, a data example is presented to illustrate the different concepts
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