101,394 research outputs found
Positive Versions of Polynomial Time
AbstractWe show that restricting a number of characterizations of the complexity classPto be positive (in natural ways) results in the same class of (monotone) problems, which we denote byposP. By a well-known result of Razborov,posPis a proper subclass of the class of monotone problems inP. We exhibit complete problems forposPvia weak logical reductions, as we do for other logically defined classes of problems. Our work is a continuation of research undertaken by Grigni and Sipser, and subsequently Stewart; indeed, we introduce the notion of a positive deterministic Turing machine and consequently solve a problem posed by Grigni and Sipser
Interpolation in Valiant's theory
We investigate the following question: if a polynomial can be evaluated at
rational points by a polynomial-time boolean algorithm, does it have a
polynomial-size arithmetic circuit? We argue that this question is certainly
difficult. Answering it negatively would indeed imply that the constant-free
versions of the algebraic complexity classes VP and VNP defined by Valiant are
different. Answering this question positively would imply a transfer theorem
from boolean to algebraic complexity. Our proof method relies on Lagrange
interpolation and on recent results connecting the (boolean) counting hierarchy
to algebraic complexity classes. As a byproduct we obtain two additional
results: (i) The constant-free, degree-unbounded version of Valiant's
hypothesis that VP and VNP differ implies the degree-bounded version. This
result was previously known to hold for fields of positive characteristic only.
(ii) If exponential sums of easy to compute polynomials can be computed
efficiently, then the same is true of exponential products. We point out an
application of this result to the P=NP problem in the Blum-Shub-Smale model of
computation over the field of complex numbers.Comment: 13 page
Computational complexity of decomposing a symmetric matrix as a sum of positive semidefinite and diagonal matrices
We study several variants of decomposing a symmetric matrix into a sum of a
low-rank positive semidefinite matrix and a diagonal matrix. Such
decompositions have applications in factor analysis and they have been studied
for many decades. On the one hand, we prove that when the rank of the positive
semidefinite matrix in the decomposition is bounded above by an absolute
constant, the problem can be solved in polynomial time. On the other hand, we
prove that, in general, these problems as well as their certain approximation
versions are all NP-hard. Finally, we prove that many of these low-rank
decomposition problems are complete in the first-order theory of the reals;
i.e., given any system of polynomial equations, we can write down a low-rank
decomposition problem in polynomial time so that the original system has a
solution iff our corresponding decomposition problem has a feasible solution of
certain (lowest) rank
Polynomial Kernels for Weighted Problems
Kernelization is a formalization of efficient preprocessing for NP-hard
problems using the framework of parameterized complexity. Among open problems
in kernelization it has been asked many times whether there are deterministic
polynomial kernelizations for Subset Sum and Knapsack when parameterized by the
number of items.
We answer both questions affirmatively by using an algorithm for compressing
numbers due to Frank and Tardos (Combinatorica 1987). This result had been
first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We
further illustrate its applicability by giving polynomial kernels also for
weighted versions of several well-studied parameterized problems. Furthermore,
when parameterized by the different item sizes we obtain a polynomial
kernelization for Subset Sum and an exponential kernelization for Knapsack.
Finally, we also obtain kernelization results for polynomial integer programs
Open problem in orthogonal polynomials
Using an algebraic method for solving the wave equation in quantum mechanics,
we encountered a new class of orthogonal polynomials on the real line. It
consists of a four-parameter polynomial with continuous spectrum on the whole
real line and two of its discrete versions; one with a finite spectrum and
another with countably infinite spectrum. A second class of these new
orthogonal polynomials appeared recently while solving a Heun-type equation.
Based on these results and on our recent study of the solution space of an
ordinary differential equation of the second kind with four singular points, we
introduce a modification of the Askey scheme of hyper-geometric orthogonal
polynomials. Up to now, these polynomials are defined by their three-term
recursion relations and initial values. However, their other properties like
the weight functions, generating functions, orthogonality, Rodrigues-type
formulas, etc. are yet to be derived analytically. Due to the prime
significance of these polynomials in physics and mathematics, we call upon
experts in the field of orthogonal polynomials to study them, derive their
properties and write them in closed form (e.g., in terms of hypergeometric
functions).Comment: 7 pages, 1 table, 17 reference
Space complexity in polynomial calculus
During the last decade, an active line of research in proof complexity has been to study space
complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of
intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused
on weak systems that are used by SAT solvers.
There has been a relatively long sequence of papers on space in resolution, which is now reasonably
well understood from this point of view. For other natural candidates to study, however, such as
polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial
space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been
for CNF formulas of unbounded width in [Alekhnovich et al. ’02], where the space lower bound is
smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent
with current knowledge that polynomial calculus could be able to refute any k-CNF formula in
constant space.
In this paper, we prove several new results on space in polynomial calculus (PC), and in the
extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. ’02]:
1. We prove an Ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole
principle formulas PHPm
n with m pigeons and n holes, and show that this is tight.
2. For PCR, we prove an Ω(n) space lower bound for a bitwise encoding of the functional pigeonhole
principle. These formulas have width O(log n), and hence this is an exponential
improvement over [Alekhnovich et al. ’02] measured in the width of the formulas.
3. We then present another encoding of the pigeonhole principle that has constant width, and
prove an Ω(n) space lower bound in PCR for these formulas as well.
4. Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential
size and linear space (which holds for resolution and thus for PCR, but was not obviously
the case for PC). We also characterize a natural class of CNF formulas for which the space
complexity in resolution and PCR does not change when the formula is transformed into 3-CNF
in the canonical way, something that we believe can be useful when proving PCR space lower
bounds for other well-studied formula families in proof complexity
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