7,337 research outputs found
Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity
This paper is motivated by questions such as P vs. NP and other questions in
Boolean complexity theory. We describe an approach to attacking such questions
with cohomology, and we show that using Grothendieck topologies and other ideas
from the Grothendieck school gives new hope for such an attack.
We focus on circuit depth complexity, and consider only finite topological
spaces or Grothendieck topologies based on finite categories; as such, we do
not use algebraic geometry or manifolds.
Given two sheaves on a Grothendieck topology, their "cohomological
complexity" is the sum of the dimensions of their Ext groups. We seek to model
the depth complexity of Boolean functions by the cohomological complexity of
sheaves on a Grothendieck topology. We propose that the logical AND of two
Boolean functions will have its corresponding cohomological complexity bounded
in terms of those of the two functions using ``virtual zero extensions.'' We
propose that the logical negation of a function will have its corresponding
cohomological complexity equal to that of the original function using duality
theory. We explain these approaches and show that they are stable under
pullbacks and base change. It is the subject of ongoing work to achieve AND and
negation bounds simultaneously in a way that yields an interesting depth lower
bound.Comment: 70 pages, abstract corrected and modifie
Graph Isomorphism and the Lasserre Hierarchy
In this paper we show lower bounds for a certain large class of algorithms
solving the Graph Isomorphism problem, even on expander graph instances.
Spielman [25] shows an algorithm for isomorphism of strongly regular expander
graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to
expf O(n^(1/5) [5]). It has since been an open question to remove the
requirement that the graph be strongly regular. Recent algorithmic results show
that for many problems the Lasserre hierarchy works surprisingly well when the
underlying graph has expansion properties. Moreover, recent work of Atserias
and Maneva [3] shows that k rounds of the Lasserre hierarchy is a
generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph
Isomorphism. These two facts combined make the Lasserre hierarchy a good
candidate for solving graph isomorphism on expander graphs. Our main result
rules out this promising direction by showing that even Omega(n) rounds of the
Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism
problem even on expander graphs.Comment: 22 pages, 3 figures, submitted to CC
Topology of two-connected graphs and homology of spaces of knots
We propose a new method of computing cohomology groups of spaces of knots in
, , based on the topology of configuration spaces and
two-connected graphs, and calculate all such classes of order As a
byproduct we define the higher indices, which invariants of knots in
define at arbitrary singular knots. More generally, for any finite-order
cohomology class of the space of knots we define its principal symbol, which
lies in a cohomology group of a certain finite-dimensional configuration space
and characterizes our class modulo the classes of smaller filtration
Inductive queries for a drug designing robot scientist
It is increasingly clear that machine learning algorithms need to be integrated in an iterative scientific discovery loop, in which data is queried repeatedly by means of inductive queries and where the computer provides guidance to the experiments that are being performed. In this chapter, we summarise several key challenges in achieving this integration of machine learning and data mining algorithms in methods for the discovery of Quantitative Structure Activity Relationships (QSARs). We introduce the concept of a robot scientist, in which all steps of the discovery process are automated; we discuss the representation of molecular data such that knowledge discovery tools can analyse it, and we discuss the adaptation of machine learning and data mining algorithms to guide QSAR experiments
Limitations of Algebraic Approaches to Graph Isomorphism Testing
We investigate the power of graph isomorphism algorithms based on algebraic
reasoning techniques like Gr\"obner basis computation. The idea of these
algorithms is to encode two graphs into a system of equations that are
satisfiable if and only if if the graphs are isomorphic, and then to (try to)
decide satisfiability of the system using, for example, the Gr\"obner basis
algorithm. In some cases this can be done in polynomial time, in particular, if
the equations admit a bounded degree refutation in an algebraic proof systems
such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on
the polynomial calculus degree over all fields of characteristic different from
2 and also linear lower bounds for the degree of Positivstellensatz calculus
derivations.
We compare this approach to recently studied linear and semidefinite
programming approaches to isomorphism testing, which are known to be related to
the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the
power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system
that lies between degree-k Nullstellensatz and degree-k polynomial calculus
Filtering graphs to check isomorphism and extracting mapping by using the Conductance Electrical Model
© 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper presents a new method of filtering graphs to check exact graph isomorphism and extracting their mapping. Each graph is modeled by a resistive electrical circuit using the Conductance Electrical Model (CEM). By using this model, a necessary condition to check the isomorphism of two graphs is that their equivalent resistances have the same values, but this is not enough, and we have to look for their mapping to find the sufficient condition. We can compute the isomorphism between two graphs in O(N-3), where N is the order of the graph, if their star resistance values are different, otherwise the computational time is exponential, but only with respect to the number of repeated star resistance values, which usually is very small. We can use this technique to filter graphs that are not isomorphic and in case that they are, we can obtain their node mapping. A distinguishing feature over other methods is that, even if there exists repeated star resistance values, we can extract a partial node mapping (of all the nodes except the repeated ones and their neighbors) in O(N-3). The paper presents the method and its application to detect isomorphic graphs in two well know graph databases, where some graphs have more than 600 nodes. (C) 2016 Elsevier Ltd. All rights reserved.Postprint (author's draft
The Cox ring of an algebraic variety with torus action
We investigate the Cox ring of a normal complete variety X with algebraic
torus action. Our first results relate the Cox ring of X to that of a maximal
geometric quotient of X. As a consequence, we obtain a complete description of
the Cox ring in terms of generators and relations for varieties with torus
action of complexity one. Moreover, we provide a combinatorial approach to the
Cox ring using the language of polyhedral divisors. Applied to smooth
k*-surfaces, our results give a description of the Cox ring in terms of
Orlik-Wagreich graphs. As examples, we explicitly compute the Cox rings of all
Gorenstein del Pezzo k*-surfaces with Picard number at most two and the Cox
rings of projectivizations of rank two vector bundles as well as cotangent
bundles over toric varieties in terms of Klyachko's description.Comment: Minor corrections, to appear in Adv. Math
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