20 research outputs found
Dynamic Complexity of Planar 3-connected Graph Isomorphism
Dynamic Complexity (as introduced by Patnaik and Immerman) tries to express
how hard it is to update the solution to a problem when the input is changed
slightly. It considers the changes required to some stored data structure
(possibly a massive database) as small quantities of data (or a tuple) are
inserted or deleted from the database (or a structure over some vocabulary).
The main difference from previous notions of dynamic complexity is that instead
of treating the update quantitatively by finding the the time/space trade-offs,
it tries to consider the update qualitatively, by finding the complexity class
in which the update can be expressed (or made). In this setting, DynFO, or
Dynamic First-Order, is one of the smallest and the most natural complexity
class (since SQL queries can be expressed in First-Order Logic), and contains
those problems whose solutions (or the stored data structure from which the
solution can be found) can be updated in First-Order Logic when the data
structure undergoes small changes.
Etessami considered the problem of isomorphism in the dynamic setting, and
showed that Tree Isomorphism can be decided in DynFO. In this work, we show
that isomorphism of Planar 3-connected graphs can be decided in DynFO+ (which
is DynFO with some polynomial precomputation). We maintain a canonical
description of 3-connected Planar graphs by maintaining a database which is
accessed and modified by First-Order queries when edges are added to or deleted
from the graph. We specifically exploit the ideas of Breadth-First Search and
Canonical Breadth-First Search to prove the results. We also introduce a novel
method for canonizing a 3-connected planar graph in First-Order Logic from
Canonical Breadth-First Search Trees
Computational Indistinguishability between Quantum States and Its Cryptographic Application
We introduce a computational problem of distinguishing between two specific
quantum states as a new cryptographic problem to design a quantum cryptographic
scheme that is "secure" against any polynomial-time quantum adversary. Our
problem, QSCDff, is to distinguish between two types of random coset states
with a hidden permutation over the symmetric group of finite degree. This
naturally generalizes the commonly-used distinction problem between two
probability distributions in computational cryptography. As our major
contribution, we show that QSCDff has three properties of cryptographic
interest: (i) QSCDff has a trapdoor; (ii) the average-case hardness of QSCDff
coincides with its worst-case hardness; and (iii) QSCDff is computationally at
least as hard as the graph automorphism problem in the worst case. These
cryptographic properties enable us to construct a quantum public-key
cryptosystem, which is likely to withstand any chosen plaintext attack of a
polynomial-time quantum adversary. We further discuss a generalization of
QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies
on similar cryptographic properties of QSCDcyc.Comment: 24 pages, 2 figures. We improved presentation, and added more detail
proofs and follow-up of recent wor
Computational Complexity and Graph Isomorphism
The graph isomorphism problem is the computational problem of determining whether two ïŹnite graphs are isomorphic, that is, structurally the same. The complexity of graph isomorphism is an open problem and it is one of the few problems in NP which is neither known to be solvable in polynomial time nor NP-complete. It is one of the most researched open problems in theoretical computer science.
The foundations of computability theory are in recursion theory and in recursive functions which are an older model of computation than Turing machines. In this masterâs thesis we discuss the basics of the recursion theory and the main theorems starting from the axioms. The aim of the second chapter is to define the most important T- and m-reductions and the implication hierarchy between reductions.
Different variations of Turing machines include the nondeterministic and oracle Turing machines. They are discussed in the third chapter. A hierarchy of different complexity classes can be created by reducing the available computational resources of recursive functions. The members of this hierarchy include for instance P and NP. There are hundreds of known complexity classes and in this work the most important ones regarding graph isomorphism are introduced.
Boolean circuits are a different method for approaching computability. Some main results and complexity classes of circuit complexity are discussed in the fourth chapter. The aim is to show that graph isomorphism is hard for the class DET.
Graph isomorphism is known to belong to the classes coAM and SPP. These classes are introduced in the fifth chapter by using theory of probabilistic classes, polynomial hierarchy, interactive proof systems and Arthur-Merlin games. Polynomial hierarchy collapses to its second level if GI is NP-complete
Graph Isomorphism for K_{3,3}-free and K_5-free graphs is in Log-space
Graph isomorphism is an important and widely studied computational problem with
a yet unsettled complexity.
However, the exact complexity is known for isomorphism of various classes of
graphs. Recently, cite{DLNTW09} proved that planar isomorphism is complete for log-space.
We extend this result %of cite{DLNTW09}
further to the classes of graphs which exclude or as
a minor, and give a log-space algorithm.
Our algorithm decomposes minor-free graphs into biconnected and those further into triconnected
components, which are known to be either planar or components cite{Vaz89}. This gives a triconnected
component tree similar to that for planar graphs. An extension of the log-space algorithm of cite{DLNTW09}
can then be used to decide the isomorphism problem.
For minor-free graphs, we consider -connected components.
These are either planar or isomorphic to the four-rung mobius ladder on vertices
or, with a further decomposition, one obtains planar -connected components cite{Khu88}.
We give an algorithm to get a unique
decomposition of minor-free graphs into bi-, tri- and -connected components,
and construct trees, accordingly.
Since the algorithm of cite{DLNTW09} does
not deal with four-connected component trees, it needs to be modified in a quite non-trivial way
Unsupervised Learning of Invariance Transformations
The need for large amounts of training data in modern machine learning is one
of the biggest challenges of the field. Compared to the brain, current
artificial algorithms are much less capable of learning invariance
transformations and employing them to extrapolate knowledge from small sample
sets. It has recently been proposed that the brain might encode perceptual
invariances as approximate graph symmetries in the network of synaptic
connections. Such symmetries may arise naturally through a biologically
plausible process of unsupervised Hebbian learning. In the present paper, we
illustrate this proposal on numerical examples, showing that invariance
transformations can indeed be recovered from the structure of recurrent
synaptic connections which form within a layer of feature detector neurons via
a simple Hebbian learning rule. In order to numerically recover the invariance
transformations from the resulting recurrent network, we develop a general
algorithmic framework for finding approximate graph automorphisms. We discuss
how this framework can be used to find approximate automorphisms in weighted
graphs in general
Logarithmic Weisfeiler--Leman and Treewidth
In this paper, we show that the -dimensional Weisfeiler--Leman
algorithm can identify graphs of treewidth in rounds. This
improves the result of Grohe & Verbitsky (ICALP 2006), who previously
established the analogous result for -dimensional Weisfeiler--Leman. In
light of the equivalence between Weisfeiler--Leman and the logic (Cai, F\"urer, & Immerman, Combinatorica 1992), we obtain an
improvement in the descriptive complexity for graphs of treewidth .
Precisely, if is a graph of treewidth , then there exists a
-variable formula in with
quantifier depth that identifies up to isomorphism