794 research outputs found
An adaptive prefix-assignment technique for symmetry reduction
This paper presents a technique for symmetry reduction that adaptively
assigns a prefix of variables in a system of constraints so that the generated
prefix-assignments are pairwise nonisomorphic under the action of the symmetry
group of the system. The technique is based on McKay's canonical extension
framework [J.~Algorithms 26 (1998), no.~2, 306--324]. Among key features of the
technique are (i) adaptability---the prefix sequence can be user-prescribed and
truncated for compatibility with the group of symmetries; (ii)
parallelizability---prefix-assignments can be processed in parallel
independently of each other; (iii) versatility---the method is applicable
whenever the group of symmetries can be concisely represented as the
automorphism group of a vertex-colored graph; and (iv) implementability---the
method can be implemented relying on a canonical labeling map for
vertex-colored graphs as the only nontrivial subroutine. To demonstrate the
practical applicability of our technique, we have prepared an experimental
open-source implementation of the technique and carry out a set of experiments
that demonstrate ability to reduce symmetry on hard instances. Furthermore, we
demonstrate that the implementation effectively parallelizes to compute
clusters with multiple nodes via a message-passing interface.Comment: Updated manuscript submitted for revie
Isomorph-free generation of 2-connected graphs with applications
Many interesting graph families contain only 2-connected graphs, which have
ear decompositions. We develop a technique to generate families of unlabeled
2-connected graphs using ear augmentations and apply this technique to two
problems. In the first application, we search for uniquely K_r-saturated graphs
and find the list of uniquely K_4-saturated graphs on at most 12 vertices,
supporting current conjectures for this problem. In the second application, we
verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at
most 12 vertices. This technique can be easily extended to more problems
concerning 2-connected graphs.Comment: 15 pages, 3 figures, 4 table
Isomorph-Free Branch and Bound Search for Finite State Controllers
The recent proliferation of smart-phones and other wearable devices has lead
to a surge of new mobile applications. Partially observable Markov decision
processes provide a natural framework to design applications that
continuously make decisions based on noisy sensor measurements. However,
given the limited battery life, there is a need to minimize the amount of
online computation. This can be achieved by compiling a policy into a
finite state controller since there is no need for belief monitoring or
online search. In this paper, we propose a new branch and bound technique
to search for a good controller. In contrast to many existing algorithms
for controllers, our search technique is not subject to local optima. We
also show how to reduce the amount of search by avoiding the enumeration of
isomorphic controllers and by taking advantage of suitable upper and lower
bounds. The approach is demonstrated on several benchmark problems as well
as a smart-phone application to assist persons with Alzheimer's to wayfind
Matroids with nine elements
We describe the computation of a catalogue containing all matroids with up to
nine elements, and present some fundamental data arising from this cataogue.
Our computation confirms and extends the results obtained in the 1960s by
Blackburn, Crapo and Higgs. The matroids and associated data are stored in an
online database, and we give three short examples of the use of this database.Comment: 22 page
Graphettes: Constant-time determination of graphlet and orbit identity including (possibly disconnected) graphlets up to size 8.
Graphlets are small connected induced subgraphs of a larger graph G. Graphlets are now commonly used to quantify local and global topology of networks in the field. Methods exist to exhaustively enumerate all graphlets (and their orbits) in large networks as efficiently as possible using orbit counting equations. However, the number of graphlets in G is exponential in both the number of nodes and edges in G. Enumerating them all is already unacceptably expensive on existing large networks, and the problem will only get worse as networks continue to grow in size and density. Here we introduce an efficient method designed to aid statistical sampling of graphlets up to size k = 8 from a large network. We define graphettes as the generalization of graphlets allowing for disconnected graphlets. Given a particular (undirected) graphette g, we introduce the idea of the canonical graphette [Formula: see text] as a representative member of the isomorphism group Iso(g) of g. We compute the mapping [Formula: see text], in the form of a lookup table, from all 2k(k - 1)/2 undirected graphettes g of size k †8 to their canonical representatives [Formula: see text], as well as the permutation that transforms g to [Formula: see text]. We also compute all automorphism orbits for each canonical graphette. Thus, given any k †8 nodes in a graph G, we can in constant time infer which graphette it is, as well as which orbit each of the k nodes belongs to. Sampling a large number N of such k-sets of nodes provides an approximation of both the distribution of graphlets and orbits across G, and the orbit degree vector at each node
The IBMAP approach for Markov networks structure learning
In this work we consider the problem of learning the structure of Markov
networks from data. We present an approach for tackling this problem called
IBMAP, together with an efficient instantiation of the approach: the IBMAP-HC
algorithm, designed for avoiding important limitations of existing
independence-based algorithms. These algorithms proceed by performing
statistical independence tests on data, trusting completely the outcome of each
test. In practice tests may be incorrect, resulting in potential cascading
errors and the consequent reduction in the quality of the structures learned.
IBMAP contemplates this uncertainty in the outcome of the tests through a
probabilistic maximum-a-posteriori approach. The approach is instantiated in
the IBMAP-HC algorithm, a structure selection strategy that performs a
polynomial heuristic local search in the space of possible structures. We
present an extensive empirical evaluation on synthetic and real data, showing
that our algorithm outperforms significantly the current independence-based
algorithms, in terms of data efficiency and quality of learned structures, with
equivalent computational complexities. We also show the performance of IBMAP-HC
in a real-world application of knowledge discovery: EDAs, which are
evolutionary algorithms that use structure learning on each generation for
modeling the distribution of populations. The experiments show that when
IBMAP-HC is used to learn the structure, EDAs improve the convergence to the
optimum
A SAT Solver and Computer Algebra Attack on the Minimum Kochen-Specker Problem
One of the foundational results in quantum mechanics is the Kochen-Specker
(KS) theorem, which states that any theory whose predictions agree with quantum
mechanics must be contextual, i.e., a quantum observation cannot be understood
as revealing a pre-existing value. The theorem hinges on the existence of a
mathematical object called a KS vector system. While many KS vector systems are
known to exist, the problem of finding the minimum KS vector system has
remained stubbornly open for over 55 years, despite significant attempts by
leading scientists and mathematicians. In this paper, we present a new method
based on a combination of a SAT solver and a computer algebra system (CAS) to
address this problem. Our approach improves the lower bound on the minimum
number of vectors in a KS system from 22 to 24, and is about 35,000 times more
efficient compared to the previous best computational methods. The increase in
efficiency derives from the fact we are able to exploit the powerful
combinatorial search-with-learning capabilities of a SAT solver together with
the isomorph-free exhaustive generation methods of a CAS. The quest for the
minimum KS vector system is motivated by myriad applications such as
simplifying experimental tests of contextuality, zero-error classical
communication, dimension witnessing, and the security of certain quantum
cryptographic protocols. To the best of our knowledge, this is the first
application of a novel SAT+CAS system to a problem in the realm of quantum
foundations
Perfect binary codes: classification and properties
An r-perfect binary code is a subset of â€2n such that for any word, there is a unique codeword at Hamming distance at most r. Such a code is r-error-correcting. Two codes are equivalent if one can be obtained from the other by permuting the coordinates and adding a constant vector. The main result of this thesis is a computer-aided classification, up to equivalence, of the 1-perfect binary codes of length 15.
In an extended 1-perfect code, the neighborhood of a codeword corresponds to a Steiner quadruple system. To utilize this connection, we start with a computational classification of Steiner quadruple systems of order 16. This classification is also used to establish the nonexistence of Steiner quintuple systems S(4, 5, 17).
The classification of the codes is used for computational examination of their properties. These properties include occurrences of Steiner triple and quadruple systems, automorphisms, ranks, structure of i-components and connections to orthogonal arrays and mixed perfect codes.
It is also proved that extended 1-perfect binary codes are equivalent if and only if their minimum distance graphs are isomorphic
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