50,597 research outputs found
Phase Transition and Network Structure in Realistic SAT Problems
A fundamental question in Computer Science is understanding when a specific
class of problems go from being computationally easy to hard. Because of its
generality and applications, the problem of Boolean Satisfiability (aka SAT) is
often used as a vehicle for investigating this question. A signal result from
these studies is that the hardness of SAT problems exhibits a dramatic
easy-to-hard phase transition with respect to the problem constrainedness. Past
studies have however focused mostly on SAT instances generated using uniform
random distributions, where all constraints are independently generated, and
the problem variables are all considered of equal importance. These assumptions
are unfortunately not satisfied by most real problems. Our project aims for a
deeper understanding of hardness of SAT problems that arise in practice. We
study two key questions: (i) How does easy-to-hard transition change with more
realistic distributions that capture neighborhood sensitivity and
rich-get-richer aspects of real problems and (ii) Can these changes be
explained in terms of the network properties (such as node centrality and
small-worldness) of the clausal networks of the SAT problems. Our results,
based on extensive empirical studies and network analyses, provide important
structural and computational insights into realistic SAT problems. Our
extensive empirical studies show that SAT instances from realistic
distributions do exhibit phase transition, but the transition occurs sooner (at
lower values of constrainedness) than the instances from uniform random
distribution. We show that this behavior can be explained in terms of their
clausal network properties such as eigenvector centrality and small-worldness
(measured indirectly in terms of the clustering coefficients and average node
distance)
Message Passing for Integrating and Assessing Renewable Generation in a Redundant Power Grid
A simplified model of a redundant power grid is used to study integration of
fluctuating renewable generation. The grid consists of large number of
generator and consumer nodes. The net power consumption is determined by the
difference between the gross consumption and the level of renewable generation.
The gross consumption is drawn from a narrow distribution representing the
predictability of aggregated loads, and we consider two different distributions
representing wind and solar resources. Each generator is connected to D
consumers, and redundancy is built in by connecting R of these consumers to
other generators. The lines are switchable so that at any instance each
consumer is connected to a single generator. We explore the capacity of the
renewable generation by determining the level of "firm" generation capacity
that can be displaced for different levels of redundancy R. We also develop
message-passing control algorithm for finding switch settings where no
generator is overloaded.Comment: 10 pages, accepted for HICSS-4
Random subcubes as a toy model for constraint satisfaction problems
We present an exactly solvable random-subcube model inspired by the structure
of hard constraint satisfaction and optimization problems. Our model reproduces
the structure of the solution space of the random k-satisfiability and
k-coloring problems, and undergoes the same phase transitions as these
problems. The comparison becomes quantitative in the large-k limit. Distance
properties, as well the x-satisfiability threshold, are studied. The model is
also generalized to define a continuous energy landscape useful for studying
several aspects of glassy dynamics.Comment: 21 pages, 4 figure
First-order transitions and the performance of quantum algorithms in random optimization problems
We present a study of the phase diagram of a random optimization problem in
presence of quantum fluctuations. Our main result is the characterization of
the nature of the phase transition, which we find to be a first-order quantum
phase transition. We provide evidence that the gap vanishes exponentially with
the system size at the transition. This indicates that the Quantum Adiabatic
Algorithm requires a time growing exponentially with system size to find the
ground state of this problem.Comment: 4 pages, 4 figures; final version accepted on Phys.Rev.Let
Satisfiability, sequence niches, and molecular codes in cellular signaling
Biological information processing as implemented by regulatory and signaling
networks in living cells requires sufficient specificity of molecular
interaction to distinguish signals from one another, but much of regulation and
signaling involves somewhat fuzzy and promiscuous recognition of molecular
sequences and structures, which can leave systems vulnerable to crosstalk. This
paper examines a simple computational model of protein-protein interactions
which reveals both a sharp onset of crosstalk and a fragmentation of the
neutral network of viable solutions as more proteins compete for regions of
sequence space, revealing intrinsic limits to reliable signaling in the face of
promiscuity. These results suggest connections to both phase transitions in
constraint satisfaction problems and coding theory bounds on the size of
communication codes
Community Structure in Industrial SAT Instances
Modern SAT solvers have experienced a remarkable progress on solving
industrial instances. Most of the techniques have been developed after an
intensive experimental process. It is believed that these techniques exploit
the underlying structure of industrial instances. However, there are few works
trying to exactly characterize the main features of this structure.
The research community on complex networks has developed techniques of
analysis and algorithms to study real-world graphs that can be used by the SAT
community. Recently, there have been some attempts to analyze the structure of
industrial SAT instances in terms of complex networks, with the aim of
explaining the success of SAT solving techniques, and possibly improving them.
In this paper, inspired by the results on complex networks, we study the
community structure, or modularity, of industrial SAT instances. In a graph
with clear community structure, or high modularity, we can find a partition of
its nodes into communities such that most edges connect variables of the same
community. In our analysis, we represent SAT instances as graphs, and we show
that most application benchmarks are characterized by a high modularity. On the
contrary, random SAT instances are closer to the classical Erd\"os-R\'enyi
random graph model, where no structure can be observed. We also analyze how
this structure evolves by the effects of the execution of a CDCL SAT solver. In
particular, we use the community structure to detect that new clauses learned
by the solver during the search contribute to destroy the original structure of
the formula. This is, learned clauses tend to contain variables of distinct
communities
Allocation in Practice
How do we allocate scarcere sources? How do we fairly allocate costs? These
are two pressing challenges facing society today. I discuss two recent projects
at NICTA concerning resource and cost allocation. In the first, we have been
working with FoodBank Local, a social startup working in collaboration with
food bank charities around the world to optimise the logistics of collecting
and distributing donated food. Before we can distribute this food, we must
decide how to allocate it to different charities and food kitchens. This gives
rise to a fair division problem with several new dimensions, rarely considered
in the literature. In the second, we have been looking at cost allocation
within the distribution network of a large multinational company. This also has
several new dimensions rarely considered in the literature.Comment: To appear in Proc. of 37th edition of the German Conference on
Artificial Intelligence (KI 2014), Springer LNC
Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses
We develop a systematic cluster expansion for dilute systems in the highly
dilute phase. We first apply it to the calculation of the entropy of the
K-satisfiability problem in the satisfiable phase. We derive a series expansion
in the control parameter, the average connectivity, that is identical to the
one obtained by using the replica approach with a replica symmetric ({\sc rs})
{\it Ansatz}, when the order parameter is calculated via a perturbative
expansion in the control parameter. As a second application we compute the
free-energy of the Viana-Bray model in the paramagnetic phase. The cluster
expansion allows one to compute finite-size corrections in a simple manner and
these are particularly important in optimization problems. Importantly enough,
these calculations prove the exactness of the {\sc rs} {\it Ansatz} below the
percolation threshold and might require its revision between this and the
easy-to-hard transition.Comment: 21 pages, 7 figs, to appear in Phys. Rev.
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