11,233 research outputs found
Algorithms for the minimum sum coloring problem: a review
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex
coloring problem which has a number of AI related applications. Due to its
theoretical and practical relevance, MSCP attracts increasing attention. The
only existing review on the problem dates back to 2004 and mainly covers the
history of MSCP and theoretical developments on specific graphs. In recent
years, the field has witnessed significant progresses on approximation
algorithms and practical solution algorithms. The purpose of this review is to
provide a comprehensive inspection of the most recent and representative MSCP
algorithms. To be informative, we identify the general framework followed by
practical solution algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications
Optimal General Matchings
Given a graph and for each vertex a subset of the
set , where denotes the degree of vertex
in the graph , a -factor of is any set such that
for each vertex , where denotes the number of
edges of incident to . The general factor problem asks the existence of
a -factor in a given graph. A set is said to have a {\em gap of
length} if there exists a natural number such that and . Without any restrictions the
general factor problem is NP-complete. However, if no set contains a gap
of length greater than , then the problem can be solved in polynomial time
and Cornuejols \cite{Cor} presented an algorithm for finding a -factor, if
it exists. In this paper we consider a weighted version of the general factor
problem, in which each edge has a nonnegative weight and we are interested in
finding a -factor of maximum (or minimum) weight. In particular, this
version comprises the minimum/maximum cardinality variant of the general factor
problem, where we want to find a -factor having a minimum/maximum number of
edges.
We present an algorithm for the maximum/minimum weight -factor for the
case when no set contains a gap of length greater than . This also
yields the first polynomial time algorithm for the maximum/minimum cardinality
-factor for this case
Engineering Data Reduction for Nested Dissection
Many applications rely on time-intensive matrix operations, such as
factorization, which can be sped up significantly for large sparse matrices by
interpreting the matrix as a sparse graph and computing a node ordering that
minimizes the so-called fill-in. In this paper, we engineer new data reduction
rules for the minimum fill-in problem, which significantly reduce the size of
the graph while producing an equivalent (or near-equivalent) instance. By
applying both new and existing data reduction rules exhaustively before nested
dissection, we obtain improved quality and at the same time large improvements
in running time on a variety of instances. Our overall algorithm outperforms
the state-of-the-art significantly: it not only yields better elimination
orders, but it does so significantly faster than previously possible. For
example, on road networks, where nested dissection algorithms are typically
used as a preprocessing step for shortest path computations, our algorithms are
on average six times faster than Metis while computing orderings with less
fill-in
Learning-Based Constraint Satisfaction With Sensing Restrictions
In this paper we consider graph-coloring problems, an important subset of
general constraint satisfaction problems that arise in wireless resource
allocation. We constructively establish the existence of fully decentralized
learning-based algorithms that are able to find a proper coloring even in the
presence of strong sensing restrictions, in particular sensing asymmetry of the
type encountered when hidden terminals are present. Our main analytic
contribution is to establish sufficient conditions on the sensing behaviour to
ensure that the solvers find satisfying assignments with probability one. These
conditions take the form of connectivity requirements on the induced sensing
graph. These requirements are mild, and we demonstrate that they are commonly
satisfied in wireless allocation tasks. We argue that our results are of
considerable practical importance in view of the prevalence of both
communication and sensing restrictions in wireless resource allocation
problems. The class of algorithms analysed here requires no message-passing
whatsoever between wireless devices, and we show that they continue to perform
well even when devices are only able to carry out constrained sensing of the
surrounding radio environment
1-Bit Matrix Completion under Exact Low-Rank Constraint
We consider the problem of noisy 1-bit matrix completion under an exact rank
constraint on the true underlying matrix . Instead of observing a subset
of the noisy continuous-valued entries of a matrix , we observe a subset
of noisy 1-bit (or binary) measurements generated according to a probabilistic
model. We consider constrained maximum likelihood estimation of , under a
constraint on the entry-wise infinity-norm of and an exact rank
constraint. This is in contrast to previous work which has used convex
relaxations for the rank. We provide an upper bound on the matrix estimation
error under this model. Compared to the existing results, our bound has faster
convergence rate with matrix dimensions when the fraction of revealed 1-bit
observations is fixed, independent of the matrix dimensions. We also propose an
iterative algorithm for solving our nonconvex optimization with a certificate
of global optimality of the limiting point. This algorithm is based on low rank
factorization of . We validate the method on synthetic and real data with
improved performance over existing methods.Comment: 6 pages, 3 figures, to appear in CISS 201
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