65,588 research outputs found
New Iterative Algorithms for Weighted Matching
Matching is an important combinatorial problem with a number ofpractical applications. Even though there exist polynomial time solutionsto most matching problems, there are settings where these are too slow.This has led to the development of several fast approximation algorithmsthat in practice compute matchings very close to the optimal.The current paper introduces a new deterministic approximationalgorithm named G 3 , for weighted matching. The algorithm is based ontwo main ideas, the first is to compute heavy subgraphs of the originalgraph on which we can compute optimal matchings. The second idea isto repeatedly merge these matchings into new matchings of even higherweight than the original ones. Both of these steps are achieved usingdynamic programming in linear or close to linear time.We compare G 3 with the randomized algorithm GPA+ROMA whichis the best known algorithm for this problem. Experiments on alarge collection of graphs show that G 3 is substantially faster thanGPA+ROMA while computing matchings of comparable weight
Mixing Color Coding-Related Techniques
Narrow sieves, representative sets and divide-and-color are three
breakthrough color coding-related techniques, which led to the design of
extremely fast parameterized algorithms. We present a novel family of
strategies for applying mixtures of them. This includes: (a) a mix of
representative sets and narrow sieves; (b) a faster computation of
representative sets under certain separateness conditions, mixed with
divide-and-color and a new technique, "balanced cutting"; (c) two mixtures of
representative sets, iterative compression and a new technique, "unbalanced
cutting". We demonstrate our strategies by obtaining, among other results,
significantly faster algorithms for -Internal Out-Branching and Weighted
3-Set -Packing, and a framework for speeding-up the previous best
deterministic algorithms for -Path, -Tree, -Dimensional -Matching,
Graph Motif and Partial Cover
On large-scale diagonalization techniques for the Anderson model of localization
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model
of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the JacobiāDavidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete
LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization
by several orders of magnitude
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
Maximum Weight Matching via Max-Product Belief Propagation
Max-product "belief propagation" is an iterative, local, message-passing
algorithm for finding the maximum a posteriori (MAP) assignment of a discrete
probability distribution specified by a graphical model. Despite the
spectacular success of the algorithm in many application areas such as
iterative decoding, computer vision and combinatorial optimization which
involve graphs with many cycles, theoretical results about both correctness and
convergence of the algorithm are known in few cases (Weiss-Freeman Wainwright,
Yeddidia-Weiss-Freeman, Richardson-Urbanke}.
In this paper we consider the problem of finding the Maximum Weight Matching
(MWM) in a weighted complete bipartite graph. We define a probability
distribution on the bipartite graph whose MAP assignment corresponds to the
MWM. We use the max-product algorithm for finding the MAP of this distribution
or equivalently, the MWM on the bipartite graph. Even though the underlying
bipartite graph has many short cycles, we find that surprisingly, the
max-product algorithm always converges to the correct MAP assignment as long as
the MAP assignment is unique. We provide a bound on the number of iterations
required by the algorithm and evaluate the computational cost of the algorithm.
We find that for a graph of size , the computational cost of the algorithm
scales as , which is the same as the computational cost of the best
known algorithm. Finally, we establish the precise relation between the
max-product algorithm and the celebrated {\em auction} algorithm proposed by
Bertsekas. This suggests possible connections between dual algorithm and
max-product algorithm for discrete optimization problems.Comment: In the proceedings of the 2005 IEEE International Symposium on
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