619 research outputs found
A path following algorithm for the graph matching problem
We propose a convex-concave programming approach for the labeled weighted
graph matching problem. The convex-concave programming formulation is obtained
by rewriting the weighted graph matching problem as a least-square problem on
the set of permutation matrices and relaxing it to two different optimization
problems: a quadratic convex and a quadratic concave optimization problem on
the set of doubly stochastic matrices. The concave relaxation has the same
global minimum as the initial graph matching problem, but the search for its
global minimum is also a hard combinatorial problem. We therefore construct an
approximation of the concave problem solution by following a solution path of a
convex-concave problem obtained by linear interpolation of the convex and
concave formulations, starting from the convex relaxation. This method allows
to easily integrate the information on graph label similarities into the
optimization problem, and therefore to perform labeled weighted graph matching.
The algorithm is compared with some of the best performing graph matching
methods on four datasets: simulated graphs, QAPLib, retina vessel images and
handwritten chinese characters. In all cases, the results are competitive with
the state-of-the-art.Comment: 23 pages, 13 figures,typo correction, new results in sections 4,5,
Using a conic bundle method to accelerate both phases of a quadratic convex reformulation
We present algorithm MIQCR-CB that is an advancement of method
MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving
mixed-integer quadratic programs and works in two phases: the first phase
determines an equivalent quadratic formulation with a convex objective function
by solving a semidefinite problem , and, in the second phase, the
equivalent formulation is solved by a standard solver. As the reformulation
relies on the solution of a large-scale semidefinite program, it is not
tractable by existing semidefinite solvers, already for medium sized problems.
To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm
within a Lagrangian duality framework for solving that substantially
speeds up the first phase. Moreover, this algorithm leads to a reformulated
problem of smaller size than the one obtained by the original MIQCR method
which results in a shorter time for solving the second phase.
We present extensive computational results to show the efficiency of our
algorithm
Product graph-based higher order contextual similarities for inexact subgraph matching
This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record Many algorithms formulate graph matching as an optimization of an objective function of pairwise quantification of nodes and edges of two graphs to be matched. Pairwise measurements usually consider local attributes but disregard contextual information involved in graph structures. We address this issue by proposing contextual similarities between pairs of nodes. This is done by considering the tensor product graph (TPG) of two graphs to be matched, where each node is an ordered pair of nodes of the operand graphs. Contextual similarities between a pair of nodes are computed by accumulating weighted walks (normalized pairwise similarities) terminating at the corresponding paired node in TPG. Once the contextual similarities are obtained, we formulate subgraph matching as a node and edge selection problem in TPG. We use contextual similarities to construct an objective function and optimize it with a linear programming approach. Since random walk formulation through TPG takes into account higher order information, it is not a surprise that we obtain more reliable similarities and better discrimination among the nodes and edges. Experimental results shown on synthetic as well as real benchmarks illustrate that higher order contextual similarities increase discriminating power and allow one to find approximate solutions to the subgraph matching problem.European Union Horizon 202
Max-sum diversity via convex programming
Diversity maximization is an important concept in information retrieval,
computational geometry and operations research. Usually, it is a variant of the
following problem: Given a ground set, constraints, and a function
that measures diversity of a subset, the task is to select a feasible subset
such that is maximized. The \emph{sum-dispersion} function , which is the sum of the pairwise distances in , is
in this context a prominent diversification measure. The corresponding
diversity maximization is the \emph{max-sum} or \emph{sum-sum diversification}.
Many recent results deal with the design of constant-factor approximation
algorithms of diversification problems involving sum-dispersion function under
a matroid constraint. In this paper, we present a PTAS for the max-sum
diversification problem under a matroid constraint for distances
of \emph{negative type}. Distances of negative type are, for
example, metric distances stemming from the and norm, as well
as the cosine or spherical, or Jaccard distance which are popular similarity
metrics in web and image search
Optimality of Treating Interference as Noise: A Combinatorial Perspective
For single-antenna Gaussian interference channels, we re-formulate the
problem of determining the Generalized Degrees of Freedom (GDoF) region
achievable by treating interference as Gaussian noise (TIN) derived in [3] from
a combinatorial perspective. We show that the TIN power control problem can be
cast into an assignment problem, such that the globally optimal power
allocation variables can be obtained by well-known polynomial time algorithms.
Furthermore, the expression of the TIN-Achievable GDoF region (TINA region) can
be substantially simplified with the aid of maximum weighted matchings. We also
provide conditions under which the TINA region is a convex polytope that relax
those in [3]. For these new conditions, together with a channel connectivity
(i.e., interference topology) condition, we show TIN optimality for a new class
of interference networks that is not included, nor includes, the class found in
[3].
Building on the above insights, we consider the problem of joint link
scheduling and power control in wireless networks, which has been widely
studied as a basic physical layer mechanism for device-to-device (D2D)
communications. Inspired by the relaxed TIN channel strength condition as well
as the assignment-based power allocation, we propose a low-complexity
GDoF-based distributed link scheduling and power control mechanism (ITLinQ+)
that improves upon the ITLinQ scheme proposed in [4] and further improves over
the heuristic approach known as FlashLinQ. It is demonstrated by simulation
that ITLinQ+ provides significant average network throughput gains over both
ITLinQ and FlashLinQ, and yet still maintains the same level of implementation
complexity. More notably, the energy efficiency of the newly proposed ITLinQ+
is substantially larger than that of ITLinQ and FlashLinQ, which is desirable
for D2D networks formed by battery-powered devices.Comment: A short version has been presented at IEEE International Symposium on
Information Theory (ISIT 2015), Hong Kon
Integrative Analysis of Many Weighted Co-Expression Networks Using Tensor Computation
The rapid accumulation of biological networks poses new challenges and calls for powerful integrative analysis tools. Most existing methods capable of simultaneously analyzing a large number of networks were primarily designed for unweighted networks, and cannot easily be extended to weighted networks. However, it is known that transforming weighted into unweighted networks by dichotomizing the edges of weighted networks with a threshold generally leads to information loss. We have developed a novel, tensor-based computational framework for mining recurrent heavy subgraphs in a large set of massive weighted networks. Specifically, we formulate the recurrent heavy subgraph identification problem as a heavy 3D subtensor discovery problem with sparse constraints. We describe an effective approach to solving this problem by designing a multi-stage, convex relaxation protocol, and a non-uniform edge sampling technique. We applied our method to 130 co-expression networks, and identified 11,394 recurrent heavy subgraphs, grouped into 2,810 families. We demonstrated that the identified subgraphs represent meaningful biological modules by validating against a large set of compiled biological knowledge bases. We also showed that the likelihood for a heavy subgraph to be meaningful increases significantly with its recurrence in multiple networks, highlighting the importance of the integrative approach to biological network analysis. Moreover, our approach based on weighted graphs detects many patterns that would be overlooked using unweighted graphs. In addition, we identified a large number of modules that occur predominately under specific phenotypes. This analysis resulted in a genome-wide mapping of gene network modules onto the phenome. Finally, by comparing module activities across many datasets, we discovered high-order dynamic cooperativeness in protein complex networks and transcriptional regulatory networks
Efficient parallel implementation of the multiplicative weight update method for graph-based linear programs
Positive linear programs (LPs) model many graph and operations research
problems. One can solve for a -approximation for positive LPs,
for any selected , in polylogarithmic depth and near-linear work via
variations of the multiplicative weight update (MWU) method. Despite extensive
theoretical work on these algorithms through the decades, their empirical
performance is not well understood.
In this work, we implement and test an efficient parallel algorithm for
solving positive LP relaxations, and apply it to graph problems such as densest
subgraph, bipartite matching, vertex cover and dominating set. We accelerate
the algorithm via a new step size search heuristic. Our implementation uses
sparse linear algebra optimization techniques such as fusion of vector
operations and use of sparse format. Furthermore, we devise an implicit
representation for graph incidence constraints. We demonstrate the parallel
scalability with the use of threading OpenMP and MPI on the Stampede2
supercomputer. We compare this implementation with exact libraries and
specialized libraries for the above problems in order to evaluate MWU's
practical standing for both accuracy and performance among other methods. Our
results show this implementation is faster than general purpose LP solvers (IBM
CPLEX, Gurobi) in all of our experiments, and in some instances, outperforms
state-of-the-art specialized parallel graph algorithms.Comment: Pre-print. 13 pages, comments welcom
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