11,322 research outputs found
Semidefinite Programming Approach for the Quadratic Assignment Problem with a Sparse Graph
The matching problem between two adjacency matrices can be formulated as the
NP-hard quadratic assignment problem (QAP). Previous work on semidefinite
programming (SDP) relaxations to the QAP have produced solutions that are often
tight in practice, but such SDPs typically scale badly, involving matrix
variables of dimension where n is the number of nodes. To achieve a speed
up, we propose a further relaxation of the SDP involving a number of positive
semidefinite matrices of dimension no greater than the number
of edges in one of the graphs. The relaxation can be further strengthened by
considering cliques in the graph, instead of edges. The dual problem of this
novel relaxation has a natural three-block structure that can be solved via a
convergent Augmented Direction Method of Multipliers (ADMM) in a distributed
manner, where the most expensive step per iteration is computing the
eigendecomposition of matrices of dimension . The new SDP
relaxation produces strong bounds on quadratic assignment problems where one of
the graphs is sparse with reduced computational complexity and running times,
and can be used in the context of nuclear magnetic resonance spectroscopy (NMR)
to tackle the assignment problem.Comment: 31 page
Discrete Signal Processing on Graphs: Frequency Analysis
Signals and datasets that arise in physical and engineering applications, as
well as social, genetics, biomolecular, and many other domains, are becoming
increasingly larger and more complex. In contrast to traditional time and image
signals, data in these domains are supported by arbitrary graphs. Signal
processing on graphs extends concepts and techniques from traditional signal
processing to data indexed by generic graphs. This paper studies the concepts
of low and high frequencies on graphs, and low-, high-, and band-pass graph
filters. In traditional signal processing, there concepts are easily defined
because of a natural frequency ordering that has a physical interpretation. For
signals residing on graphs, in general, there is no obvious frequency ordering.
We propose a definition of total variation for graph signals that naturally
leads to a frequency ordering on graphs and defines low-, high-, and band-pass
graph signals and filters. We study the design of graph filters with specified
frequency response, and illustrate our approach with applications to sensor
malfunction detection and data classification
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