19,286 research outputs found
Message passing for the coloring problem: Gallager meets Alon and Kahale
Message passing algorithms are popular in many combinatorial optimization
problems. For example, experimental results show that {\em survey propagation}
(a certain message passing algorithm) is effective in finding proper
-colorings of random graphs in the near-threshold regime. In 1962 Gallager
introduced the concept of Low Density Parity Check (LDPC) codes, and suggested
a simple decoding algorithm based on message passing. In 1994 Alon and Kahale
exhibited a coloring algorithm and proved its usefulness for finding a
-coloring of graphs drawn from a certain planted-solution distribution over
-colorable graphs. In this work we show an interpretation of Alon and
Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus
showing a connection between the two problems - coloring and decoding. This
also provides a rigorous evidence for the usefulness of the message passing
paradigm for the graph coloring problem. Our techniques can be applied to
several other combinatorial optimization problems and networking-related
issues.Comment: 11 page
Phase Transitions in Semidefinite Relaxations
Statistical inference problems arising within signal processing, data mining,
and machine learning naturally give rise to hard combinatorial optimization
problems. These problems become intractable when the dimensionality of the data
is large, as is often the case for modern datasets. A popular idea is to
construct convex relaxations of these combinatorial problems, which can be
solved efficiently for large scale datasets.
Semidefinite programming (SDP) relaxations are among the most powerful
methods in this family, and are surprisingly well-suited for a broad range of
problems where data take the form of matrices or graphs. It has been observed
several times that, when the `statistical noise' is small enough, SDP
relaxations correctly detect the underlying combinatorial structures.
In this paper we develop asymptotic predictions for several `detection
thresholds,' as well as for the estimation error above these thresholds. We
study some classical SDP relaxations for statistical problems motivated by
graph synchronization and community detection in networks. We map these
optimization problems to statistical mechanics models with vector spins, and
use non-rigorous techniques from statistical mechanics to characterize the
corresponding phase transitions. Our results clarify the effectiveness of SDP
relaxations in solving high-dimensional statistical problems.Comment: 71 pages, 24 pdf figure
Scaling and Universality in Continuous Length Combinatorial Optimization
We consider combinatorial optimization problems defined over random
ensembles, and study how solution cost increases when the optimal solution
undergoes a small perturbation delta. For the minimum spanning tree, the
increase in cost scales as delta^2; for the mean-field and Euclidean minimum
matching and traveling salesman problems in dimension d>=2, the increase scales
as delta^3; this is observed in Monte Carlo simulations in d=2,3,4 and in
theoretical analysis of a mean-field model. We speculate that the scaling
exponent could serve to classify combinatorial optimization problems into a
small number of distinct categories, similar to universality classes in
statistical physics.Comment: 5 pages; 3 figure
Systems approaches and algorithms for discovery of combinatorial therapies
Effective therapy of complex diseases requires control of highly non-linear
complex networks that remain incompletely characterized. In particular, drug
intervention can be seen as control of signaling in cellular networks.
Identification of control parameters presents an extreme challenge due to the
combinatorial explosion of control possibilities in combination therapy and to
the incomplete knowledge of the systems biology of cells. In this review paper
we describe the main current and proposed approaches to the design of
combinatorial therapies, including the empirical methods used now by clinicians
and alternative approaches suggested recently by several authors. New
approaches for designing combinations arising from systems biology are
described. We discuss in special detail the design of algorithms that identify
optimal control parameters in cellular networks based on a quantitative
characterization of control landscapes, maximizing utilization of incomplete
knowledge of the state and structure of intracellular networks. The use of new
technology for high-throughput measurements is key to these new approaches to
combination therapy and essential for the characterization of control
landscapes and implementation of the algorithms. Combinatorial optimization in
medical therapy is also compared with the combinatorial optimization of
engineering and materials science and similarities and differences are
delineated.Comment: 25 page
Provably Good Solutions to the Knapsack Problem via Neural Networks of Bounded Size
The development of a satisfying and rigorous mathematical understanding of
the performance of neural networks is a major challenge in artificial
intelligence. Against this background, we study the expressive power of neural
networks through the example of the classical NP-hard Knapsack Problem. Our
main contribution is a class of recurrent neural networks (RNNs) with rectified
linear units that are iteratively applied to each item of a Knapsack instance
and thereby compute optimal or provably good solution values. We show that an
RNN of depth four and width depending quadratically on the profit of an optimum
Knapsack solution is sufficient to find optimum Knapsack solutions. We also
prove the following tradeoff between the size of an RNN and the quality of the
computed Knapsack solution: for Knapsack instances consisting of items, an
RNN of depth five and width computes a solution of value at least
times the optimum solution value. Our results
build upon a classical dynamic programming formulation of the Knapsack Problem
as well as a careful rounding of profit values that are also at the core of the
well-known fully polynomial-time approximation scheme for the Knapsack Problem.
A carefully conducted computational study qualitatively supports our
theoretical size bounds. Finally, we point out that our results can be
generalized to many other combinatorial optimization problems that admit
dynamic programming solution methods, such as various Shortest Path Problems,
the Longest Common Subsequence Problem, and the Traveling Salesperson Problem.Comment: A short version of this paper appears in the proceedings of AAAI 202
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