Extremal Optimization at the Phase Transition of the 3-Coloring Problem

We investigate the phase transition of the 3-coloring problem on random graphs, using the extremal optimization heuristic. 3-coloring is among the hardest combinatorial optimization problems and is closely related to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the ``backbone'', an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size $n$ up to 512. For graphs up to this size, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about $O(n^{3.5})$ update steps. Finite size scaling gives a critical mean degree value $\alpha_{\rm c}=4.703(28)$. Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available at http://www.physics.emory.edu/faculty/boettcher

Predict or classify: The deceptive role of time-locking in brain signal classification

Several experimental studies claim to be able to predict the outcome of simple decisions from brain signals measured before subjects are aware of their decision. Often, these studies use multivariate pattern recognition methods with the underlying assumption that the ability to classify the brain signal is equivalent to predict the decision itself. Here we show instead that it is possible to correctly classify a signal even if it does not contain any predictive information about the decision. We first define a simple stochastic model that mimics the random decision process between two equivalent alternatives, and generate a large number of independent trials that contain no choice-predictive information. The trials are first time-locked to the time point of the final event and then classified using standard machine-learning techniques. The resulting classification accuracy is above chance level long before the time point of time-locking. We then analyze the same trials using information theory. We demonstrate that the high classification accuracy is a consequence of time-locking and that its time behavior is simply related to the large relaxation time of the process. We conclude that when time-locking is a crucial step in the analysis of neural activity patterns, both the emergence and the timing of the classification accuracy are affected by structural properties of the network that generates the signal.Comment: 23 pages, 5 figure

Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications

In computer vision, many problems such as image segmentation, pixel labelling, and scene parsing can be formulated as binary quadratic programs (BQPs). For submodular problems, cuts based methods can be employed to efficiently solve large-scale problems. However, general nonsubmodular problems are significantly more challenging to solve. Finding a solution when the problem is of large size to be of practical interest, however, typically requires relaxation. Two standard relaxation methods are widely used for solving general BQPs--spectral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high, especially for large scale problems. In this work, we present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, compared with conventional SDP methods, the new SDP formulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the dual problem. Both of them are significantly more efficiently than standard interior-point methods. In practice, the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems. Our experiments on a few computer vision applications including clustering, image segmentation, co-segmentation and registration show the potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern Analysis and Machine Intelligenc

A Domain-Independent Algorithm for Plan Adaptation

The paradigms of transformational planning, case-based planning, and plan debugging all involve a process known as plan adaptation - modifying or repairing an old plan so it solves a new problem. In this paper we provide a domain-independent algorithm for plan adaptation, demonstrate that it is sound, complete, and systematic, and compare it to other adaptation algorithms in the literature. Our approach is based on a view of planning as searching a graph of partial plans. Generative planning starts at the graph's root and moves from node to node using plan-refinement operators. In planning by adaptation, a library plan - an arbitrary node in the plan graph - is the starting point for the search, and the plan-adaptation algorithm can apply both the same refinement operators available to a generative planner and can also retract constraints and steps from the plan. Our algorithm's completeness ensures that the adaptation algorithm will eventually search the entire graph and its systematicity ensures that it will do so without redundantly searching any parts of the graph.Comment: See http://www.jair.org/ for any accompanying file

Newton-Raphson Consensus for Distributed Convex Optimization

We address the problem of distributed uncon- strained convex optimization under separability assumptions, i.e., the framework where each agent of a network is endowed with a local private multidimensional convex cost, is subject to communication constraints, and wants to collaborate to compute the minimizer of the sum of the local costs. We propose a design methodology that combines average consensus algorithms and separation of time-scales ideas. This strategy is proved, under suitable hypotheses, to be globally convergent to the true minimizer. Intuitively, the procedure lets the agents distributedly compute and sequentially update an approximated Newton- Raphson direction by means of suitable average consensus ratios. We show with numerical simulations that the speed of convergence of this strategy is comparable with alternative optimization strategies such as the Alternating Direction Method of Multipliers. Finally, we propose some alternative strategies which trade-off communication and computational requirements with convergence speed.Comment: 18 pages, preprint with proof

The structure of Inter-Urban traffic: A weighted network analysis

We study the structure of the network representing the interurban commuting traffic of the Sardinia region, Italy, which amounts to 375 municipalities and 1,600,000 inhabitants. We use a weighted network representation where vertices correspond to towns and the edges to the actual commuting flows among those. We characterize quantitatively both the topological and weighted properties of the resulting network. Interestingly, the statistical properties of commuting traffic exhibit complex features and non-trivial relations with the underlying topology. We characterize quantitatively the traffic backbone among large cities and we give evidences for a very high heterogeneity of the commuter flows around large cities. We also discuss the interplay between the topological and dynamical properties of the network as well as their relation with socio-demographic variables such as population and monthly income. This analysis may be useful at various stages in environmental planning and provides analytical tools for a wide spectrum of applications ranging from impact evaluation to decision-making and planning support.Comment: 12 pages, 12 figures, 4 tables; 1 missing ref added and minor revision