Postponing Branching Decisions

Solution techniques for Constraint Satisfaction and Optimisation Problems often make use of backtrack search methods, exploiting variable and value ordering heuristics. In this paper, we propose and analyse a very simple method to apply in case the value ordering heuristic produces ties: postponing the branching decision. To this end, we group together values in a tie, branch on this sub-domain, and defer the decision among them to lower levels of the search tree. We show theoretically and experimentally that this simple modification can dramatically improve the efficiency of the search strategy. Although in practise similar methods may have been applied already, to our knowledge, no empirical or theoretical study has been proposed in the literature to identify when and to what extent this strategy should be used.Comment: 11 pages, 3 figure

Decomposition Based Search - A theoretical and experimental evaluation

In this paper we present and evaluate a search strategy called Decomposition Based Search (DBS) which is based on two steps: subproblem generation and subproblem solution. The generation of subproblems is done through value ranking and domain splitting. Subdomains are explored so as to generate, according to the heuristic chosen, promising subproblems first. We show that two well known search strategies, Limited Discrepancy Search (LDS) and Iterative Broadening (IB), can be seen as special cases of DBS. First we present a tuning of DBS that visits the same search nodes as IB, but avoids restarts. Then we compare both theoretically and computationally DBS and LDS using the same heuristic. We prove that DBS has a higher probability of being successful than LDS on a comparable number of nodes, under realistic assumptions. Experiments on a constraint satisfaction problem and an optimization problem show that DBS is indeed very effective if compared to LDS.Comment: 16 pages, 8 figures. LIA Technical Report LIA00203, University of Bologna, 200

The motion of a deformable drop in a second-order fluid

The cross-stream migration of a deformable drop in a unidirectional shear flow of a second-order fluid is considered. Expressions for the particle velocity due to the separate effects of deformation and viscoelastic rheology are obtained. The direction and magnitude of migration are calculated for the particular cases of Poiseuille flow and simple shear flow and compared with experimental data

Statistical Mechanics of High-Dimensional Inference

To model modern large-scale datasets, we need efficient algorithms to infer a set of $P$ unknown model parameters from $N$ noisy measurements. What are fundamental limits on the accuracy of parameter inference, given finite signal-to-noise ratios, limited measurements, prior information, and computational tractability requirements? How can we combine prior information with measurements to achieve these limits? Classical statistics gives incisive answers to these questions as the measurement density $\alpha = \frac{N}{P}\rightarrow \infty$. However, these classical results are not relevant to modern high-dimensional inference problems, which instead occur at finite $\alpha$. We formulate and analyze high-dimensional inference as a problem in the statistical physics of quenched disorder. Our analysis uncovers fundamental limits on the accuracy of inference in high dimensions, and reveals that widely cherished inference algorithms like maximum likelihood (ML) and maximum-a posteriori (MAP) inference cannot achieve these limits. We further find optimal, computationally tractable algorithms that can achieve these limits. Intriguingly, in high dimensions, these optimal algorithms become computationally simpler than MAP and ML, while still outperforming them. For example, such optimal algorithms can lead to as much as a 20% reduction in the amount of data to achieve the same performance relative to MAP. Moreover, our analysis reveals simple relations between optimal high dimensional inference and low dimensional scalar Bayesian inference, insights into the nature of generalization and predictive power in high dimensions, information theoretic limits on compressed sensing, phase transitions in quadratic inference, and connections to central mathematical objects in convex optimization theory and random matrix theory.Comment: See http://ganguli-gang.stanford.edu/pdf/HighDimInf.Supp.pdf for supplementary materia