How Good are Genetic Algorithms at Finding Large Cliques: An Experimental Study

This paper investigates the power of genetic algorithms at solving the MAX-CLIQUE problem. We measure the performance of a standard genetic algorithm on an elementary set of problem instances consisting of embedded cliques in random graphs. We indicate the need for improvement, and introduce a new genetic algorithm, the multi-phase annealed GA, which exhibits superior performance on the same problem set. As we scale up the problem size and test on \hard" benchmark instances, we notice a degraded performance in the algorithm caused by premature convergence to local minima. To alleviate this problem, a sequence of modi cations are implemented ranging from changes in input representation to systematic local search. The most recent version, called union GA, incorporates the features of union cross-over, greedy replacement, and diversity enhancement. It shows a marked speed-up in the number of iterations required to find a given solution, as well as some improvement in the clique size found. We discuss issues related to the SIMD implementation of the genetic algorithms on a Thinking Machines CM-5, which was necessitated by the intrinsically high time complexity (O(n3)) of the serial algorithm for computing one iteration. Our preliminary conclusions are: (1) a genetic algorithm needs to be heavily customized to work "well" for the clique problem; (2) a GA is computationally very expensive, and its use is only recommended if it is known to find larger cliques than other algorithms; (3) although our customization e ort is bringing forth continued improvements, there is no clear evidence, at this time, that a GA will have better success in circumventing local minima.

On the Effectiveness of Genetic Search in Combinatorial Optimization

In this paper, we study the efficacy of genetic algorithms in the context of combinatorial optimization. In particular, we isolate the effects of cross-over, treated as the central component of genetic search. We show that for problems of nontrivial size and difficulty, the contribution of cross-over search is marginal, both synergistically when run in conjunction with mutation and selection, or when run with selection alone, the reference point being the search procedure consisting of just mutation and selection. The latter can be viewed as another manifestation of the Metropolis process. Considering the high computational cost of maintaining a population to facilitate cross-over search, its marginal benefit renders genetic search inferior to its singleton-population counterpart, the Metropolis process, and by extension, simulated annealing. This is further compounded by the fact that many problems arising in practice may inherently require a large number of state transitions for a near-optimal solution to be found, making genetic search infeasible given the high cost of computing a single iteration in the enlarged state-space.

The Specialized Mappings Architecture

A probabilistic, nonlinear supervised learning model is proposed: the Specialized Mappings Architecture (SMA). The SMA employs a set of several forward mapping functions that are estimated automatically from training data. Each specialized function maps certain domains of the input space (e.g., image features) onto the output space (e.g., articulated body parameters). The SMA can model ambiguous, one-to-many mappings that may yield multiple valid output hypotheses. Once learned, the mapping functions generate a set of output hypotheses for a given input via a statistical inference procedure. The SMA inference procedure incorporates an inverse mapping or feedback function in evaluating the likelihood of each of the hypothesis. Possible feedback functions include computer graphics rendering routines that can generate images for given hypotheses. The SMA employs a variant of the Expectation-Maximization algorithm for simultaneous learning of the specialized domains along with the mapping functions, and approximate strategies for inference. The framework is demonstrated in a computer vision system that can estimate the articulated pose parameters of a human’s body or hands, given silhouettes from a single image. The accuracy and stability of the SMA are also tested using synthetic images of human bodies and hands, where ground truth is known.

Stochastic Refinement of the Visual Hull to Satisfy Photometric and Silhouette Consistency Constraints

An iterative method for reconstructing a 3D polygonal mesh and color texture map from multiple views of an object is presented. In each iteration, the method first estimates a texture map given the current shape estimate. The texture map and its associated residual error image are obtained via maximum a posteriori estimation and reprojection of the multiple views into texture space. Next, the surface shape is adjusted to minimize residual error in texture space. The surface is deformed towards a photometrically-consistent solution via a series of 1D epipolar searches at randomly selected surface points. The texture space formulation has improved computational complexity over standard image-based error approaches, and allows computation of the reprojection error and uncertainty for any point on the surface. Moreover, shape adjustments can be constrained such that the recovered model's silhouette matches those of the input images. Experiments with real world imagery demonstrate the validity of the approach.

3D Hand Pose Estimation by Finding Appearance-Based Matches in a Large Database of Training Views

Ongoing work towards appearance-based 3D hand pose estimation from a single image is presented. A large database of synthetic hand views is generated using a 3D hand model and computer graphics. The views display different hand shapes as seen from arbitrary viewpoints. Each synthetic view is automatically labeled with parameters describing its hand shape and viewing parameters. Given an input image, the system retrieves the most similar database views, and uses the shape and viewing parameters of those views as candidate estimates for the parameters of the input image. Preliminary results are presented, in which appearance-based similarity is defined in terms of the chamfer distance between edge images.

3D Hand Pose Reconstruction Using Specialized Mappings

A system for recovering 3D hand pose from monocular color sequences is proposed. The system employs a non-linear supervised learning framework, the specialized mappings architecture (SMA), to map image features to likely 3D hand poses. The SMA's fundamental components are a set of specialized forward mapping functions, and a single feedback matching function. The forward functions are estimated directly from training data, which in our case are examples of hand joint configurations and their corresponding visual features. The joint angle data in the training set is obtained via a CyberGlove, a glove with 22 sensors that monitor the angular motions of the palm and fingers. In training, the visual features are generated using a computer graphics module that renders the hand from arbitrary viewpoints given the 22 joint angles. We test our system both on synthetic sequences and on sequences taken with a color camera. The system automatically detects and tracks both hands of the user, calculates the appropriate features, and estimates the 3D hand joint angles from those features. Results are encouraging given the complexity of the task.

Dynamic Programming for Optimal Control of Set-Up Scheduling with Neural Network Modifications

This paper demonstrates an optimal control solution to change of machine set-up scheduling based on dynamic programming average cost per stage value iteration as set forth by Cararnanis et. al. [2] for the 2D case. The difficulty with the optimal approach lies in the explosive computational growth of the resulting solution. A method of reducing the computational complexity is developed using ideas from biology and neural networks. A real time controller is described that uses a linear-log representation of state space with neural networks employed to fit cost surfaces.