Greedy algorithm

About: Greedy algorithm is a research topic. Over the lifetime, 15347 publications have been published within this topic receiving 393945 citations.

A fundamental problem in vehicle routing

Abstract: An important but difficult combinatorial problem, in general, is to find the optimal route for a single vehicle on a given network. This paper defines a problem type, called the General Routing Problem, and gives an algorithm for its solution. The classical Traveling Salesman Problem and the Chinese Postman Problem are shown to be special limiting cases of the General Routing Problem. The algorithm provides a unified approach to both node and arc oriented routing problems, and exploits special properties of most real transportation networks such as sparsity of the associated adjacency matrix, and the tendency for arc symmetry at many nodes. For node oriented routing problems, this approach tends to produce large reduction in effective problem size.

ADMiRA: Atomic Decomposition for Minimum Rank Approximation

Abstract: We address the inverse problem that arises in compressed sensing of a low-rank matrix. Our approach is to pose the inverse problem as an approximation problem with a specified target rank of the solution. A simple search over the target rank then provides the minimum rank solution satisfying a prescribed data approximation bound. We propose an atomic decomposition that provides an analogy between parsimonious representations of a sparse vector and a low-rank matrix. Efficient greedy algorithms to solve the inverse problem for the vector case are extended to the matrix case through this atomic decomposition. In particular, we propose an efficient and guaranteed algorithm named ADMiRA that extends CoSaMP, its analogue for the vector case. The performance guarantee is given in terms of the rank-restricted isometry property and bounds both the number of iterations and the error in the approximate solution for the general case where the solution is approximately low-rank and the measurements are noisy. With a sparse measurement operator such as the one arising in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. The numerical experiments for the matrix completion problem show that, although the measurement operator in this case does not satisfy the rank-restricted isometry property, ADMiRA is a competitive algorithm for matrix completion.

Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems

Abstract: A greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters. Large-scale models are too costly to evaluate repeatedly, as is required in the statistical setting. Furthermore, these models often have high-dimensional parametric input spaces, which compounds the difficulty of effectively exploring the uncertainty space. We simultaneously address both challenges by constructing a projection-based reduced model that accepts low-dimensional parameter inputs and whose model evaluations are inexpensive. The associated parameter and state bases are obtained through a greedy procedure that targets the governing equations, model outputs, and prior information. The methodology and results are presented for groundwater inverse problems in one and two dimensions.