Searching for the M Best Solutions in Graphical Models

Abstract

The paper focuses on finding the m best solutions to combinatorial optimization problems using best-first or depth-first branch and bound search. Specifically, we present a new algorithm m-A*, extending the well-known A* to the m-best task, and for the first time prove that all its desirable properties, including soundness, completeness and optimal efficiency, are maintained. Since bestfirst algorithms require extensive memory, we also extend the memory-efficient depth-first branch and bound to the m-best task.We adapt both algorithms to optimization tasks over graphical models (e.g., Weighted CSP and MPE in Bayesian networks), provide complexity analysis and an empirical evaluation. Our experiments confirm theory that the best-first approach is largely superior when memory is available, but depth-first branch and bound is more robust. We also show that our algorithms are competitive with related schemes recently developed for the m-best task.

Tree projections and constraint optimization problems: Fixed-parameter tractability and parallel algorithms

“…Moreover, we might be interested in the Top-K problem of enumerating the best (w.r.t. Max or Min) K solutions in form of a ranked list (see, e.g., [25,10]), 1 or even in the Next problem of computing the next solution (w.r.t. such an ordering) following one that is at hand [9].…”

“…All previous algorithms proposed in the literature for computing the best CSP solutions in polynomial time [25,37,53,8,67,30,40,49,1] (or, more generally, for optimizing functions in different application domains-see, e.g., [58]) require the knowledge of some suitable tree projection, which provides at each node a list of potentially good partial evaluations with their associated values to be propagated within a dynamic programming scheme. The main conceptual contribution of the present paper is to show that this knowledge is not necessary, since promise-free algorithms can be exhibited in the tree projection framework even when dealing with optimization problems.…”

“…The result can be established by exploiting a method proposed by Lawler [59] for ranking solutions to discrete optimization problems. In fact, the method has been already discussed in the context of inference in graphical models [25] and in conjunctive query evaluation [54]. Reformulated in the CSP context, for a CSP instance over n variables, the idea is to first compute the optimal solution (w.r.t.…”

Predicting building façade structures with multilinear Gaussian graphical models based on few observations

“…They extended the formalism of interval bounds by CDFs that enable to represent a degree of knowledge for uncertain data. Flerova and Dechter (2010) solve the problem to find the m best solutions for optimization tasks in graphical models. To this end, combination and marginalization operators are adapted in order to generate a sorted list of solutions in tree decompositions.…”

Pattern Recognition as an Implementation of a Certain Type of Thought Processes

“…Задача в этом случае заключается в поиске не произвольного решения системы ограничений, а наилучшего решения, оптимизирующего заданную функцию качества. Следующий этап обобщения состоит в том, что для заданной системы ограничений и функции качества на множестве ее решений требуется найти не единственное ее решение, пусть и наилучшее, а заданное количество наилучших решений [19].…”