Marcus Poggi de Aragão

Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem

The best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) have been based on either branch-and-cut or Lagrangean relaxation/column generation. This paper presents an algorithm that combines both approaches: it works over the intersection of two polytopes, one associated with a traditional Lagrangean relaxation over q-routes, the other defined by bound, degree and capacity constraints. This is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods. The resulting branch-and-cut-and-price algorithm can solve to optimality all instances from the literature with up to 135 vertices. This more than doubles the size of the instances that can be consistently solved.

A hybrid approach for searching in the semantic web

This paper presents a search architecture that combines classical search techniques with spread activation techniques applied to a semantic model of a given domain. Given an ontology, weights are assigned to links based on certain properties of the ontology, so that they measure the strength of the relation. Spread activation techniques are used to find related concepts in the ontology given an initial set of concepts and corresponding initial activation values. These initial values are obtained from the results of classical search applied to the data associated with the concepts in the ontology. Two test cases were implemented, with very positive results. It was also observed that the proposed hybrid spread activation, combining the symbolic and the sub-symbolic approaches, achieved better results when compared to each of the approaches alone.

Column Generation Methods for Probabilistic Logic

Nilsson recently introduced a rigorous semantic generalization of logic in which the truth values of sentences are probability values. This led to state precisely several basic problems of artificial intelligence, a paradigm of which is probabilistic satisfiability (PSAT): determine, given a set of clauses and probabilities that these clauses are true, whether these probabilities are consistent. We consider several extensions of this model involving intervals on probability values, conditional probabilities and minimal modifications of probability values to ensure satisfiability. Investigating further an approach of G. Georgakopoulos, D. Kavvadias and C. H. Papadimitriou, we propose a column generation algorithm which allows to solve exactly all these extensions. Computational experience shows that large problems, with up to 140 variables and 300 clauses, may be solved in reasonable time. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Solving capacitated arc routing problems using a transformation to the CVRP

A well-known transformation by Pearn, Assad and Golden reduces a capacitated arc routing problem (CARP) into an equivalent capacitated vehicle routing problem (CVRP). However, that transformation is regarded as unpractical, since an original instance with r required edges is turned into a CVRP over a complete graph with 3r + 1 vertices. We propose a similar transformation that reduces this graph to 2r + 1 vertices, with the additional restriction that a previously known set of r pairwise disconnected edges must belong to every solution. Using a recent branch-and-cut-and-price algorithm for the CVRP, we observed that it yields an effective way of attacking the CARP, being significantly better than the exact methods created specifically for that problem. Computational experiments obtained improved lower bounds for almost all open instances from the literature. Several such instances could be solved to optimality.Scope and purpose The scope of this paper is transforming arc routing problems into node routing problems. The paper shows that this approach can be effective and, in particular, that the original instances may generate node routing instances that behave as if the size is not increased. This result is obtained by slightly modifying the well-known transformation by Pearn, Assad and Golden from capacitated arc routing problem (CARP) to the capacitated vehicle routing problem (CVRP), that is regarded as unpractical. The paper provides a computational experience using a recent branch-and-cut-and-price algorithm for the CVRP. The results are significantly better than the exact methods created specifically for that problem, improving lower bounds for almost all open instances from the literature. Several such instances could be solved to optimality.

Hyperbolic 0-1 programming and query optimization in information retrieval

Unconstrained hyperbolic 0–1 programming can be solved in linear time when the numerator and the denominator are linear and the latter is always positive. It is NP-hard, and finding an approximate solution with a value equal to a positive multiple of the optimal one is also NP-hard, if this last hypothesis does not hold. Determining the optimal logical form of a query in information retrieval, given the attributes to be used, can be expressed as a parametric hyperbolic 0–1 program and solved in O(n logn) time, wheren is the number of elementary logical conjunctions of the attributes. This allows to characterize the optimal queries for the Van Rijsbergen synthetic criterion.

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms: powerful new cuts expressed over a very large set of variables are added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.

Dual Heuristics on the Exact Solution of Large Steiner Problems

Abstract Abstract We present dual heuristics for the directed cut formulation of the Steiner problem in graphs. These heuristics usually give tight lower and upper bounds, and are enough to quickly solve two thirds of the instances from the literature. For harder instances, we propose two exact algorithms using those heuristics: branch-and-ascent, an implicit enumeration without LP solving; and a branch-and-cut that starts from bases provided by dual heuristics, which may be called afterwards to improve convergence. These algorithms have a good practical performance and solved several open instances, including the 1320 series and very large and degenerated problems from VLSI layout.

Stabilized branch-and-cut-and-price for the generalized assignment problem

The Generalized Assignment Problem (GAP) is a classic scheduling problem with many applications. We propose a branch-and-cut-andprice for that problem featuring a stabilization mechanism to accelerate column generation convergence. We also propose ellipsoidal cuts, a new way of transforming the exact algorithm into a powerful heuristic, in the same spirit of the cuts recently proposed by Fischetti and Lodi. The improved solutions found by this heuristic can, in turn, help the task of the exact algorithm. The resulting algorithms showed a very good performance and were able to solve three among the last five open instances from the OR-Library.

Preprocessing Steiner problems from VLSI layout

VLSI layout applications yield instances of the Steiner tree problem over grid graphs with holes, which are considered hard to be solved by current methods. In particular, preprocessing techniques developed for Steiner problems over general graphs are not likely to reduce, significantly, such VLSI instances. We propose a new preprocessing procedure, extending earlier ideas from the literature and improving their application, so as to make them effective for VLSI problems. We report significant reductions within reasonable computational times, obtained with the application of this procedure to 116 instances of the SteinLib. These reductions allowed a branch and cut to solve 28 of 32 open instances of the SteinLib, some with more than 10,000 vertices and 20,000 edges.

Local optima topology for the k -coloring problem

Abstract In a first attempt to explain the good behaviour of local improvement heuristics (such as tabu search and simulated annealing) for the k-coloring problem, we study the topology of the local optima. We first propose an efficient procedure for generating all local optima. Then, we devise statistical tests for analyzing their topology, e.g., the number of local optima with respect to their value or the depth of the valleys. Finally, computational results are presented.