Willem Jan van Hoeve

On global warming: Flow-based soft global constraints

In case a CSP is over-constrained, it is natural to allow some constraints, called soft constraints, to be violated. We propose a generic method to soften global constraints that can be represented by a flow in a graph. Such constraints are softened by adding violation arcs to the graph and then computing a minimum-weight flow in the extended graph to measure the violation. We present efficient propagation algorithms, based on different violation measures, achieving domain consistency for the alldifferent constraint, the global cardinality constraint, the regular constraint and the same constraint.

A systematic approach to MDD-based constraint programming

Fixed-width MDDs were introduced recently as a more refined alternative for the domain store to represent partial solutions to CSPs. In this work, we present a systematic approach to MDD-based constraint programming. First, we introduce a generic scheme for constraint propagation in MDDs. We show that all previously known propagation algorithms for MDDs can be expressed using this scheme. Moreover, we use the scheme to produce algorithms for a number of other constraints, including Among, Element, and unary resource constraints. Finally, we discuss an implementation of our MDD-based CP solver, and provide experimental evidence of the benefits of MDD-based constraint programming.

Manipulating MDD relaxations for combinatorial optimization

We study the application of limited-width MDDs (multivalued decision diagrams) as discrete relaxations for combinatorial optimization problems. These relaxations are used for the purpose of generating lower bounds. We introduce a new compilation method for constructing such MDDs, as well as algorithms that manipulate the MDDs to obtain stronger relaxations and hence provide stronger lower bounds. We apply our methodology to set covering problems, and evaluate the strength of MDD relaxations to relaxations based on linear programming. Our experimental results indicate that the MDD relaxation is particularly effective on structured problems, being able to outperform state-of-the-art integer programming technology by several orders of magnitude.

A hyper-arc consistency algorithm for the soft alldifferent constraint

This paper presents an algorithm that achieves hyper-arc consistency for the soft alldifferent constraint. To this end, we prove and exploit the equivalence with a minimum-cost flow problem. Consistency of the constraint can be checked in O(nm) time, and hyper-arc consistency is achieved in O(m) time, where n is the number of variables involved and m is the sum of the cardinalities of the domains. It improves a previous method that did not ensure hyper-arc consistency.

Variable ordering for the application of BDDs to the maximum independent set problem

The ordering of variables can have a significant effect on the size of the reduced binary decision diagram (BDD) that represents the set of solutions to a combinatorial optimization problem. It also influences the quality of the objective function bound provided by a limited-width relaxation of the BDD. We investigate these effects for the maximum independent set problem. By identifying variable orderings for the BDD, we show that the width of an exact BDD can be given a theoretical upper bound for certain classes of graphs. In addition, we draw an interesting connection between the Fibonacci numbers and the width of exact BDDs for general graphs. We propose variable ordering heuristics inspired by these results, as well as a k-layer look-ahead heuristic applicable to any problem domain. We find experimentally that orderings that result in smaller exact BDDs have a strong tendency to produce tighter bounds in relaxation BDDs.

Connections in networks: a hybrid approach

This paper extends our previous work by exploring the use of a hybrid solution method for solving the connection subgraph problem. We employ a two phase solution method, which drastically reduces the cost of testing for infeasibility and also helps prune the search space for MIP-based optimization. Overall, this provides a much more scalable solution than simply optimizing a MIP model of the problem with Cplex. We report results for semi-structured lattice instances as well as on real data used for the construction of a wildlife corridor for grizzly bears in the Northern Rockies region.

Vehicle routing for food rescue programs: a comparison of different approaches

The 1-Commodity Pickup and Delivery Vehicle Routing Problem (1-PDVRP) asks to deliver a single commodity from a set of supply nodes to a set of demand nodes, which are unpaired. That is, a demand node can be served by any supply node. In this paper, we further assume that the supply and demand is unsplittable, which implies that we can visit each node only once. The 1-PDVRP arises in several practical contexts, ranging from bike-sharing programs in which bikes at each station need to be redistributed at various points in time, to food rescue programs in which excess food is collected from, e.g., restaurants and schools, and redistributed through agencies to people in need. The latter application is the main motivation of our study.

Open constraints in a closed world

We study domain filtering algorithms for open constraints, i.e., constraints that are not a priori defined on specific sets of variables. We present an efficient filtering algorithm, achieving set-domain consistency, for open global cardinality constraints. We extend this result to conjunctions of them, in case they are defined on disjoint sets of variables. We also analyze the case when the sets of variables may overlap. As establishing set-domain consistency is NP-complete in that case, we propose a weaker, though efficient, filtering algorithm instead. Finally, we extend our results to conjunctions of similar open constraints.

Discrete Optimization with Decision Diagrams

We propose a general branch-and-bound algorithm for discrete optimization in which binary decision diagrams (BDDs) play the role of the traditional linear programming relaxation. In particular, relaxed BDD representations of the problem provide bounds and guidance for branching, and restricted BDDs supply a primal heuristic. Each problem is given a dynamic programming model that allows one to exploit recursive structure, even though the problem is not solved by dynamic programming. A novel search scheme branches within relaxed BDDs rather than on values of variables. Preliminary testing shows that a rudimentary BDD-based solver is competitive with or superior to a leading commercial integer programming solver for the maximum stable set problem, the maximum cut problem on a graph, and the maximum 2-satisfiability problem. Specific to the maximum cut problem, we tested the BDD-based solver on a classical benchmark set and identified tighter relaxation bounds than have ever been found by any technique, nearly closing the entire optimality gap on four large-scale instances.

New filtering algorithms for combinations of among constraints

Several combinatorial problems, such as car sequencing and rostering, feature sequence constraints, restricting the number of occurrences of certain values in every subsequence of a given length. We present three new filtering algorithms for the sequence constraint, including the first that establishes domain consistency in polynomial time. The filtering algorithms have complementary strengths: One borrows ideas from dynamic programming; another reformulates it as a regular constraint; the last is customized. The last two algorithms establish domain consistency, and the customized one does so in polynomial time. We provide experimental results that demonstrate the practical usefulness of each. We also show that the customized algorithm applies naturally to a generalized version of the sequence constraint that allows subsequences of varied lengths. The significant computational advantage of using a single generalized sequence constraint over a semantically equivalent collection of among or sequence constraints is demonstrated empirically.