We propose a new global SPACING constraint that is useful in modeling events that are distributed over time, like learning units scheduled over a study program or repeated patterns in music compositions. First, we investigate theoretical properties o
f the constraint and identify tractable special cases. We propose efficient DC filtering algorithms for these cases. Then, we experimentally evaluate performance of the proposed algorithms on a music composition problem and demonstrate that our filtering algorithms outperform the state-of-the-art approach for solving this problem.
We propose a simple method for combining together voting rules that performs a run-off between the different winners of each voting rule. We prove that this combinator has several good properties. For instance, even if just one of the base voting rul
es has a desirable property like Condorcet consistency, the combination inherits this property. In addition, we prove that combining voting rules together in this way can make finding a manipulation more computationally difficult. Finally, we study the impact of this combinator on approximation methods that find close to optimal manipulations.
Nansons and Baldwins voting rules select a winner by successively eliminating candidates with low Borda scores. We show that these rules have a number of desirable computational properties. In particular, with unweighted votes, it is NP-hard to manip
ulate either rule with one manipulator, whilst with weighted votes, it is NP-hard to manipulate either rule with a small number of candidates and a coalition of manipulators. As only a couple of other voting rules are known to be NP-hard to manipulate with a single manipulator, Nansons and Baldwins rules appear to be particularly resistant to manipulation from a theoretical perspective. We also propose a number of approximation methods for manipulating these two rules. Experiments demonstrate that both rules are often difficult to manipulate in practice. These results suggest that elimination style voting rules deserve further study.
Dealing with large numbers of symmetries is often problematic. One solution is to focus on just symmetries that generate the symmetry group. Whilst there are special cases where breaking just the symmetries in a generating set is complete, there are
also cases where no irredundant generating set eliminates all symmetry. However, focusing on just generators improves tractability. We prove that it is polynomial in the size of the generating set to eliminate all symmetric solutions, but NP-hard to prune all symmetric values. Our proof considers row and column symmetry, a common type of symmetry in matrix models where breaking just generator symmetries is very effective. We show that propagating a conjunction of lexicographical ordering constraints on the rows and columns of a matrix of decision variables is NP-hard.
We propose a new family of constraints which combine together lexicographical ordering constraints for symmetry breaking with other common global constraints. We give a general purpose propagator for this family of constraints, and show how to improv
e its complexity by exploiting properties of the included global constraints.
An attractive mechanism to specify global constraints in rostering and other domains is via formal languages. For instance, the Regular and Grammar constraints specify constraints in terms of the languages accepted by an automaton and a context-free
grammar respectively. Taking advantage of the fixed length of the constraint, we give an algorithm to transform a context-free grammar into an automaton. We then study the use of minimization techniques to reduce the size of such automata and speed up propagation. We show that minimizing such automata after they have been unfolded and domains initially reduced can give automata that are more compact than minimizing before unfolding and reducing. Experimental results show that such transformations can improve the size of rostering problems that we can model and run.
A wide range of constraints can be compactly specified using automata or formal languages. In a sequence of recent papers, we have shown that an effective means to reason with such specifications is to decompose them into primitive constraints. We ca
n then, for instance, use state of the art SAT solvers and profit from their advanced features like fast unit propagation, clause learning, and conflict-based search heuristics. This approach holds promise for solving combinatorial problems in scheduling, rostering, and configuration, as well as problems in more diverse areas like bioinformatics, software testing and natural language processing. In addition, decomposition may be an effective method to propagate other global constraints.
