Automatically improving constraint models in Savile Row

Abstract

When solving a combinatorial problem using Constraint Programming (CP) or Satisfiability (SAT), modelling and formulation are vital and difficult tasks. Even an expert human may explore many alternatives in modelling a single problem. We make a number of contributions in the automated modelling and reformulation of constraint models. We study a range of automated reformulation techniques, finding combinations of techniques which perform particularly well together. We introduce and describe in detail a new algorithm, X-CSE, to perform Associative-Commutative Common Subexpression Elimination (AC-CSE) in constraint problems, significantly improving existing CSE techniques for associative and commutative operators such as +. We demonstrate that these reformulation techniques can be integrated in a single automated constraint modelling tool, called Savile Row, whose architecture we describe. We use Savile Row as an experimental testbed to evaluate each reformulation on a set of 50 problem classes, with 596 instances in total. Our recommended reformulations are well worthwhile even including overheads, especially on harder instances where solver time dominates. With a SAT solver we observed a geometric mean of 2.15 times speedup compared to a straightforward tailored model without recommended reformulations. Using a CP solver, we obtained a geometric mean of 5.96 times speedup for instances taking over 10 seconds to solve.

Closed Frequent Itemset Mining with Arbitrary Side Constraints

“…forAll (sol, sup) in solutions_so_far . 17 (fis [1] subset sol) -> (fis [2] > sup) frequent itemsets and to produce more focused results. There are broadly two categories of these extensions in the literature:…”

Automatic Generation and Selection of Streamlined Constraint Models via Monte Carlo Search on a Model Lattice

“…Hence, streamliners related to sets and functions, such as that given in the introduction, can be generated straightforwardly. In contrast an equivalent constraint model in, for example, MiniZinc [29] or Essence Prime [31] has to represent these abstract decision variables with constrained collections of more primitive (e.g. integer domain) variables, such as the matrix model [11,12] proposed by Smith et al [35].…”

“…Using training instances drawn from the problem class under consideration, streamliner candidates are evaluated via a toolchain consisting of the automated constraint modelling tools Conjure [1][2][3][4] and Savile Row [30][31][32], and the constraint solver Minion [20]. Promising candidates, which retain at least one solution to the training instances while significantly reducing search, are used to solve more difficult instances from the same problem class.…”

Complexity of n-Queens Completion

“…Further, since we can generate hard instances of the Excluded Diagonals Problem, this approach gives us confidence that we can also generate hard instances of n-Queens Completion. The third approach uses the constraint solver Minion 1.8 (Gent et al, 2006), with a constraint model optimised by Savile Row 1.6.4 (Nightingale et al, 2017). For each row β, the model has three variables representing the column, sum-and difference-diagonal of the queen placed on row β.…”

“…These are respectively {0, 4, 5} (the 'red diagonals') and {1, 2, 6} (the 'blue diagonals'). The gadget G 0 was found by a search using Savile Row (Nightingale et al, 2017) and Minion (Gent et al, 2006) from a model available online. 6 For a variable v i in a clause c we will use one red diagonal and one blue diagonal from the i th copy of G 0 .…”