Faster Symmetry Breaking Constraints for Abstract Structures

In constraint programming and related paradigms, a modeller specifies their problem in a modelling language for a solver to search and return its solution(s). Using high-level modelling languages such as Essence, a modeller may express their problems in terms of abstract structures. These are structures not natively supported by the solvers, and so they have to be transformed into or represented as other structures before solving. For example, nested sets are abstract structures, and they can be represented as matrices in constraint solvers. Many problems contain symmetries and one very common and highly successful technique used in constraint programming is to "break" symmetries, to avoid searching for symmetric solutions. This can speed up the solving process by many orders of magnitude. Most of these symmetry-breaking techniques involve placing some kind of ordering for the variables of the problem, and picking a particular member under the symmetries, usually the smallest. Unfortunately, applying this technique to abstract variables produces a very large number of complex constraints that perform poorly in practice. In this paper, we demonstrate a new incomplete method of breaking the symmetries of abstract structures by better exploiting their representations. We apply the method in breaking the symmetries arising from indistinguishable objects, a commonly occurring type of symmetry, and show that our method is faster than the previous methods proposed in (Akgün et al. 2025).

Breaking the Symmetries of Indistinguishable Objects

Indistinguishable objects often occur when modelling problems in constraint programming, as well as in other related paradigms. They occur when objects can be viewed as being drawn from a set of unlabelled objects, and the only operation allowed on them is equality testing. For example, the golfers in the social golfer problem are indistinguishable. If we do label the golfers, then any relabelling of the golfers in one solution gives another valid solution. Therefore, we can regard the symmetric group of size $n$ as acting on a set of $n$ indistinguishable objects. In this paper, we show how we can break the symmetries resulting from indistinguishable objects. We show how symmetries on indistinguishable objects can be defined properly in complex types, for example in a matrix indexed by indistinguishable objects. We then show how the resulting symmetries can be broken correctly. In Essence, a high-level modelling language, indistinguishable objects are encapsulated in "unnamed types". We provide an implementation of complete symmetry breaking for unnamed types in Essence.

Towards Automatic Design of Factorio Blueprints

Factorio is a 2D construction and management simulation video game about building automated factories to produce items of increasing complexity. A core feature of the game is its blueprint system, which allows players to easily save and replicate parts of their designs. Blueprints can reproduce any layout of objects in the game, but are typically used to encapsulate a complex behaviour, such as the production of a non-basic object. Once created, these blueprints are then used as basic building blocks, allowing the player to create a layer of abstraction. The usage of blueprints not only eases the expansion of the factory but also allows the sharing of designs with the game's community. The layout in a blueprint can be optimised using various criteria, such as the total space used or the final production throughput. The design of an optimal blueprint is a hard combinatorial problem, interleaving elements of many well-studied problems such as bin-packing, routing or network design. This work presents a new challenging problem and explores the feasibility of a constraint model to optimise Factorio blueprints, balancing correctness, optimality, and performance.