A Global Constraint for the Exact Cover Problem: Application to Conceptual Clustering

Abstract


 
 
 We introduce the exactCover global constraint dedicated to the exact cover problem, the goal of which is to select subsets such that each element of a given set belongs to exactly one selected subset. This NP-complete problem occurs in many applications, and we more particularly focus on a conceptual clustering application.
 We introduce three propagation algorithms for exactCover, called Basic, DL, and DL+: Basic ensures the same level of consistency as arc consistency on a classical decomposition of exactCover into binary constraints, without using any specific data structure; DL ensures the same level of consistency as Basic but uses Dancing Links to efficiently maintain the relation between elements and subsets; and DL+ is a stronger propagator which exploits an extra property to filter more values than DL.
 We also consider the case where the number of selected subsets is constrained to be equal to a given integer variable k, and we show that this may be achieved either by combining exactCover with existing constraints, or by designing a specific propagator that integrates algorithms designed for the NValues constraint.
 These different propagators are experimentally evaluated on conceptual clustering problems, and they are compared with state-of-the-art declarative approaches. In particular, we show that our global constraint is competitive with recent ILP and CP models for mono-criterion problems, and it has better scale-up properties for multi-criteria problems.
 
 

Solving Exact Cover Instances with Molecular-Motor-Powered Network-Based Biocomputation

“…Many sophisticated algorithms have been developed to solve Exact Cover problems. One of these, DLX, uses specific data structures (Dancing links) and is widely used to address applications such as data clustering . However, in the worst case, the processing times and energy consumption of these algorithms still increase exponentially with the number of elements in S . , …”

Compressing Exact Cover Problems with Zero-suppressed Binary Decision Diagrams

“…Knuth [Knuth, 2019] shows a few crucial extensions of exact cover problems, including exact cover with colors (XCC) problems and XCC with multiplicity problems. Another extension is made in [Chabert and Solnon, 2020] to cope with different types of constraints. [Nishino et al, 2017] also combines the ZDD with DLX.…”

“…An important target is graph substructures [Kawahara et al, 2017]. Thus, ZDDs and binary decision diagrams [Bryant, 1986] have been used in graph-related problems like network analyses [Hardy et al, 2007], probabilistic inference in a structured space [Choi et al, 2016], and combinatorial optimization [Inoue et al, 2014]. The problems we addressed in the experiments are typical exact cover problems related to graphs.…”

“…DLX performs an exhaustive depth-first backtracking-based search by exploiting doubly linked list structures called dancing links. DLX is known as the state-of-the-art method for finding all solutions of an exact cover problem [Junttila and Kaski, 2010]. Enumerating solutions is beneficial if we consider exact covers appearing in practical situations, such as designing electric circuits [Chang and Jiang, 2016], designing 3D shapes from small fragments [Hu et al, 2014], and timetabling [Nguyen et al, 2018].…”

“…DLX is known as the state-of-the-art method for finding all solutions of an exact cover problem [Junttila and Kaski, 2010]. Enumerating solutions is beneficial if we consider exact covers appearing in practical situations, such as designing electric circuits [Chang and Jiang, 2016], designing 3D shapes from small fragments [Hu et al, 2014], and timetabling [Nguyen et al, 2018]. In such practical situations, users may want to compare multiple exact covers.…”