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

2020 ◽  
Vol 67 ◽  
pp. 509-547
Author(s):  
Maxime Chabert ◽  
Christine Solnon
Keyword(s):  
Scale Up ◽  
Global Constraint ◽  
Conceptual Clustering ◽  
Specific Data ◽  
Binary Constraints ◽  
Arc Consistency ◽  
Selected Subset ◽  
Cover Problem ◽  
Clustering Problems ◽  
Exact Cover

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.

scholarly journals EXISTENCE OF -ANALOGS OF STEINER SYSTEMS

2016 ◽  
Vol 4 ◽  
Author(s):  
MICHAEL BRAUN ◽  
TUVI ETZION ◽  
PATRIC R. J. ÖSTERGÅRD ◽  
ALEXANDER VARDY ◽  
ALFRED WASSERMANN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Automorphism Groups ◽  
Steiner System ◽  
Dimensional Subspace ◽  
Steiner Systems ◽  
Cover Problem ◽  
Exact Cover

Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$. A $q$-analog of a Steiner system (also known as a $q$-Steiner system), denoted ${\mathcal{S}}_{q}(t,\!k,\!n)$, is a set ${\mathcal{S}}$ of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that each $t$-dimensional subspace of $\mathbb{F}_{q}^{n}$ is contained in exactly one element of ${\mathcal{S}}$. Presently, $q$-Steiner systems are known only for $t\,=\,1\!$, and in the trivial cases $t\,=\,k$ and $k\,=\,n$. In this paper, the first nontrivial $q$-Steiner systems with $t\,\geqslant \,2$ are constructed. Specifically, several nonisomorphic $q$-Steiner systems ${\mathcal{S}}_{2}(2,3,13)$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\text{GL}(13,2)$. This approach leads to an instance of the exact cover problem, which turns out to have many solutions.

An Improved Multilinear Map and its Applications

2015 ◽  
Vol 10 (3) ◽  
pp. 64-81 ◽  
Author(s):  
Chunsheng Gu
Keyword(s):  
Random Matrix ◽  
Principal Ideal ◽  
Key Exchange ◽  
Ideal Lattice ◽  
Witness Encryption ◽  
Lattice Problem ◽  
Multilinear Maps ◽  
Multilinear Map ◽  
Cover Problem ◽  
Exact Cover

Cryptographic multilinear maps have extensive applications. However, current constructions of multilinear maps suffer from the zeroizing attacks. For a candidate construction of multilinear maps described by Garg, Gentry, and Halevi (GGH13), Hu & Jia recently presented an efficient attack, which broke the GGH13-based applications of multipartite key exchange (MPKE) and witness encryption (WE) based on the hardness of 3-exact cover problem. By introducing random matrix, the author presents an improvement of the GGH13 map, which supports the applications for public tools of encoding in the GGH13 map, such as MPKE and WE. The security of the construction depends upon new hardness assumption. Moreover, the author's improvement destroys the structure of the ring element in the principal ideal lattice problem, and avoids potential attacks using algorithm of solving short principal ideal lattice generator.

scholarly journals Design of Network-Based Biocomputation Circuits for the Exact Cover Problem

2021 ◽  
Author(s):  
Till Korten ◽  
Stefan Diez ◽  
Heiner Linke ◽  
Dan Nicolau Jr. ◽  
Hillel Kugler
Keyword(s):  
Cover Problem ◽  
Exact Cover

scholarly journals On polynomial reduction of problems based on diagonal Latin squares to the exact cover problem

2020 ◽  
Author(s):  
E. Vatutin ◽  
N. Nikitina ◽  
A. Belyshev ◽  
M. Manzyuk
Keyword(s):  
Distributed Computing ◽  
Latin Squares ◽  
Computational Experiments ◽  
Polynomial Reduction ◽  
Software Implementations ◽  
Cover Problem ◽  
Exact Cover

The paper discusses the reduction of problems based on Latin squares to the exact cover problem aiming at its subsequent solution using the dancing links algorithm. The former problems include generation of Latin squares and diagonal Latin squares of a general form/with a given normalization, generation of orthogonal Latin and diagonal Latin squares directly/through the set of transversals, obtaining a set of transversals for a given square, forming a subset of disjoint transversals. For each subproblem, we describe in detail the process of forming the corresponding binary coverage matrices. We show that the use of the proposed approach in comparison with the classical one, i.e. the formation of sets of transversals and their coverages using exhaustive enumeration, allows one to increase the eective processing pace of diagonal Latin squares by 2.5{5.6 times. The developed software implementations of the algorithms are used in computational experiments as part of the Gerasim@Home volunteer distributed computing project on the BOINC platform

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