scholarly journals Ultrafast adiabatic quantum algorithm for the NP-complete exact cover problem

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
Hefeng Wang ◽  
Lian-Ao Wu
Keyword(s):  
Quantum Algorithm ◽  
Np Complete ◽  
Cover Problem ◽  
Exact Cover

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.

A Note on the Ohya-Masuda Quantum Algorithm

2002 ◽  
Vol 09 (02) ◽  
pp. 115-123
Author(s):  
Miroljub Dugić
Keyword(s):  
Density Matrix ◽  
Complexity Theory ◽  
Quantum Measurement ◽  
Pure State ◽  
Quantum Algorithm ◽  
Satisfiability Problem ◽  
Second Step ◽  
Coherent Superposition ◽  
Complete Problem ◽  
Np Complete

We analyze the Ohya-Masuda quantum algorithm that solves the so-called “satisfiability” problem, which is an NP-complete problem of the complexity theory. We distinguish three steps in the algorithm, and analyze the second step, in which a coherent superposition of states (a “pure” state) transforms into an “incoherent” mixture presented by a density matrix. We show that, if “nonideal” (in analogy with “nonideal” quantum measurement), this transformation can make the algorithm to fail in some cases. On this basis we give some general notions on the physical implementation of the Ohya-Masuda algorithm.

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.

NEAREST NEIGHBOR PROBLEMS

1992 ◽  
Vol 02 (04) ◽  
pp. 383-416 ◽  
Author(s):  
GORDON WILFONG
Keyword(s):  
Nearest Neighbor ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Minimum Cardinality ◽  
Nearest Neighbor Rule ◽  
Np Complete ◽  
Cover Problem ◽  
Consistent Subset ◽  
Special Case

Suppose E is a set of labeled points (examples) in some metric space. A subset C of E is said to be a consistent subset ofE if it has the property that for any example e∈E, the label of the closest example in C to e is the same as the label of e. We consider the problem of computing a minimum cardinality consistent subset. Consistent subsets have applications in pattern classification schemes that are based on the nearest neighbor rule. The idea is to replace the training set of examples with as small a consistent subset as possible so as to improve the efficiency of the system while not significantly affecting its accuracy. The problem of finding a minimum size consistent subset of a set of examples is shown to be NP-complete. A special case is described and is shown to be equivalent to an optimal disc cover problem. A polynomial time algorithm for this optimal disc cover problem is then given.

Rigorous Estimation of Computational Complexity for OMV SAT Algorithm

2008 ◽  
Vol 15 (02) ◽  
pp. 173-187 ◽  
Author(s):  
Satoshi Iriyama ◽  
Masanori Ohya
Keyword(s):  
Computational Complexity ◽  
Boolean Function ◽  
Polynomial Time ◽  
Quantum Algorithm ◽  
Sat Problem ◽  
Amplification Process ◽  
Np Complete

For SAT problem, which is known to be NP-complete, Ohya and Masuda found a quantum algorithm calculating a given boolean function for all truth assignments. They showed that we can decide whether this boolean function is satisfiable or not in polynomial time if a certain superposed state can be detected physically, which turns out to be related to an amplification process. Then Ohya and Volovich realized this amplification by means of chaos dynamics [4, 5]. In this paper, we study the complexity of the SAT algorithm by rigorously counting the steps of OMV algorithm discussed previously in [1, 2, 4, 5].

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