scholarly journals NP-hardness of quadratic Euclidean 1-Mean and 1-Median 2-Clustering problem with the constraints on the cluster sizes

2019 ◽  
Vol 489 (4) ◽  
pp. 339-343
Author(s):  
A. V. Kel’manov ◽  
A. V. Pyatkin ◽  
V. I. Khandeev
Keyword(s):  
Euclidean Space ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Clustering Problem ◽  
Geometric Center ◽  
Finite Set ◽  
The Given ◽  
Np Hardness

In the paper, we consider a problem of clustering a finite set of N points in d-dimensional Euclidean space into two clusters minimizing the sum over all clusters of the intracluster sums of the distances between clusters elements and their centers. The center of one cluster is defined as centroid (geometric center). The center of the other one is a sought point in the input set. We analyze the variant of the problem with the given clusters sizes. We have proved the strong NP-hardness of this problem.

ON THE STEINER RATIO IN $\mathcal{R}_{n}$

2011 ◽  
Vol 03 (04) ◽  
pp. 473-489
Author(s):  
HAI DU ◽  
WEILI WU ◽  
ZAIXIN LU ◽  
YINFENG XU
Keyword(s):  
Euclidean Space ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Minimum Length ◽  
Steiner Ratio ◽  
Line Segments ◽  
Finite Set ◽  
Steiner Minimum Tree ◽  
Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

scholarly journals Constructions of Maximum Few-Distance Sets in Euclidean Spaces

10.37236/8565 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ferenc Szöllősi ◽  
Patric R.J. Östergård
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Dimensional Euclidean Space ◽  
Combined Approach ◽  
Euclidean Spaces ◽  
Grobner Basis ◽  
Finite Set ◽  
Distance Set ◽  
Distance Sets ◽  
Basis Computation

A finite set of vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets in $\mathbb{R}^d$ for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.

Cell Complexes, Valuations, and the Euler Relation

1970 ◽  
Vol 22 (2) ◽  
pp. 235-241 ◽  
Author(s):  
M. A. Perles ◽  
G. T. Sallee
Keyword(s):  
Euclidean Space ◽  
Hausdorff Metric ◽  
Dimensional Euclidean Space ◽  
Cell Complexes ◽  
Finite Dimensional ◽  
Finite Set ◽  
Euler Relation ◽  
Set Of Points ◽  
The Relationship ◽  
Dimensional Face

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.

scholarly journals The number of partitions of a set of N points in k dimensions induced by hyperplanes

1967 ◽  
Vol 15 (4) ◽  
pp. 285-289 ◽  
Author(s):  
E. F. Harding
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
General Position ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Unordered Pair

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

scholarly journals On some sufficient conditions for the blow-up solutions of the nonlinear Ginzburg-Landau-Schrödinger evolution equation

2004 ◽  
Vol 2004 (1) ◽  
pp. 23-35 ◽  
Author(s):  
Sh. M. Nasibov
Keyword(s):  
Boundary Condition ◽  
Euclidean Space ◽  
Initial Data ◽  
Evolution Equation ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Homogeneous Boundary Condition ◽  
Dimensional Euclidean Space ◽  
Ginzburg Landau ◽  
The Given

Investigation of the blow-up solutions of the problem in finite time of the first mixed-value problem with a homogeneous boundary condition on a bounded domain ofn-dimensional Euclidean space for a class of nonlinear Ginzburg-Landau-Schrödinger evolution equation is continued. New simple sufficient conditions have been obtained for a wide class of initial data under which collapse happens for the given new values of parameters.

Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space

2018 ◽  
Vol 15 (06) ◽  
pp. 1850092 ◽  
Author(s):  
Djavvat Khadjiev ◽  
İdri̇s Ören ◽  
Ömer Pekşen
Keyword(s):  
Euclidean Space ◽  
Existence And Uniqueness ◽  
Dimensional Euclidean Space ◽  
Uniqueness Theorems ◽  
Two Dimensional ◽  
Existence And Uniqueness Theorems ◽  
Global Invariants ◽  
The Given

Let [Formula: see text] be the [Formula: see text]-dimensional Euclidean space, [Formula: see text] be the group of all linear similarities of [Formula: see text] and [Formula: see text] be the group of all orientation-preserving linear similarities of [Formula: see text]. The present paper is devoted to solutions of problems of global [Formula: see text]-equivalence of paths and curves in [Formula: see text] for the groups [Formula: see text]. Complete systems of global [Formula: see text]-invariants of a path and a curve in [Formula: see text] are obtained. Existence and uniqueness theorems are given. Evident forms of a path and a curve with the given global invariants are obtained.

scholarly journals Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space

2019 ◽  
Vol 27 (2) ◽  
pp. 37-65 ◽  
Author(s):  
Djavvat Khadjiev ◽  
İdris Ören
Keyword(s):  
Euclidean Space ◽  
Orthogonal Group ◽  
Dimensional Euclidean Space ◽  
Plane Curves ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Special Orthogonal Group ◽  
The Common ◽  
Global Invariants ◽  
The Given

AbstractIn this paper, for the orthogonal group G = O(2) and special orthogonal group G = O+(2) global G-invariants of plane paths and plane curves in two-dimensional Euclidean space E2 are studied. Using complex numbers, a method to detect G-equivalences of plane paths in terms of the global G-invariants of a plane path is presented. General evident form of a plane path with the given G-invariants are obtained. For given two plane paths x(t) and y(t) with the common G-invariants, evident forms of all transformations g ∈ G, carrying x(t) to y(t), are obtained. Similar results have obtained for plane curves.

How to see a cube moving into its mirror image

2012 ◽  
Vol 96 (537) ◽  
pp. 471-479
Author(s):  
Ester Dalvit ◽  
Domenico Luminati
Keyword(s):  
Euclidean Space ◽  
Dimensional Space ◽  
Rigid Motion ◽  
Mirror Image ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Other Hand ◽  
Rigid Motions

In n-dimensional Euclidean space no reflection with respect to a hyperplane can be realised by a rigid motion. But this is possible if we allow rigid motions in (n + 1)-dimensional space. These notes show a way to visualise a rigid motion of a cube in 4-dimensional space that flips the cube ‘as the page of a book’.The two terms rigid motion and isometry are sometimes used as synonyms. Yet they do refer to different concepts. The first one has a purely kinematic connotation: the swing of a door or the movement of a piece of furniture pushed over the floor are described by rigid motions. On the other hand to ensure that two figures are isometric it is enough that there exists a correspondence between their points that maintains the relative distances.

scholarly journals NP-Hardness of the Problem of Optimal Box Positioning

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 711
Author(s):  
Alexei V. Galatenko ◽  
Stepan A. Nersisyan ◽  
Dmitriy N. Zhuk
Keyword(s):  
Reduction Technique ◽  
Satisfiability Problem ◽  
Np Hard ◽  
Polynomial Reduction ◽  
Finite Set ◽  
The Given ◽  
Np Hardness

We consider the problem of finding a position of a d-dimensional box with given edge lengths that maximizes the number of enclosed points of the given finite set P ⊂ R d , i.e., the problem of optimal box positioning. We prove that while this problem is polynomial for fixed values of d, it is NP-hard in the general case. The proof is based on a polynomial reduction technique applied to the considered problem and the 3-CNF satisfiability problem.

scholarly journals An exact algorithm for stable instances of the $ k $-means problem with penalties in fixed-dimensional Euclidean space

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fan Yuan ◽  
Dachuan Xu ◽  
Donglei Du ◽  
Min Li
Keyword(s):  
Euclidean Space ◽  
Local Search ◽  
Heuristic Algorithms ◽  
Search Algorithm ◽  
Optimal Solution ◽  
Exact Algorithm ◽  
Dimensional Euclidean Space ◽  
Local Search Algorithm ◽  
Clustering Problem ◽  
Penalty Costs

<p style='text-indent:20px;'>We study stable instances of the <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-means problem with penalties in fixed-dimensional Euclidean space. An instance of the problem is called <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-stable if this instance exists a sole optimal solution and the solution keeps unchanged when distances and penalty costs are scaled by a factor of no more than <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>. Stable instances of clustering problem have been used to explain why certain heuristic algorithms with poor theoretical guarantees perform quite well in practical. For any fixed <inline-formula><tex-math id="M5">\begin{document}$ \epsilon &gt; 0 $\end{document}</tex-math></inline-formula>, we show that when using a common multi-swap local-search algorithm, a <inline-formula><tex-math id="M6">\begin{document}$ (1+\epsilon) $\end{document}</tex-math></inline-formula>-stable instance of the <inline-formula><tex-math id="M7">\begin{document}$ k $\end{document}</tex-math></inline-formula>-means problem with penalties in fixed-dimensional Euclidean space can be solved accurately in polynomial time.</p>

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