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### Covering a set by subsets

R.J. Clarke
1990 Discrete Mathematics
We give formulae for determining the number of ways of writing a finite set as the union of a given number of subsets, in such a way that none of the subsets may be omitted.  ...  In particular, we consider the case in which the elements of the set are identical.  ...  Introduction Let X be a set with n elements (an n-set) and let k be a positive integer. A k-cover of X is a collection of k not necessarily distinct subsets of X whose union is X.  ...

### THE NUMBER OF DISORDERED COVERS OF A FINITE SET BY SUBSETS HAVING FIXED CARDINALITIES ЧИСЛО НЕУПОРЯДОЧЕННЫХ ПОКРЫТИЙ КОНЕЧНОГО МНОЖЕСТВА ПОДМНОЖЕСТВАМИ ФИКСИРОВАННОГО РАЗМЕРА

R. M. Ganopolsky
Рассматривается новый вид комбинаторных чисел, исчисляющих количество покрытий конечного множества подмножествами с заданными мощностями. Доказывается ряд соотношений и тождеств. Вычисляются некоторые суммы этих чисел. Приводятся частные случаи новых комбинаторных чисел при определенных значениях коэффициентов и интерпретация этих чисел. Ключевые слова: покрытие, конечное множество, комбинаторные числа. В качестве примера конечного множества X можно взять множество целых чисел от 1 до n:
more » ... рим множество X мощности 3 в виде набора чисел X 3 = {1, 2, 3}, а также все различные покрытия X 3 , состоящие только из подмножеств мощности 2: Таким образом, комбинаторные числа вида 3 N k 2 = 3 N (0, k, 0) равны 3 N (0, 1, 0) = 0; 3 N (0, 2, 0) = 3; 3 N (0, 3, 0) = 1. Рассмотрим теперь все покрытия X 3 , состоящие из подмножеств мощности 2 и 3: Следовательно, комбинаторные числа вида 3 N k l 2 3 = 3 N (0, k, l), где k = 0 и l = 0, равны 3 N (0, 1, 1) = 3; 3 N (0, 2, 1) = 3; 3 N (0, 3, 1) = 1. С помощью перебора всех вариантов покрытий можно получить и остальные коэффициенты вида 3 N (m, k, l):

### Page 794 of Annals of Mathematics Vol. 39, Issue 4 [page]

1938 Annals of Mathematics
More specifically: By calling a set B’ whose elements are subsets of A (thus a subset B’ of B)a covering system of the subset A’ of A if each point of A’ is an interior point of at least one subset of  ...  consisting of subsets of the first set, and not necessarily a mapping by means of the relation “interior point of a set’’).  ...

### Exact algorithms and applications for Tree-like Weighted Set Cover

Jiong Guo, Rolf Niedermeier
2006 Journal of Discrete Algorithms
We introduce an NP-complete special case of the WEIGHTED SET COVER problem and show its fixed-parameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant  ...  More precisely, in this practically relevant variant we require that the given collection C of subsets of a base set S should be "tree-like".  ...  We thank an editor of Journal of Discrete Algorithms for pointing to the connection with set covering with consecutive ones property. 4 The corresponding parameters are the maximum subset size and the  ...

### The Minimal Description of Formal Concept Analysis

Li Fu
2010 Journal of Mathematics Research
The theory of rough sets as a tool for processing uncertain and incomplete information, proposed by Z.  ...  the corresponding relation between the covering of objects set and the attributes set.  ...  Theorem 3.2 The covering C M is formed by the subset of attributes set M, every attributes subset which shares maximal objects set is corresponding a covering of the objects set.  ...

### Dual parameterization and parameterized approximability of subset graph problems

Édouard Bonnet, Vangelis Th. Paschos
2017 Reserche operationelle
We discuss approximability in FPT-time for the class of subset optimization graph problems where a feasible solution S is a subset of the vertex set of the input graph.  ...  This class encompasses many well-known problems, such as min dominating set, min vertex cover, max independent set, min feedback vertex set.  ...  On the other hand, when parameterized by the dual parameter, min set cover is W-hard as being a generalization of max independent set.  ...

### Analysis of the greedy approach in problems of maximumk-coverage

Dorit S. Hochbaum, Anu Pathria
1998 Naval Research Logistics
In this paper, we consider a general covering problem in which k subsets are to be selected such that their union covers as large a weight of objects from a universal set of elements as possible.  ...  Our results also yield a simple proof that the number of subsets used by the greedy approach to achieve entire coverage of the universal set is within a logarithmic factor of the optimal number of subsets  ...  Covering Graphs by Subgraphs Covering Edges by Cut Sets Consider the problem of covering the (possibly weighted) edges of a graph, G, by a minimum cardinality set of cuts.  ...

### Page 119 of American Journal of Mathematics Vol. 44, Issue 2 [page]

1922 American Journal of Mathematics
If a closed compact enclosable set Q is covered by a set [R] of regions (Q), then the set Q is covered by some finite subset of the set | R].  ...  If an enclosable set Q is covered by a set [R] of regions (Q), then it is covered by some denumerable subset of the set [R ].  ...

### An exact algorithm for the weighed mutually exclusive maximum set cover problem [article]

Songjian Lu, Xinghua Lu
2014 arXiv   pre-print
In this paper, we introduce an exact algorithm with a time complexity of O^*(1.325^m) for the weighted mutually exclusive maximum set cover problem, where m is the number of subsets in the problem.  ...  By providing a relatively efficient exact algorithm, our approach will like increase the capability of finding better solutions in the application of cancer research.  ...  Very recently, people also studied the extension version of the set cover problem that find a sub-set F ′ of F such that each element in X is covered by at least t subsets in F ′ .  ...

### Stateful Subset Cover [chapter]

Mattias Johansson, Gunnar Kreitz, Fredrik Lindholm
2006 Lecture Notes in Computer Science
This paper describes a method to convert stateless key revocation schemes based on the subset cover principle into stateful schemes.  ...  A concrete new scheme based on the Subset Difference scheme  is presented, accomplishing a bandwidth overhead of Δm + 2Δr + 1 compared to e.g.  ...  An Intuitive Description Recall that, as presented in Section 2, a subset cover scheme by covering members using a static family of subsets of users.  ...

### Set covering number for a finite set

H.-C. Chang, N. Prabhu
1996 Bulletin of the Australian Mathematical Society
Given a finite set S of cardinality N, the minimum number of j-subsets of S needed to cover all the r-subsets of 5 is called the covering number C (N, j , r).  ...  In this note we obtain a nontrivial upper bound by showing that for finite N, Let 5 be a set with N elements.  ...  For N ^ j >r, M PROOF: Let S be a finite set of cardinality N.  ...

### Semiring of Sets: Examples

Roland Coghetto
2014 Formalized Mathematics
This article proposes the formalization of some examples of semiring of sets proposed by Goguadze  and Schmets .  ...  Y is \ f p -closed by [1, (10) ], [11, (39) ], [4, (8) ], [11, (45) ]. (8) Let us consider non empty sets X 1 , X 2 , a family S 1 of subsets of X 1 with countable cover, a family S 2 of subsets of  ...  by [6, (96) ]. (2) Let us consider sets X 1 , X 2 , a non empty family S 1 of subsets of X 1 , and a non empty family S 2 of subsets of X 2 .  ...

### An exact algorithm with the time complexity of O^*(1.299^m) for the weighed mutually exclusive set cover problem [article]

Songjian Lu, Xinghua Lu
2013 arXiv   pre-print
In this paper, we will introduce an exact algorithm with a time complexity of O^*(1.299^m) for the weighted mutually exclusive set cover problem, where m is the number of subsets in the problem.  ...  covered by a subset in F ′′ , needs a subset in F ′ to cover it), then |P| = k + (m − k)(n + 1) + n 1 , and |P ′ | = k ′ + (m − k ′ )(n + 1) + n 2 .  ...  Introduction The set cover problem is that: given a ground set X of n elements and a collection F of m subsets of X, try to find a minimum number of subsets S 1 , S 2 , . . . , S h in F such that ∪ h i  ...

### On nearly S-paracompactness

José Sanabria, Osmin Ferrer, Clara Blanco
2021 WSEAS Transactions on Mathematics
The objective of the present work is to introduce the notion of α-nearly S-paracompact subset, which is closely related to α-nearly paracompact and αS-paracompact subsets.  ...  Thus, V = {V x : x ∈ A} is a cover of A by open subsets of (X, τ ) and, as V x ⊂ Int(Cl(V x )), we have V ′ = {Int(Cl(V x )) : x ∈ A} is a cover of A by regular open sets in (X, τ ).  ...  cover is formed by regular open sets and the refinement is formed by semi-open sets.  ...

### A note on 'Algorithms for connected set cover problem and fault-tolerant connected set cover problem'

Wei Ren, Qing Zhao
2011 Theoretical Computer Science
A flaw in the greedy approximation algorithm proposed by Zhang et al. (2009)  for the minimum connected set cover problem is corrected, and a stronger result on the approximation ratio of the modified  ...  The results are now consistent with the existing results on the connected dominating set problem which is a special case of the minimum connected set cover problem.  ...  A connected set cover (CSC) R with respect to (V , S, G) is a set cover of V such that R induces a connected subgraph of G.  ...
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