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Set Covering
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. It is one of Karp's 21 NP-complete problems shown to be NP-complete in 1972. Given a set of elements (called the universe) and a collection of sets whose union equals the universe, the set cover problem is to identify the smallest sub-collection of whose union equals the universe. For example, consider the universe and the collection of sets Clearly the union of is . However, we can cover all of the elements with the following, smaller number of sets: More formally, given a universe \mathcal and a family \mathcal of subsets of \mathcal, a ''cover'' is a subfamily \mathcal\subseteq\mathcal of sets whose union is \mathcal. In the set covering decision problem, the input is a pair (\mathcal,\mathcal) and an integer k; the question is whether there is a set covering of size k or less. In the set covering optimization problem, the input is a pair (\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Cover Problem
In graph theory, an edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. In computer science, the minimum edge cover problem is the problem of finding an edge cover of minimum size. It is an optimization problem that belongs to the class of covering problems and can be solved in polynomial time. Definition Formally, an edge cover of a graph ''G'' is a set of edges ''C'' such that each vertex in ''G'' is incident with at least one edge in ''C''. The set ''C'' is said to ''cover'' the vertices of ''G''. The following figure shows examples of edge coverings in two graphs (the set ''C'' is marked with red). : A minimum edge covering is an edge covering of smallest possible size. The edge covering number \rho(G) is the size of a minimum edge covering. The following figure shows examples of minimum edge coverings (again, the set ''C'' is marked with red). : Note that the figure on the right is not only an edge c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Cover Problem
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a minimum vertex cover is a classical optimization problem. It is NP-hard, so it cannot be solved by a polynomial-time algorithm if P ≠ NP. Moreover, it is hard to approximate – it cannot be approximated up to a factor smaller than 2 if the unique games conjecture is true. On the other hand, it has several simple 2-factor approximations. It is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory. The minimum vertex cover problem can be formulated as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomized Rounding
Within computer science and operations research, many combinatorial optimization problems are computationally intractable to solve exactly (to optimality). Many such problems do admit fast (polynomial time) approximation algorithms—that is, algorithms that are guaranteed to return an approximately optimal solution given any input. Randomized rounding is a widely used approach for designing and analyzing such approximation algorithms. The basic idea is to use the probabilistic method to convert an optimal solution of a relaxation of the problem into an approximately optimal solution to the original problem. Overview The basic approach has three steps: # Formulate the problem to be solved as an integer linear program (ILP). # Compute an optimal fractional solution x to the linear programming relaxation (LP) of the ILP. # Round the fractional solution x of the LP to an integer solution x' of the ILP. (Although the approach is most commonly applied with linear programs, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uriel Feige
Uriel Feige ( he, אוריאל פייגה) is an Israeli computer scientist who was a doctoral student of Adi Shamir. Life Uriel Feige currently holds the post of Professor at the Department of Computer Science and Applied Mathematics, the Weizmann Institute of Science, Rehovot in Israel. Work He is notable for co-inventing the Feige–Fiat–Shamir identification scheme along with Amos Fiat and Adi Shamir. Honors and awards He won the Gödel Prize in 2001 "for the PCP theorem and its applications to hardness of approximation In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems. Scope Hardness of approximation complements the study of approximation algorithms by prov ...". References {{DEFAULTSORT:Feige, Uriel Living people Gödel Prize laureates Theoretical computer scientists Modern cryptographers Public-key cryptographers 20th-century Israeli mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-polynomial Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expressed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Linear Program Formulation
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Cover Problem
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. It is one of Karp's 21 NP-complete problems shown to be NP-complete in 1972. Given a set of elements (called the universe) and a collection of sets whose union equals the universe, the set cover problem is to identify the smallest sub-collection of whose union equals the universe. For example, consider the universe and the collection of sets Clearly the union of is . However, we can cover all of the elements with the following, smaller number of sets: More formally, given a universe \mathcal and a family \mathcal of subsets of \mathcal, a ''cover'' is a subfamily \mathcal\subseteq\mathcal of sets whose union is \mathcal. In the set covering decision problem, the input is a pair (\mathcal,\mathcal) and an integer k; the question is whether there is a set covering of size k or less. In the set covering optimization problem, the input is a pair (\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dots Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
