What is the objective of set covering problem?
Given a collection of elements, the set covering problem aims to find the minimum number of sets that incorporate (cover) all of these elements.
What is a set covering constraint?
The set covering problem is a specific type of a discrete location model. In this model, a facility can serve all demand nodes that are within a given coverage distance Dc from the facility. The problem is the place the minimum number of facilities so as to ensure that all demand nodes can be served.
Why Is set cover NP-hard?
An instance of the Set Cover problem is a Ground set X, an integer k and a collection of subsets Si formed out of X. Since an NP-complete problem, by definition, is a problem which is both NP and NP-Hard, the proof or statement that a problem is NP-Complete consists of two parts: The problem itself is NP-Complete.
What is minimum set cover?
Assume that every item in X appears in some set, i.e., ⋃j Sj = X. A set cover of X with S is a set I ⊆ {1, 2,…,m} such that ⋃j∈I Sj = X. The solution for MIN-SET-COVER problem is a set cover I of minimum size. That is, a minimum set cover is the smallest set of sets {Si1 ,Si2 ,…,Sik } that covers X. Example 1.
Is the set cover problem NP-complete?
Theorem: Set Cover is NP-Complete. Proof: First, we argue that Set Cover is in NP, since given a collection of sets C, a certifier can efficiently check that C indeed contains at most k elements, and that the union of all sets listed in C does include all elements from the ground set U.
Is the set cover problem NP-Complete?
What is a vertex cover problem?
The vertex cover problem is an NP-Complete problem, which means that there is no known polynomial-time solution for finding the minimum vertex cover of a graph unless it can be proven that P = NP. There, however, exists polynomial-time approximate algorithms to find the vertex cover of a graph.
Is vertex cover a NP?
The minimum vertex cover problem is the optimization problem of finding a smallest vertex cover in a given graph. The vertex cover problem is an NP-complete problem: it was one of Karp’s 21 NP-complete problems. It is often used in computational complexity theory as a starting point for NP-hardness proofs.
What is set covering?
Set covering is equivalent to the hitting set problem. That is seen by observing that an instance of set covering can be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, the universe represented by vertices on the right, and edges representing the inclusion of elements in sets.
What is a cover model called?
The cover model is generally a fashion model, celebrity, or contest winner. Generally, cover models are depicted solitarily; however, on occasion magazines will present a front cover with multiple cover models. Female cover models are often referred to as cover girls .
What is a cover set in topology?
Cover in topology. Covers are commonly used in the context of topology. If the set X is a topological space, then a cover C of X is a collection of subsets U α of X whose union is the whole space X. In this case we say that C covers X, or that the sets U α cover X.
What is the mathematical formulation of the set covering problem?
The mathematical formulation of the set covering problem is define as follows. We define ). Addionally, each set . The objective is to find the minimum cost sub-collection of sets . An integer linear program (ILP) model can be formulated for the minimum set covering problem as follows: