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| The '''set covering problem''' ('''SCP''') is a classical question in [[combinatorics]], [[computer science]] and [[Computational complexity theory|complexity theory]].
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| It is a problem "whose study has led to the development of fundamental techniques for the entire field" of [[approximation algorithms]].<ref>{{harvtxt|Vazirani|2001|p=15}}</ref> It was also one of [[Karp's 21 NP-complete problems]] shown to be [[NP-complete]] in 1972.
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| Given a set of elements <math>\{1,2,...,m\}</math> (called the universe) and a set <math>S</math> of <math>n</math> sets whose union equals the universe, the set cover problem is to identify the smallest subset of <math>S</math> whose union equals the universe. For example, consider the universe <math>U = \{1, 2, 3, 4, 5\}</math> and the set of sets <math>S = \{\{1, 2, 3\}, \{2, 4\}, \{3, 4\}, \{4, 5\}\}</math>. Clearly the union of <math>S</math> is <math>U</math>. However, we can cover all of the elements with the following, smaller number of sets: <math>\{\{1, 2, 3\}, \{4, 5\}\}</math>.
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| More formally, given a universe <math>\mathcal{U}</math> and a family <math>\mathcal{S}</math> of subsets of <math>\mathcal{U}</math>,
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| a ''cover'' is a subfamily <math>\mathcal{C}\subseteq\mathcal{S}</math> of sets whose union is <math>\mathcal{U}</math>. In the set covering [[decision problem]], the input is a pair <math>(\mathcal{U},\mathcal{S})</math> and an integer <math>k</math>; the question is whether
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| there is a set covering of size <math>k</math> or less. In the set covering [[optimization problem]], the input is a pair <math>(\mathcal{U},\mathcal{S})</math>, and the task is to find a set covering that uses the fewest sets.
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| The decision version of set covering is [[NP-complete]], and the optimization version of set cover is [[NP-hard]] .{{sfn |Korte|Vygen|2012|p=414}}
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| {{Covering-Packing Problem Pairs}}
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| ==Integer linear program formulation==
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| The minimum set cover problem can be formulated as the following [[integer linear program]] (ILP).<ref>{{harvtxt|Vazirani|2001|p=108}}</ref>
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| {|
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| | minimize
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| | <math>\sum_{S \in \mathcal S} x_S</math>
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| | (minimize the number of sets)
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| |-
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| | subject to
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| | <math>\sum_{S\colon e \in S} x_S \geqslant 1 </math>
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| | for all <math>e\in \mathcal U</math>
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| | (cover every element of the universe)
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| |-
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| | <math>x_S \in \{0,1\}</math>
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| | for all <math>S\in \mathcal S</math>.
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| | (every set is either in the set cover or not)
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| |}
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| This ILP belongs to the more general class of ILPs for [[covering problem]]s.
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| The [[Linear programming relaxation#Approximation and integrality gap|integrality gap]] of this ILP is at most <math>\scriptstyle \log n</math>, so its [[Linear programming relaxation|relaxation]] gives a factor-<math>\scriptstyle \log n</math> [[approximation algorithm]] for the minimum set cover problem (where <math>\scriptstyle n</math> is the size of the universe).<ref>{{harvtxt|Vazirani|2001|pp=110–112}}</ref>
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| == Hitting set formulation ==
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| Set covering is equivalent to the '''hitting set problem'''. It is easy to see this by observing that an instance of set covering can
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| be viewed as an arbitrary [[bipartite graph]], with sets represented by vertices on the left, the universe represented by vertices on the
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| right, and edges representing the inclusion of elements in sets. The task is then to find a minimum cardinality subset of left-vertices which covers all of the right-vertices. In the Hitting set problem, the objective is to cover the left-vertices using a minimum subset of the right vertices. Converting from one problem to the other is therefore achieved by interchanging the two sets of vertices.
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| == Greedy algorithm ==
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| The [[greedy algorithm]] for set covering chooses sets according to one rule: at each stage, choose the set that contains the largest number of uncovered elements. It can be shown<ref>Chvatal, V. [http://www.jstor.org/stable/3689577 A Greedy Heuristic for the Set-Covering Problem]. Mathematics of Operations Research
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| Vol. 4, No. 3 (Aug., 1979), pp. 233-235</ref> that this algorithm achieves an approximation ratio of <math>H(s)</math>, where <math>s</math> is the size of the set to be covered, <math>H(n)</math> is the <math>n</math>-th [[harmonic number]]:
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| :<math> H(n) = \sum_{k=1}^{n} \frac{1}{k} \le \ln{n} +1</math>
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| This greedy algorithm actually achieves an approximation ratio of <math>H(s^\prime)</math> where <math>s^\prime</math> is the maximum cardinality set of <math>S</math>.
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| [[Image:SetCoverGreedy.gif|frame|Tight example for the greedy algorithm with k=3]]
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| There is a standard example on which the greedy algorithm achieves an approximation ratio of <math>\log_2(n)/2</math>.
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| The universe consists of <math>n=2^{(k+1)}-2</math> elements. The set system consists of <math>k</math> pairwise disjoint sets
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| <math>S_1,\ldots,S_k</math> with sizes <math>2,4,8,\ldots,2^k</math> respectively, as well as two additional disjoint sets <math>T_0,T_1</math>,
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| each of which contains half of the elements from each <math>S_i</math>. On this input, the greedy algorithm takes the sets
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| <math>S_k,\ldots,S_1</math>, in that order, while the optimal solution consists only of <math>T_0</math> and <math>T_1</math>. | |
| An example of such an input for <math>k=3</math> is pictured on the right.
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| Inapproximability results show that the greedy algorithm is essentially the best-possible polynomial time approximation algorithm for set cover
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| (see [[Set cover problem#Inapproximability results|Inapproximability results]] below), under plausible complexity assumptions.
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| == Low-frequency systems ==
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| If each element occurs in at most ''f'' sets, then a solution can be found in polynomial time that approximates the optimum to within a factor of ''f'' using [[Linear programming relaxation|LP relaxation]].<ref>{{harvtxt|Vazirani|2001|pp=118–119}}</ref>
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| == Inapproximability results ==
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| When <math> n</math> refers to the size of the universe, {{harvtxt |Lund|Yannakakis|1994}} showed that set covering cannot be approximated in polynomial time to within a factor of <math>\tfrac{1}{2}\log_2{n} \approx 0.72\ln{n}</math>, unless '''NP''' has [[quasi-polynomial time]] algorithms. Feige (1998) improved this lower bound to <math>\bigl(1-o(1)\bigr)\cdot\ln{n}</math> under the same assumptions, which essentially matches the approximation ratio achieved by the greedy algorithm. {{harvtxt |Raz|Safra|1997}} established a lower bound
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| of <math>c\cdot\ln{n}</math>, where <math>c</math> is a constant, under the weaker assumption that '''P'''<math>\not=</math>'''NP'''.
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| A similar result with a higher value of <math>c</math> was recently proved by {{harvtxt |Alon|Moshkovitz|Safra|2006}}.
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| == Related problems ==
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| * Hitting set is an equivalent reformulation of Set Cover.
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| * [[Vertex cover problem|Vertex cover]] is a special case of Hitting Set.
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| * [[Edge cover problem|Edge cover]] is a special case of Set Cover.
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| * [[Set packing]] is the dual problem of Set Cover.
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| * [[Maximum coverage problem]] is to choose at most k sets to cover as many elements as possible.
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| * [[Dominating set]] is the problem of selecting a set of vertices (the dominating set) in a graph such that all other vertices are adjacent to at least one vertex in the dominating set. The Dominating set problem was shown to be NP complete through a reduction from Set cover.
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| * [[Exact cover problem]] is to choose a set cover with no element included in more than one covering set.
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| * [[Closest pair of points problem]]
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| * [[Nearest neighbor search]]
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| ==See also==
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| *[[MinHash]]
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| *[[Sequence assembly]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{Citation | last1=Alon | first1=Noga | author1-link=Noga Alon | last2=Moshkovitz | first2=Dana | last3=Safra | first3=Shmuel | author3-link=Shmuel Safra | title=Algorithmic construction of sets for k-restrictions | publisher=ACM | year=2006 | journal=ACM Trans. Algorithms | issn=1549-6325 | volume=2 | issue=2 | pages=153–177 | doi=10.1145/1150334.1150336}}.
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| * {{Citation
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| | last=Cormen | first=Thomas H. | authorlink=Thomas H. Cormen
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| | last2=Leiserson | first2=Charles E. | authorlink2=Charles E. Leiserson
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| | last3=Rivest | first3=Ronald L. | authorlink3=Ronald L. Rivest
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| | last4=Stein | first4=Clifford | authorlink4=Clifford Stein
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| | title=Introduction to Algorithms | year=2001 | publisher=MIT Press and McGraw-Hill | location=Cambridge, Mass.
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| | isbn=0-262-03293-7 | pages=1033–1038}}
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| * {{Citation | last1=Feige | first1=Uriel | author1-link=Uriel Feige | title=A threshold of ln n for approximating set cover | publisher=ACM | year=1998 | journal=[[Journal of the ACM]] | issn=0004-5411 | volume=45 | issue=4 | pages=634–652 | doi=10.1145/285055.285059}}.
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| * {{Citation | last1=Lund | first1=Carsten | author1-link=Carsten Lund | last2=Yannakakis | first2=Mihalis | author2-link=Mihalis Yannakakis | title=On the hardness of approximating minimization problems | publisher=ACM | year=1994 | journal=[[Journal of the ACM]] | issn=0004-5411 | volume=41 | issue=5 | pages=960–981 | doi=10.1145/185675.306789}}.
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| * {{Citation | last1=Raz | first1=Ran | author1-link=Ran Raz | last2=Safra | first2=Shmuel | author2-link=Shmuel Safra | title=STOC '97: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing | publisher=ACM | isbn=978-0-89791-888-6 | year=1997 | chapter=A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP | pages=475–484}}.
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| * {{cite book
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| | last = Vazirani | first = Vijay V.
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| | authorlink = Vijay Vazirani
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| | title = Approximation Algorithms
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| | year = 2001
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| | publisher = Springer-Verlag
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| | isbn = 3-540-65367-8
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| | url = http://www.cc.gatech.edu/fac/Vijay.Vazirani/book.pdf
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| | ref = harv
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| }}
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| * {{cite book
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| | last1 = Korte | first1 = Bernhard
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| | last2 = Vygen | first2 = Jens
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| | title = Combinatorial Optimization: Theory and Algorithms
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| | year = 2012
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| | isbn = 978-3-642-24487-2
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| | edition = 5
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| | publisher = Springer
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| | ref = harv
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| }}
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| == External links ==
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| * [http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/set-benchmarks.htm Benchmarks with Hidden Optimum Solutions for Set Covering, Set Packing and Winner Determination]
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| * [http://www.csc.kth.se/~viggo/wwwcompendium/node146.html A compendium of NP optimization problems - Minimum Set Cover]
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| {{DEFAULTSORT:Set Cover Problem}}
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| [[Category:Set families]]
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| [[Category:NP-complete problems]]
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While this dream of yours could very well become a reality in the next several years, you might want to consider a few things before becoming a full-time work from home employee. Working from home might seem like a dream - and since many companies are going the virtual route, more employees are now expecting flexible work options - such as telecommuting and working virtually or from home. According to a survey commissioned by Mom Corps, a national flexible staffing firm, 51 percent of working adults (ages 18 to 44) plan to look for a new job within the next three years with an employer that offers flexible work options.
For most workers, this means a computer, reliable Internet access and a telephone. Companies often opt to allow workers to stay at home to save costs; so don't expect that these types of things will be provided to you as they normally are in an office setting. Do you have the necessary tools to get work done? If something else is required to successfully complete your daily tasks, add that to the list as well.
When working from home, your internal motivation must be strong enough to overcome your desire to sleep in, do laundry or talk to your friend on the phone for an hour. No one will be around to remind you to get that project done or to tell you to stop playing online games. Can you motivate yourself?
Working from home might seem like a dream - and since many companies are going the virtual route, more employees are now expecting flexible work options - such as telecommuting and working virtually or from home. According to a survey commissioned by Mom Corps, a national flexible staffing firm, 51 percent of working adults (ages 18 to 44) plan to look for a new job within the next three years with an employer that offers flexible work options. While this dream of yours could very well become a reality in the next several years, you might want to consider a few things before becoming a full-time work from home employee.
Companies often opt to allow workers to stay at home to save costs; so don't expect that these types of things will be provided to you as they normally are in an office setting. For most workers, this means a computer, reliable Internet access and a telephone. If something else is required to successfully complete your daily tasks, add that to the list as well. Do you have the necessary tools to get work done?
Can you motivate yourself? No one will be around to remind you to get that project done or to tell you to stop playing online games. When working from home, your internal motivation must be strong enough to overcome your desire to sleep in, do laundry or talk to your friend on the phone for an hour.
If you have any questions relating to where and the best ways to utilize activite a domicile, you could call us at our own website.