Maximum coverage problem: Difference between revisions

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Formally, (unweighted) Maximum Coverage

: Instance: A number <math> k </math> and a collection of sets <math> S = \{S_1, S_2, \ldots, S_m\} </math>.

: Objective: Find a subset <math> S^{'} \subseteq S</math> of sets, such that <math> \left| S^{'} \right| \leq k</math> and the number of covered elements <math> \left| \bigcup_{S_i \in S^{'}}{S_i} \right| </math> is maximized.

The maximum coverage problem is [[NP-hard]], and cannot be approximated to within <math>1 - \frac{1}{e} + o(1) \approx 0.632</math> under standard assumptions.

This result essentially matches the approximation ratio achieved by the generic greedy algorithm used for [[Submodular_set_functionSubmodular set function#Optimization_problemsOptimization problems|maximization of submodular functions with a cardinality constraint]].<ref name="NVF"> [[George Nemhauser|G. L. Nemhauser]], L. A. Wolsey and M. L. Fisher. An analysis of approximations for maximizing submodular set functions I, Mathematical Programming 14 (1978), 265–294</ref>

==ILP formulation==

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== Greedy algorithm ==

The [[greedy algorithm]] for maximum coverage chooses sets according to one rule: at each stage, choose a set which contains the largest number of uncovered elements. It can be shown that this algorithm achieves an approximation ratio of <math>1 - \frac{1}{e}</math>.<ref>{{cite book | last=Hochbaum | first=Dorit S. | authorlinkauthor-link=Dorit S. Hochbaum | editor-first=Dorit S. | editor-last=Hochbaum | year=1997 | chapter=Approximating Covering and Packing Problems: Set Cover, Vertex Cover, Independent Set, and Related Problems | title=Approximation Algorithms for NP-Hard Problems | publisher=PWS Publishing Company | ___location=Boston | isbn=978-053494968-16 | pages=94–143}}</ref> Inapproximabilityln-approximability results show that the greedy algorithm is essentially the best-possible polynomial time approximation algorithm for maximum coverage unless <math>P = NP</math>.<ref>{{cite articlejournal | last = Feige | first = Uriel | authorlinkauthor-link = Uriel Feige | title = A Threshold of ln ''n'' for Approximating Set Cover | journal = Journal of the ACM | volume = 45 | number = 4 |date=July 1998 | issn = 0004-5411 | pages = 634–652 | doi = 10.1145/285055.285059 | publisher = Association for Computing Machinery | ___location = New York, NY, USA| s2cid = 52827488 | doi-access = free }}</ref>

== Known extensions ==

The inapproximability results apply to all extensions of the maximum coverage problem since they hold the maximum coverage problem as a special case.

The Maximum Coverage Problem can be applied to road traffic situations; one such example is selecting which bus routes in a public transportation network should be installed with pothole detectors to maximise coverage, when only a limited number of sensors is available. This problem is a known extension of the Maximum Coverage Problem and was first explored in literature by Junade Ali and Vladimir Dyo.<ref>{{cite book|last1=Ali|first1=Junade|last2=Dyo|first2=Vladimir|title=Proceedings of the 14th International Joint Conference on e-Business and Telecommunications |chapter=Coverage and Mobile Sensor Placement for Vehicles on Predetermined Routes: A Greedy Heuristic Approach |date=2017|volume=2: WINSYS|pages=83–88|doi=10.5220/0006469800830088|url=http://www.scitepress.org/DigitalLibrary/PublicationsDetail.aspx?ID=ddWw1NMB3VI%3d|isbn=978-989-758-261-5}}</ref>

== Weighted version ==

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::<math>x_i \in \{0,1\}</math> (if <math>x_i=1</math> then <math>S_i</math> is selected for the cover).

A greedy algorithm will no longer produce solutions with a performance guarantee. Namely, the worst case behavior of this algorithm might be very far from the optimal solution. The approximation algorithm is extended by the following way. First, define a modified greedy algorithm, that selects the set <math>S_i</math> that has the best ratio of weighted uncovered elements to cost. Second, among covers of cardinality <math>1, 2, ..., k-1</math>, find the best cover that does not violate the budget. Call this cover <math>H_1</math>. Third, find all covers of cardinality <math>k</math> that do not violate the budget. Using these covers of cardinality <math>k</math> as starting points, apply the modified greedy algorithm, maintaining the best cover found so far. Call this cover <math>H_2</math>. At the end of the process, the approximate best cover will be either <math>H_1</math> or <math>H_2</math>. This algorithm achieves an approximation ratio of <math>1- {1 \over e}</math> for values of <math>k \geq 3</math>. This is the best possible approximation ratio unless <math>NP \subseteq DTIME(n^{O(\log\log n)})</math>.<ref>Khuller,{{Cite S.,journal Moss,|doi A.,= and Naor, J. 1999. [http://dx.doi.org/10.1016/S0020-0190(99)00031-9|title = The budgeted maximum coverage problem].|journal ''Inf.= Process.Information Lett''.Processing Letters|volume = 70,|pages 1= (Apr.39–45|year = 1999|last1 = Khuller|first1 = Samir|last2 = Moss|first2 = Anna|last3 = Naor|first3 = Joseph (Seffi),|citeseerx 39-45= 10.1.1.49.5784}}</ref>

== Generalized maximum coverage ==

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The algorithm uses the concept of residual cost/weight. The residual cost/weight is measured against a tentative solution and it is the difference of the cost/weight from the cost/weight gained by a tentative solution.

The algorithm has several stages. First, find a solution using greedy algorithm. In each iteration of the greedy algorithm the tentative solution is added the set which contains the maximum residual weight of elements divided by the residual cost of these elements along with the residual cost of the set. Second, compare the solution gained by the first step to the best solution which uses a small number of sets. Third, return the best out of all examined solutions. This algorithm achieves an approximation ratio of <math>1-1/e - o(1)</math>.<ref>Cohen,{{Cite R.journal and|doi Katzir,= L. 2008. [http://dx.doi.org/10.1016/j.ipl.2008.03.017|title = The Generalized Maximum Coverage Problem].|journal ''Inf.= Process.Information Lett''.Processing Letters|volume = 108,|pages 1= (Sep.15–22|year = 2008),|last1 15-22= Cohen|first1 = Reuven|last2 = Katzir|first2 = Liran|citeseerx = 10.1.1.156.2073}}</ref>

== Related problems ==

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== References ==

* {{Cite book | last=Vazirani | first=Vijay V. | authorlinkauthor-link=Vijay Vazirani | title=Approximation Algorithms | year=2001 | publisher=Springer-Verlag | isbn=978-3-540-65367-87 | pages=}}

== External links ==

[[Category:SetFamilies familiesof sets]]

[[Category:NP-complete problems]]