DOCSLIB.ORG

Approximation Algorithms - Weighted Set Cover Problem Lecturer: Kavitha Telikepalli, Naveen Garg Scribe: Rahul Aggarwal, Tarun Aggarwal

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Approximation Algorithms - Weighted Set Cover Problem Lecturer: Kavitha Telikepalli, Naveen Garg Scribe: Rahul Aggarwal, Tarun Aggarwal CSL758: Advanced Algorithms April 16, 2008 Approximation Algorithms - Weighted Set Cover Problem Lecturer: Kavitha Telikepalli, Naveen Garg Scribe: Rahul Aggarwal, Tarun Aggarwal 1 Introduction In this lecture we will discuss a NP-hard problem and try to find a good approximation algorithm for it. The problem we consider is the weighted set cover problem. In the process of analysing this problem we will also discuss a closely related problem of finding the vertex cover both weighted and non-weighted. 2 Weighted Set Cover Problem Given a finite universe U = fe1; e2; : : : ; eng of n members , S = fs1; s2; : : : ; smg ^ 8i si ⊆ U a collection of subsets of U and a weight function w : s ! <+ that assigns a positive real weight to each subset of U, the goal is to find the minimum weight subcollection of S whose union is U or a minimum weight set cover. 3 Proof of NP completeness We will show a reduction from the vertex cover problem to the weighted set cover problem. The decision version of set cover problem is wether there exists a set cover with number of sets in it ≤ c; c is an integer and w : s ! 1. Vertex Cover Problem - Given a graph G = (V; E), A set S ⊆ V is a vertex cover if 8e = (u; v) 2 E atleast one of u or v is in S. The problem is to find the minimum car- dinality vertex cover. The corresponding decision problem is wether there exists a vertex cover of size ≤ k; k is an integer. Proof - Assuming we have a polynomial time algorithm for each instance of the set cover problem and the vertex cover problem to be NP-Complete. We consider an instance of vertex cover problem G = (V; E) and k. Let us define U = E, Sv = fcollection of all edges incident on vg, S = fSv : v 2 V g and c = k. It is easy to see that 9 a set cover of size ≤ k , 9 a vertex cover of size ≤ k. Therefore it is a valid polynomial time reduction. So, we can solve vertex cover problem in polynomial time which is a contradiction. Hence subset sum problem is NP complete. Now we prove the NP completeness of the vertex cover problem. An independent set S in G = (V; E) is a subset of vertices such that 8e = (u; v) 2 E atmost one of u or v is in S. If S ⊆ V is a vertex cover of G than V − S is an independent set. 13-1 Claim - S is an independent set iff V − S is a vertex cover. Proof - ) Let S be any independent set and consider any edge (u; v). As S is independent ) either u 2 V − S or v 2 V − S. Thus V − S covers (u; v). So, it is a vertex cover. ( Let V − S be any vertex cover and consider 2 vertices u 2 S and v 2 S. As V − S is a vertex cover ) (u; v) 2= E. Thus no 2 nodes in S are joined by an edge. So, it is an independent set. So, finding a vertex cover of size ≤ k is same as finding an independent set of size ≥ k. So, we will prove independent set problem NP complete here. Proof - Assume we have a polynomial time algorithm for the vertex cover problem we consider an instance φ of 3-SAT (a well known NP complete problem). We construct an instance of independent set problem (G; k) as follows: 1. G contains 3 vertices for each clause, one for each literal. 2. Connect 3 literals in a clause in a triangle. 3. Connect a literal to each of its negations in other clauses. 4. Set k = number of clauses in φ. Example : φ = (x1 _ x2 _ x3) ^ (x1 _ x2 _ x3) ^ (x1 _ x2 _ x4). So, k = 3 and G is: Claim - G contains an independent set of size k = jφj iff φ is satisfiable. Proof - ) Let S be any independent set of size k. S must contains exactly one vertex in each triangle. Set these literals to true and any other variables (taking care of negations) in a consistent way. So, all clauses are satisfied ) φ is satisfiable. ( Given a satisfying assignment select a true literal from each triangle which forms an independent set of size k So, we have got a valid polynomial time reduction. Hence by contradiction the independent set problem is NP complete. 13-2 4 f-Approximation algorithm for weighted set cover problem We try to formulate the problem as a linear programming problem. Let S = fs1; s2; : : : ; sng be the given set of subsets of U, associate xj with each sj suct that it is 1 if sj is in mini- mum weight set cover otherwise 0. Let w : s ! <+ be the weight function. So, the linear program is: Minimize X xjwj j Subject to For each e 2 U such that e 2 si1; si2; : : : ; sik xi1 + xi2 + ::: + xik ≥ 1 xj 2 f0; 1g 8j The constrains an also be represented in matrix form Ax ≥ b, where A(i; j) = 1 if ei 2 sj other wise 0, x is a column vector of n elements and b is a colun vector of all 1s. The linear program presented here is an integer program (IP) which is hard to solve. So, we can get an approximation to the solution by relaxing the constraints to 0 ≤ xj ≤ 1. The solution to the relaxed problem will always be less or equal to the solution to the integer program (as some constraints are removed so we can achieve a lesser value). Now we can solve the relaxed program by using any LP solver and get a solution x. The task is to convert this real solution to an integral solution considering the fact that the sets we get unite to give U. While doing so, we have to round off the fractional values of the LP solution. We follow the following rounding strategy. Suppose each element e is present in atmost f subsets, we call f as the frequency of occu- rance. So, 8 xi, which are the solutions of the LP, if xi ≥ 1=f, then set xi = 1 otherwise xi = 0. In this way, the fractional solution will be converted to an integer solution. Now, we prove that this integer solution gives a set cover. For any solution to be a set cover the above constraints must be satisfied. Consider the constraint, xi1 + xi2 + ::: + xik ≥ 1. If the frequnecy is atmost f, then the LHS can not contain more than f terms. So, 9 some xij ≥ 1=f in the solution of LP to satisfy the constraint. So, this xij is set to 1 after rounding and the constraint remains satisfied. This is true for all constraints. So, after rounding we will get a valid set cover. P Also the rounded solution i.e. j xjwj can be atmost f times the solution to the LP as we can increase any xi by a factor of f atmost. The optimal solution to the LP is less than or equal to the optimal solution of the IP. So, the rounded solution will be atmost f times the optimal solution to the integer program. 13-3 5 2-Approximation algorithm for Cardinality Vertex Cover Recall the definition of the cardinality vertex cover problem. Input: An undirected graph G = (V; E). Goal: Find a minimum cardinality set of vertices S ⊆ V such that every edge in E has at least one endpoint in S. Definition: A matching in a graph G = (V; E) is a subset of edges M ⊆ E such that no two edges in M share a common endpoint. A matching M is called maximal if it is not strictly contained in any other matching (i.e. no edges in E − M can be added to M). Our vertex cover algorithm simply finds any maximal matching M. The cover is defined as all endpoints of edges in M. Algorithm VC2 S φ while E 6= φ do let (u; v) be any edge in E S S [ u; v delete vertices u; v and all their incident edges from G return S Lemma 1: The set S returned by VC2 is a vertex cover. Proof: Consider any edge e deleted in an iteration of the loop. If e was selected as the edge in the first line of the loop, then both of its endpoints were added to S. Otherwise, e must share an endpoint with the edge selected in the first line of the loop so one of its endpoints is added to S. Since all edges are eventually deleted in some iteration, the final set S is a vertex cover. Lemma 2: Let M be any matching and S be any vertex cover. Then, jMj ≤ jSj. Proof: Each e 2 M must have at least one of its endpoints covered by S. Since M is a matching then no two edges in e share an endpoint. Therefore, each e 2 M is covered by a vertex that covers no other edge , f 2 M, so jMj ≤ jSj. Lemma 3: VC2 is a 2 − approximation: Proof: Let M be the set of all edges selected in the first line of the loop. Then M is a matching since any edge sharing an endpoint with some e 2 M was deleted in the third line of the loop. Since S consists exactly of the endpoints of edges in M then jSj = 2jMj.
Load more

Recommended publications
  • Vertex Cover Might Be Hard to Approximate to Within 2 Ε − Subhash Khot ∗ Oded Regev †
    Vertex Cover Might be Hard to Approximate to within 2 ε − Subhash Khot ∗ Oded Regev † Abstract Based on a conjecture regarding the power of unique 2-prover-1-round games presented in [Khot02], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k-uniform hypergraphs is hard to approximate within any constant factor better than k. 1 Introduction Minimum vertex cover is the problem of finding the smallest set of vertices that touches all the edges in a given graph. This is one of the most fundamental NP-complete problems. A simple 2- approximation algorithm exists for this problem: construct a maximal matching by greedily adding edges and then let the vertex cover contain both endpoints of each edge in the matching. It can be seen that the resulting set of vertices indeed touches all the edges and that its size is at most twice the size of the minimum vertex cover. However, despite considerable efforts, state of the art techniques can only achieve an approximation ratio of 2 o(1) [16, 21]. − Given this state of affairs, one might strongly suspect that vertex cover is NP-hard to approxi- mate within 2 ε for any ε> 0. This is one of the major open questions in the field of approximation − algorithms. In [18], H˚astad showed that approximating vertex cover within constant factors less 7 than 6 is NP-hard. This factor was recently improved by Dinur and Safra [10] to 1.36.
    [Show full text]
  • 3.1 Matchings and Factors: Matchings and Covers
    1 3.1 Matchings and Factors: Matchings and Covers This copyrighted material is taken from Introduction to Graph Theory, 2nd Ed., by Doug West; and is not for further distribution beyond this course. These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553. 2 Matchings 3.1.1 Definition A matching in a graph G is a set of non-loop edges with no shared endpoints. The vertices incident to the edges of a matching M are saturated by M (M-saturated); the others are unsaturated (M-unsaturated). A perfect matching in a graph is a matching that saturates every vertex. perfect matching M-unsaturated M-saturated M Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 3 Perfect Matchings in Complete Bipartite Graphs a 1 The perfect matchings in a complete b 2 X,Y-bigraph with |X|=|Y| exactly c 3 correspond to the bijections d 4 f: X -> Y e 5 Therefore Kn,n has n! perfect f 6 matchings. g 7 Kn,n The complete graph Kn has a perfect matching iff… Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 4 Perfect Matchings in Complete Graphs The complete graph Kn has a perfect matching iff n is even. So instead of Kn consider K2n. We count the perfect matchings in K2n by: (1) Selecting a vertex v (e.g., with the highest label) one choice u v (2) Selecting a vertex u to match to v K2n-2 2n-1 choices (3) Selecting a perfect matching on the rest of the vertices.
    [Show full text]
  • Theoretical Computer Science Efficient Approximation of Min Set
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 410 (2009) 2184–2195 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Efficient approximation of min set cover by moderately exponential algorithms N. Bourgeois, B. Escoffier ∗, V.Th. Paschos LAMSADE, CNRS FRE 3234 and Université Paris-Dauphine, France article info a b s t r a c t Article history: We study the approximation of min set cover combining ideas and results from polynomial Received 13 October 2008 approximation and from exact computation (with non-trivial worst case complexity upper Received in revised form 30 January 2009 bounds) for NP-hard problems. We design approximation algorithms for min set cover Accepted 1 February 2009 achieving ratios that cannot be achieved in polynomial time (unless problems in NP could Communicated by G. Ausiello be solved by slightly super-polynomial algorithms) with worst-case complexity much lower (though super-polynomial) than those of an exact computation. Keywords: 2009 Elsevier B.V. All rights reserved. Exponential algorithms ' Approximation algorithms min set cover 1. Introduction C Given a ground set C of cardinality n and a system S D fS1;:::; Smg ⊂ 2 , min set cover consists of determining a 0 minimum size subsystem S such that [S2S0 S D C. min set cover is a famous NP-hard problem dealt with in the seminal paper [22]. For the last ten years, the issue of exact resolution of NP-hard problems by algorithms having provably non-trivial upper time-complexity bounds has been very actively studied (see, for instance, the surveys [16,26,30]).
    [Show full text]
  • Set Cover Algorithms for Very Large Datasets
    Set Cover Algorithms For Very Large Datasets ∗ Graham Cormode Howard Karloff Anthony Wirth AT&T Labs–Research AT&T Labs–Research Department of Computer [email protected] [email protected] Science and Software Engineering The University of Melbourne [email protected] ABSTRACT 1. INTRODUCTION The problem of Set Cover—to find the smallest subcol- The problem of Set Cover arises in a surprisingly broad lection of sets that covers some universe—is at the heart number of places. Although presented as a somewhat ab- of many data and analysis tasks. It arises in a wide range stract problem, it captures many different scenarios that of settings, including operations research, machine learning, arise in the context of data management and knowledge planning, data quality and data mining. Although finding an mining. The basic problem is that we are given a collec- optimal solution is NP-hard, the greedy algorithm is widely tion of sets, each of which is drawn from a common universe used, and typically finds solutions that are close to optimal. of possible items. The goal is to find a subcollection of sets However, a direct implementation of the greedy approach, so that their union includes every item from the universe. which picks the set with the largest number of uncovered Places where Set Cover occurs include: items at each step, does not behave well when the in- • In operations research, the problem of choosing where put is very large and disk resident. The greedy algorithm to locate a number of facilities so that all sites are must make many random accesses to disk, which are unpre- within a certain distance of the closest facility can be dictable and costly in comparison to linear scans.
    [Show full text]
  • Approximation Algorithms
    Lecture 21 Approximation Algorithms 21.1 Overview Suppose we are given an NP-complete problem to solve. Even though (assuming P = NP) we 6 can’t hope for a polynomial-time algorithm that always gets the best solution, can we develop polynomial-time algorithms that always produce a “pretty good” solution? In this lecture we consider such approximation algorithms, for several important problems. Specific topics in this lecture include: 2-approximation for vertex cover via greedy matchings. • 2-approximation for vertex cover via LP rounding. • Greedy O(log n) approximation for set-cover. • Approximation algorithms for MAX-SAT. • 21.2 Introduction Suppose we are given a problem for which (perhaps because it is NP-complete) we can’t hope for a fast algorithm that always gets the best solution. Can we hope for a fast algorithm that guarantees to get at least a “pretty good” solution? E.g., can we guarantee to find a solution that’s within 10% of optimal? If not that, then how about within a factor of 2 of optimal? Or, anything non-trivial? As seen in the last two lectures, the class of NP-complete problems are all equivalent in the sense that a polynomial-time algorithm to solve any one of them would imply a polynomial-time algorithm to solve all of them (and, moreover, to solve any problem in NP). However, the difficulty of getting a good approximation to these problems varies quite a bit. In this lecture we will examine several important NP-complete problems and look at to what extent we can guarantee to get approximately optimal solutions, and by what algorithms.
    [Show full text]
  • Complexity and Approximation Results for the Connected Vertex Cover Problem in Graphs and Hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot
    Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot To cite this version: Bruno Escoffier, Laurent Gourvès, Jérôme Monnot. Complexity and approximation results forthe connected vertex cover problem in graphs and hypergraphs. 2007. hal-00178912 HAL Id: hal-00178912 https://hal.archives-ouvertes.fr/hal-00178912 Preprint submitted on 12 Oct 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024 CAHIER DU LAMSADE 262 Juillet 2007 Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier∗ Laurent Gourv`es∗ J´erˆome Monnot∗ Abstract We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).
    [Show full text]
  • Approximation Algorithms Instructor: Richard Peng Nov 21, 2016
    CS 3510 Design & Analysis of Algorithms Section B, Lecture #34 Approximation Algorithms Instructor: Richard Peng Nov 21, 2016 • DISCLAIMER: These notes are not necessarily an accurate representation of what I said during the class. They are mostly what I intend to say, and have not been carefully edited. Last week we formalized ways of showing that a problem is hard, specifically the definition of NP , NP -completeness, and NP -hardness. We now discuss ways of saying something useful about these hard problems. Specifi- cally, we show that the answer produced by our algorithm is within a certain fraction of the optimum. For a maximization problem, suppose now that we have an algorithm A for our prob- lem which, given an instance I, returns a solution with value A(I). The approximation ratio of algorithm A is dened to be OP T (I) max : I A(I) This is not restricted to NP-complete problems: we can apply this to matching to get a 2-approximation. Lemma 0.1. The maximal matching has size at least 1=2 of the optimum. Proof. Let the maximum matching be M ∗. Consider the process of taking a maximal matching: an edge in M ∗ can't be chosen only after one (or both) of its endpoints are picked. Each edge we add to the maximal matching only has 2 endpoints, so it takes at least M ∗ 2 edges to make all edges in M ∗ ineligible. This ratio is indeed close to tight: consider a graph that has many length 3 paths, and the maximal matching keeps on taking the middle edges.
    [Show full text]
  • What Is the Set Cover Problem
    18.434 Seminar in Theoretical Computer Science 1 of 5 Tamara Stern 2.9.06 Set Cover Problem (Chapter 2.1, 12) What is the set cover problem? Idea: “You must select a minimum number [of any size set] of these sets so that the sets you have picked contain all the elements that are contained in any of the sets in the input (wikipedia).” Additionally, you want to minimize the cost of the sets. Input: Ground elements, or Universe U= { u1, u 2 ,..., un } Subsets SSSU1, 2 ,..., k ⊆ Costs c1, c 2 ,..., ck Goal: Find a set I ⊆ {1,2,...,m} that minimizes ∑ci , such that U SUi = . i∈ I i∈ I (note: in the un-weighted Set Cover Problem, c j = 1 for all j) Why is it useful? It was one of Karp’s NP-complete problems, shown to be so in 1972. Other applications: edge covering, vertex cover Interesting example: IBM finds computer viruses (wikipedia) elements- 5000 known viruses sets- 9000 substrings of 20 or more consecutive bytes from viruses, not found in ‘good’ code A set cover of 180 was found. It suffices to search for these 180 substrings to verify the existence of known computer viruses. Another example: Consider General Motors needs to buy a certain amount of varied supplies and there are suppliers that offer various deals for different combinations of materials (Supplier A: 2 tons of steel + 500 tiles for $x; Supplier B: 1 ton of steel + 2000 tiles for $y; etc.). You could use set covering to find the best way to get all the materials while minimizing cost.
    [Show full text]
  • 1 Bipartite Matching and Vertex Covers
    ORF 523 Lecture 6 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a [email protected]. In this lecture, we will cover an application of LP strong duality to combinatorial optimiza- tion: • Bipartite matching • Vertex covers • K¨onig'stheorem • Totally unimodular matrices and integral polytopes. 1 Bipartite matching and vertex covers Recall that a bipartite graph G = (V; E) is a graph whose vertices can be divided into two disjoint sets such that every edge connects one node in one set to a node in the other. Definition 1 (Matching, vertex cover). A matching is a disjoint subset of edges, i.e., a subset of edges that do not share a common vertex. A vertex cover is a subset of the nodes that together touch all the edges. (a) An example of a bipartite (b) An example of a matching (c) An example of a vertex graph (dotted lines) cover (grey nodes) Figure 1: Bipartite graphs, matchings, and vertex covers 1 Lemma 1. The cardinality of any matching is less than or equal to the cardinality of any vertex cover. This is easy to see: consider any matching. Any vertex cover must have nodes that at least touch the edges in the matching. Moreover, a single node can at most cover one edge in the matching because the edges are disjoint. As it will become clear shortly, this property can also be seen as an immediate consequence of weak duality in linear programming. Theorem 1 (K¨onig). If G is bipartite, the cardinality of the maximum matching is equal to the cardinality of the minimum vertex cover.
    [Show full text]
  • 1 Vertex Cover on Hypergraphs
    15-859FF: Coping with Intractability CMU, Fall 2019 Lecture #3: Reductions for Problems in FPT September 11, 2019 Lecturer: Venkatesan Guruswami Scribe: Anish Sevekari 1 Vertex Cover on Hypergraphs We define a d-regular hypergraph H = (V; F) where F is a family of subsets of V of size d. The vertex cover problem on hypergraphs is to find the smallest set S such that every edge X 2 F intersects S at least once. We parameterize the vertex cover problem by size of the optimum solution. Then the parameterized version of the problem is given a hypergraph H and an integer k, does H have a vertex cover of size k? We will now give a dkpoly(n) algorithm that solves the vertex cover problem and a O(d2d!kd) kernel for d-regular Hypergraph Vertex Cover problem. 1.1 An algorithm for Hypergraph Vertex Cover The algorithm is a simple generalization of the 2kpoly(n) algorithm for vertex cover over graphs. We pick any edge, then branch on the vertex on this edge which is in the vertex cover. Algorithm 1 VertexCover(H = (V; F); k) 1: if k = 0 and jFj > 0 then 2: return ? 3: end if 4: Pick any edge e 2 F. 5: for v 2 e do 6: H0 = (V n fvg; F n ff 2 F : v 2 fg) 7: if VertexCover(H0; k) 6=? then 8: return fvg [ VertexCover(H0; k − 1) 9: end if 10: end for 11: return ? Note that the recursion depth of this algorithm is k, and we branch on at most d points inside each call.
    [Show full text]
  • Reductions and Satisfiability
    Reductions and Satisfiability 1 Polynomial-Time Reductions reformulating problems reformulating a problem in polynomial time independent set and vertex cover reducing vertex cover to set cover 2 The Satisfiability Problem satisfying truth assignments SAT and 3-SAT reducing 3-SAT to independent set transitivity of reductions CS 401/MCS 401 Lecture 18 Computer Algorithms I Jan Verschelde, 30 July 2018 Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 1 / 45 Reductions and Satifiability 1 Polynomial-Time Reductions reformulating problems reformulating a problem in polynomial time independent set and vertex cover reducing vertex cover to set cover 2 The Satisfiability Problem satisfying truth assignments SAT and 3-SAT reducing 3-SAT to independent set transitivity of reductions Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 2 / 45 reformulating problems The Ford-Fulkerson algorithm computes maximum flow. By reduction to a flow problem, we could solve the following problems: bipartite matching, circulation with demands, survey design, and airline scheduling. Because the Ford-Fulkerson is an efficient algorithm, all those problems can be solved efficiently as well. Our plan for the remainder of the course is to explore computationally hard problems. Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 3 / 45 imagine a meeting with your boss ... From Computers and intractability. A Guide to the Theory of NP-Completeness by Michael R. Garey and David S. Johnson, Bell Laboratories, 1979. Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 4 / 45 what you want to say is From Computers and intractability.
    [Show full text]
  • Жиз © © 9D%')( X` Yb Acedtac Dd Fgaf H Acedd F Gaf [3, 13]. X
    Covering Problems with Hard Capacities Julia Chuzhoy Joseph (Seffi) Naor Computer Science Department Technion, Haifa 32000, Israel E-mail: cjulia,naor ¡ @cs.technion.ac.il. - Abstract elements of . A cover is a multi-set of input sets that * + can cover all the elements, while - contains at most '() We consider the classical vertex cover and set cover copies of each ./ , and each copy covers at most problems with the addition of hard capacity constraints. elements. We assume that the capacity constraints are hard, This means that a set (vertex) can only cover a limited i.e., the number of copies and the capacity of a set cannot number of its elements (adjacent edges) and the number of be exceeded. The capacitated set cover problem is a natu- available copies of each set (vertex) is bounded. This is a ral generalization of a basic and well-studied problem that natural generalization of the classical problems that also captures practical scenarios where resource limitations are captures resource limitations in practical scenarios. present. We obtain the following results. For the unweighted A special case of the capacitated set cover problem that vertex cover problem with hard capacities we give a ¢ - we consider is the capacitated vertex cover problem, de- © £ approximation algorithm which is based on randomized fined as follows. An undirected graph 01¤1+2 is given !36 rounding with alterations. We prove that the weighted ver- and each vertex 3452 is associated with a cost , a * !36 sion is at least as hard as the set cover problem. This is capacity ',!36 , and a multiplicity .
    [Show full text]