V. Chv'atal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233--235, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2 represent edges in the graph. So, starting with an initial A yes instance is a recognition problem instance in which the answer is yes. matching, augmenting path techniques can be used to nd the best matching (i.e. the matching that covers the maximum number of vertices, or elements of S) [1]. Having completed this step, some of the elements of S may still be uncovered. Any element c j 2 C not in the cover will have one of the three following characteristics: 1. jc j j = 1 2. jc j j = 2, one of the two elements of c j is already contained in some c k 2 C , and the other element ....

V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, August 1979.

....Forwarding Set problem assuming no knowledge of the geographic location of the nodes. In this case, the Minimum Forwarding Set problem is essentially the well studied Set Cover problem. Not surprisingly, the heuristic proposed in [13] see also [12] is a translation of Chvatal s greedy algorithm [4] for Set Cover, and thus guarantees an approximation factor of O(log m) where m is the maximum neighborhood size. The greedy algorithm iteratively selects a 1 hop neighbor covering the maximum number of 2 hop neighbors not yet covered, and terminates when all 2 hop neighbors have been covered. ....

V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operation Research, 4(3):233--235, 1979.

....h essentially encodes the set of positive regions. Thus our goal is to find the union of as few boxes as possible that cover all of the positive regions. We now describe how to formulate this problem as a set covering problem for which we can then use the standard greedy set covering heuristic [13] to perform the conversion. The set X of objects to cover will simply contain all positive regions in h. Thus jX j (4m 1) Then the set F of subsets of X will be made as follows. Consider the set B of boxes where each box in B is formed by picking a minimum and maximum coordinate in each ....

V. Chvatal. A greedy heuristic for the set covering problem. Mathematics of Operations Research, 4(3):233--235, 1979.

....efficient than corresponding algorithms designed and analyzed non probabilistically. For instance, although approximation algorithms for packing and covering problems can be obtained by derandomizing probabilistic proofs [14, 3] the most efficient algorithms are not derived by derandomization [8, 10, 4, 11, 13, 16]. However, these algorithms often have suggestive similarities to their probabilistic counterparts, both in their design and in the results obtained. In this paper we introduce oblivious derandomi ation, a technique that uses the probabilistic framework to derive algorithms that don t solve ....

V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233--235, 1979.

....on instances where each set has size at most B is hard to approximate to within a factor ln B O(ln ln B) unless P = NP . The result follows from an appropriate setting of parameters in Feige s reduction [11] This is essentially tight in light of the existence of (1 ln B) approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner veri er, and we provide a tight analysis for it. Using the new construction, and some modi cations to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to ....

....applies to other constraint satisfaction problems as well. Set Cover with sets of bounded size For Set Cover, the natural parameter of interest is the size of the sets. The standard greedy algorithm achieves an approximation ratio of 1 ln B, if B is an upper bound on the size of the sets [20, 23, 9]. If sets can have arbitrary size then the greedy algorithm achieves at least an approximation 1 lnn, where n is the size of the instance, and Feige [11] proved that, for any 0, it is infeasible to achieve a (1 ) ln n approximation, unless NP has quasi polynomial time algorithms. It was also ....

V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4:233-235, 1979.

....factor would be O(ln jSj) O(log m logN ) by the well known performance of greedy heuristic. The j WTSP is approximated to within a constant factor in time O( mN ) using the algorithm in Refs. 4,18] An inspection of the analysis of greedy heuristic for weighted set cover (see, e.g. Refs. [9,15]) reveals that this extra constant factor only increases the constant hidden in O(logm log N ) 2 32 Acknowledgments We thank Ajay Joneja of the Industrial Engineering Department at the Hong Kong University of Science and Technology for suggesting the multiple tool milling problem and for ....

V. Chv'atal. A greedy heuristic for the set-covering problem. Math. Oper. Res., 4:233--235, 1979.

....We can now present a proof of Theorem 4.2. Proof: of Theorem 4.2) We cast the problem of finding a minimum 2 hop cover of P as a minimum set cover problem. We then apply the greedy algorithm and find a cover that is larger than the optimal cover by at most a logarithmic factor (Chvatal [1], Johnson [6] Lovasz [9] One di#culty which arises is that the resulting set cover instance is huge. We show, however, that it is possible to apply the greedy algorithm to this set cover instance without generating it explicitly. We first recall the flow of the greedy algorithm of the set ....

....subgraph of Gw . We solve this problem, computing S(w) for each center V , and finally choose the vertex w for which S(w) has the best ratio. We then add the corresponding set S(w) to the cover, update T # , and repeat until T # is empty. It is shown by Johnson [6] Lovasz [9] and Chvatal [1] that the greedy heuristic achieves a performance ratio of H t for the set cover problem, where t is the number of elements to be covered and H t is the Harmonic number. For our problem, the number of elements is equal to the number of vertex pairs such that there is at least one path in P between ....

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V. Chvatal. A greedy heuristic for the set-covering problem. Math. of Oper. Res., 4:233--235, 1979.

....We can now present a proof of Theorem 4.2. Proof: of Theorem 4.2) We cast the problem of finding a minimum 2 hop cover of P as a minimum set cover problem. We then apply the greedy algorithm and find a cover that is larger than the optimal cover by at most a logarithmic factor (Chv atal [1], Johnson [6] Lov asz [9] One difficulty which arises is that the resulting set cover instance is huge. We show, however, that it is possible to apply the greedy algorithm to this set cover instance without generating it explicitly. We first recall the flow of the greedy algorithm of the set ....

....of Gw . We solve this problem, computing S(w) for each center w 2 V , and finally choose the vertex w for which S(w) has the best ratio. We then add the corresponding set S(w) to the cover, update T , and repeat until T is empty. It 7 is shown by Johnson [6] Lov asz [9] and Chv atal [1] that the greedy heuristic achieves a performance ratio of H t for the set cover problem, where t is the number of elements to be covered and H t is the Harmonic number. For our problem, the number of elements is equal to the number of vertex pairs such that there is at least one path in P between ....

V. Chv'atal. A greedy heuristic for the set-covering problem. Math. of Oper. Res., 4:233--235, 1979.

....C for further discussion of some of the relevant results. The best known polynomial time computable approximation ratio (essentially ln n, where n is the number of elements in the underlying universe) for (the weighted) set cover problem is achieved by a most natural greedy algorithm [7] 8] [2] and quite good approximation ratios (namely, 1:61 [6] and 1:861 [9] have been derived by greedy algorithms for the metric uncapacitated facility location problem. For these two optimization problems, we apply the priority algorithm framework in [1] and derive lower bounds on the approximation ....

V. Chv'atal. A greedy heuristic for the set covering problem. Mathematics of Operations Research, 4(3):233--235, 1979.

.... problem entails nding the cover with the minimum cost, and the set partitioning problem is de ned correspondingly [8] Some heuristic and greedy algorithms have been proposed for set covering problem by di erent researchers, for example Chvatal have succeeded to gain a logarithmic ratio bound [18]. Many researches have been done on using di erent genetic algorithms based methods for solving set partitioning problem [19,20] Although many advanced algorithms have been presented for set partitioning problem, but none of them consider the cost of forming a team of agents. Some mathematical ....

V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, 1979.

....bound is incrementally strengthened by adding suitable rows to the set and determining the optimum of this reduced problem. As for heuristics and the computation of upper bounds, a simple greedy algorithm for unate covering problems was proposed by Johnson and Lovasz [16] and extended by Chvatal [9] to non uniform cost problems. Improved variants, were proposed by Balas and Ho, who combined them with dual heuristics in a cutting plane exact algorithm [1] Beasley [2] adopted several reduction procedures, in addition to a lagrangian approach similar to the one we describe in Section 3. Fisher ....

V. Chvatal. A greedy heuristic for the Set Covering problem. Mathematics of Operations Research, 4(3):233--235, August 1979.

....reach all other points. Thus, the minimum energy consumed byall broadcasting methods is at most 1. So the approximation ratio of SPT is at least ffl . As ffl = m. p 1 q 1 1 Figure 2: A bad instance for SPT. The second greedy heuristic is similar to Chvatal s algorithm [1] for the Set Cover Problem and is a variation of BIP. Like BIP, an arborescence, which starts with the source node, is maintained throughout the execution of the algorithm. However, unlikeBIP,many new nodes can be added one at a time. Similar to Chvatal s algorithm [1] the new nodes added are ....

....to Chvatal s algorithm [1] for the Set Cover Problem and is a variation of BIP. Like BIP, an arborescence, which starts with the source node, is maintained throughout the execution of the algorithm. However, unlikeBIP,many new nodes can be added one at a time. Similar to Chvatal s algorithm [1], the new nodes added are chosen to havethe minimal average incremental cost, which is defined as the ratio of the minimum additional power increased by some node in the current arborescence to reach these new nodes to the number of these new nodes. We refer to this heuristic as the Broadcast ....

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V. Chv'atal, "A Greedy Heuristic for the Set-Covering Problem", Mathematics of Operations Research,Vol. 4, No. 3, pp. 233--235, 1979.

....algorithm proposed by Das et al. consists of three stages. The first stage finds an approximation to Minimum Dominating Set, which is essentially the well studied Set Cover problem. Not surprisingly, the heuristic proposed by das et al. in [1] 7] 10] is a translation of Chvatal s greedy algorithm [4] for Set Cover, and thus guarantees an approximation factor of H ( Delta) where Delta is the maximum degree and H is the harmonic function. Let U denote the dominatg set output in this stage. The second stage constructs a spanning forest F . Each tree component in F is a union of stars centered ....

V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operation Research, 4(3):233--235, 1979.

.... Before the LCE is started, the record is assumed to be 1 , or it may be determined by means of some heuristic algorithm (for example, in the LCE algorithm described below the initial admissible solution and the corresponding record value are obtained by the well known Chvatal greedy algorithm [8]) Every time when a new integer point is found, the LCE excludes it by the record inequality. The LCE can be started from any point x(0) The process stops if it is impossible to find the next suitable L class. After this i is the optimal solution to the SCP, given that x(0) 0 . Let i ....

....to exclude the redundant columns from the solution. The most simple heuristic Prime starts with a given cover discarding the columns in increasing order of indices. A column is discarded if the remaining solution is still a cover. The second heuristic is the well known Greedy algorithm [8]. This algorithm might find a solution which is not minimal, therefore Prime is run after it to eliminate the redundant columns. The next heuristic is the Dual Greedy algorithm, which combines the successive columns elimination and the adaptive columns pricing. Let s denote the set of columns in ....

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Chv'atal V. A Greedy Heuristic for the Set Covering Problem, Mathematics of Operations Research, vol. 4, No 3, 1979, pp. 233-235.

....where we still require the triangle inequality, but allow the possibility that c ij #= c ji . The asymmetric versions of UFL and k median are intimately tied to the set cover problem. The best algorithms known for these two clustering problems [4, 7] are based on the greedy set cover algorithm [2, 6, 8], yielding approximation algorithms with O(log N) performance guarantees (though for k median this is only a bicriterion approximation) This note shows, by a simple reduction from set cover, that these factors are the best possible, up to a constant. First we define the problems. The input to ....

V. Chvatal, "A greedy heuristic for the set covering problem," Math. Oper. Res., 4 (1979) 233-235.

....(Here the usual function H is extended to fractional arguments as follows: for i 2 Z and f 2 [0; 1) define H(i f) 1 : f i 1 . We eliminate the dependence on n by using the LP to guide our choice of a 2 cover for A. The greedy set cover algorithm is well studied (see [3, 8, 11]) and the following theorem is known. Theorem 11 If there exists a fractional set cover for S using p centers, then GREEDYSETCOVER (G; S) outputs a cover of size at most pH( jSj ) This follows easily from Lemma 4 with z j 1 and induction on jSj. Now the following three lemmas yield ....

V. Chvatal, "A greedy heuristic for the set covering problem," Math. Oper. Res., 4 (1979) 233-235.

.... A large fraction of the theory of approximation algorithms, as we know it today, is built around the theory of linear programming, which o ers the two fundamental algorithm design techniques of rounding and the primal dual schema (see [44] Interestingly enough, the LP duality based analysis [30, 10] for perhaps the most central problem of this theory, the set cover problem, did not use either of these techniques. Moreover, the analysis used for set cover does not seem to have found use outside of this problem and its generalizations [38] leading to a somewhat unsatisfactory state of ....

V. Chvatal. A greedy heuristic for the set covering problem. Math. Oper. Res., 4:233-235, 1979.

....Since the problem of finding the minimum integer k so that a set of objects S is k pierceable was shown to be NP complete [Kar72, FPT81] many authors have focused on the problem of approximating k. There exist some polynomial time algorithms for the latter problem with bounded error ratio [Chv79, Hoc82]. Bellare et al. BGLR93] show that no polynomial time algorithm can approximate the optimal solution within a factor of ( Gamma ffl) log jSj, unless NP DT IME[n log log n ] where ffl 0. This problem (in its non geometric formulation) is called in the literature the set cover problem. ....

V. Chv'atal. A greedy heuristic for the set-covering problem. Math. Oper. Res., 4:233--235, 1979.

....the deviations from the predicting line, as well. The other algorithms behaves in very similar way The algorithm have been obtained by substituting the stochastic sphere inducer with a deterministic version that always acquire the sphere that cover the largest amount of positive examples. [2] describes the very same algorithm. 4.2 Relation between the number of experiments and the estimation accuracy In all the experiments performed so far, we estimated the regression line using 1000 GH pairs. An interesting point concerns the stability of the line as the number of experiments ....

V. Chvatal. A Greedy Heuristic for the Set Covering Problem. OMEGA, 4(3):233--235, 1979.

....that a weighted set cover problem is defined by a domain D, an index set I and corresponding sets D i ; i 2 I, with positive real costs cost(D i ) A solution to the problem is the subset J I such that j2J D j = D and the sum Sigma j2J cost(D j ) of costs for sets D j is minimized. Chvatal ([2]) showed that if the minimal solution has cost M , then the greedy method obtains in polynomial time a solution with a cost at most M Delta H(jDj) where H(n) P n i=1 1 i = Theta(log n) Greedy algorithm builds the approximately minimal set cover incrementally adding one set (that covers ....

V. Chv'atal. A greedy heuristic for the set-covering problem. Mathematics of the Operation Research, 4(3):233--235, August 1979.

....i = 1; m compute D i = j=1 dX (x i ; x j )p ij . 67 3. Given a relative error bound ffl 0, for each j such that q j 0, construct a set S j : A point x i is in S j iff dX (x i ; x j ) 1 ffl) D i . Note that x j 2 S j for all S j . 4. Apply the greedy set cover algorithm [19, 36] to the covering of by the sets fS j g, choosing iteratively the set S j that covers the most uncovered points. Repeat this process until all points of are covered. Let I U be the set of indices of sets chosen by the greedy set covering heuristic. Output U = fx i g i2I U as the median set. The ....

V. Chv'atal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233--235, 1979. 114

....the k Set Cover problem with performance ratio H k Gamma 5=12 for = 4 and k 4. Proof : We want to minimize the number of chosen sets covering the universe. Therefore we charge a cost of 1 for each set chosen by the algorithm. The cost is uniformly distributed to the elements of a chosen set. [2, 11, 14] We consider any fixed k set S from an optimal solution B. If this optimal solution also contains k sets for some k, then the same investigation can be carried out with an even better result. Let S = fs 1 ; s k g with the elements ordered by nonincreasing cost. Then s k is ....

V. Chvatal, A greedy heuristic for the set-covering problem, Mathematics Of Operations Research (1979), no. 3, 233--235.

....largest number of remaining elements. Johnson [55] and Lov asz [64] independently showed that the performance ratio of the greedy method is no worse than H(m) where H(m) 1 Delta Delta Delta 1=m is the m harmonic number, a value which is clearly between ln m and ln m 1. Chv atal in [24] extended their results to the weighted version of the problem. The original classical analysis of the greedy algorithm has remained essentially unchanged for the last 20 years, despite the fact that H(m) was not known to be a lower bound on the performance ratio of the greedy algorithm. In fact, ....

....example we could make U smaller, namely we could take m=d to be the least common multiple of all numbers between 1 and d. However, it is clear that the harmonic bound can be made tight only when the ground set is much larger than d. The result of Theorem 3. 2 was later generalized by Chv atal [24]. He proved that the weighted version of d Set Cover can be approximated within H(d) using a modified greedy algorithm see Chapter 4 for details. Other, less efficient algorithms for approximating the d Set Cover were able to slightly improve the harmonic bound. Halld orsson [45] showed that an ....

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V. Chv'atal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233--235, Aug. 1979.

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V. Chv'atal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233--235, 1979.

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V. Chvatal. A greedy heuristic for the set covering problem. Math. Operations Research, 1979.

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Chvatal, V. (1979), A Greedy Heuristic for the Set Covering Problem, Math. of Operations Research, 4, 233-235.

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Chvatal, V. A Greedy Heuristic for the Set Covering Problem. Mathematics of Operations Research, 1979, 4:233-235.

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Chvatal, V. A Greedy Heuristic for the Set Covering Problem. Mathematics of Operations Research, 1979, 4:233-235.

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Chvatal, V. A Greedy Heuristic for the Set Covering Problem. Mathematics of Operations Research, 1979, 4:233-235.

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Vasek Chvatal, "A greedy heuristic for the set covering problem," Mathematics of Operations Research, vol. 4, no. 3, pp. 233-- 235, 1979.

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Chvatal, V. A Greedy Heuristic for the Set Covering Problem. Mathematics of Operations Research, 1979, 4:233-235.

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Chvatal, V. A Greedy Heuristic for the Set Covering Problem. Mathematics of Operations Research, 1979, 4:233-235.

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C. CHVATAL, A greedy heuristic for the set-covering problem, Math. of Operation Research, 4 (1979), 233--235.

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V. Chvatal, A greedy heuristic for the set-covering problem, Mathematics of Operations Research, 4 (3) (1979), 233-235.

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V. Chvatal. A greedy heuristic for the set covering problem. Math. Operations Research, 1979.

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V. Chvatal. A greedy heuristic for the set covering problem. Mathematics of Operations Research, 4(3), 1979, pp. 233-235.

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V. Chvatal, A greedy heuristic for the set-covering problem, Mathematics of Operations Research, 4 (1979), pp. 233--235.

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V. Chvatal. A greedy heuristic for the set covering problem. Mathematics of Operations Research, 4(3):233--235, 1979.

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V. Chvatal. A greedy heuristic for the set covering problem. Mathematics of Operations Research, 4:233--235, 1979.

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V. Chvatal. A greedy heuristic for the set-covering problem. Math. of Oper. Res., 4:233--235, 1979.

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V. Chvatal. "A greedy heuristics for the set-covering problem." Mathematics of Operations Research, vol. 4(3), pp. 233-235, 1979.

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V. Chv'atal, "A greedy heuristic for the set covering problem," Mathematics of Operations Research, 4(1979), pp. 233--235.

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V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233--235, 1979.

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V. Chv'atal, "A greedy heuristic for the set-covering problem", Mathematics of Operations Research 4 (1979) 233-235.

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V. Chvatal. A Greedy Heuristic for the Set-Covering Problem. Mathematics of Operations Research, 4(3), August 1979.

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V. Chvatal, \A greedy heuristic for the set covering problem," Mathematics of Operation Research, vol. 4, pp. 233-235, 1979.

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V. Chvatal, \A greedy heuristic for the set-covering problem," Mathematics of Operation Research, vol. 4, no. 3, pp. 233-235, 1979.

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V. Chvatal. A greedy heuristic for the set covering problem. Math. Operations Research, 1979.

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V. Chvatal. A greedy heuristic for the set covering problem. Mathematics of Operations Research, 4(3):233-- 235, 1979.

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) Chvatal, V., "A Greedy Heuristic for the Set Covering Problem," Mathematics of Operations Research, 4, 3, pp. 233-235, 1979.

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