L. A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2:385-393, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....This was generalized by Chv atal [3] to the weighted uncapacitated case, and further generalized by Dobson [6] to approximating with logarithmic ratio the integer linear program min c x subject to Ax b with all the entries in A nonnegative. A much more general result is given by Wolsey [18], giving a logarithmic ratio approximation algorithm for submodular cover problems. Both the vertex cover problem with hard capacities, and set cover problem with hard capacities are an example of a submodular cover problem. Hence [18] gave the rst nontrivial approximation for both problems. See ....

....in A nonnegative. A much more general result is given by Wolsey [18] giving a logarithmic ratio approximation algorithm for submodular cover problems. Both the vertex cover problem with hard capacities, and set cover problem with hard capacities are an example of a submodular cover problem. Hence [18] gave the rst nontrivial approximation for both problems. See also the work by Bar Ilan et al. 2] for a generalization of the method including, e.g. generalization of the set cover problem with hard capacities problem, facility location problems under ow constraints and the 2 layered ....

L. A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2:385-393, 1982.

....spanning subhypergraph problem is even harder than the Steiner tree problem. Input a weighted hypergraph (R; FC ; j j) i 0 While (R; ft 1 ; t i g) is not connected: Select t i 1 2 FC that minimizes f i i i 1 p i Output t 1 ; t p Figure 1: Algorithm Greedy MSS. Wolsey [17] showed that the greedy algorithm achieves a performance ratio of H(k 1) if the size of every hyperedge is bounded by k. Here H(k) P k 1=i = ln k O(1) denotes the k th harmonic number. This result is best possible [5] and does not immediately lead to a good approximation algorithm for the ....

L.A. Wolsey, An analysis of the greedy algorithm for the submodular set covering problem, Combinatorica 2 (1982), 385--393.

....i ) and (R, t, t 1 , t i ) Note that in quasi bipartite graphs, algorithm Greedy MSS is identical to Rayward Smith s average distance heuristic [14] 3 Performance Analysis In general the minimum spanning subhypergraph problem is even harder than the Steiner tree problem. Wolsey [17] showed that the greedy algorithm achieves a performance ratio of H(k 1) if the size of every hyperedge is bounded by k. Here H(k) # k i=1 1 i = Input a weighted hypergraph (R, FC , i # 0 While (R, t 1 , t i ) is not connected: Select t i 1 # FC that minimizes ....

L.A. Wolsey, An analysis of the greedy algorithm for the submodular set covering problem, Combinatorica 2 (1982), 385--393.

.... a partial spanning tree of minimum weight among those spanning at least k nodes in the graph [RSM 94] Yet another related tree problem is studied in [KR95] Other kinds of greedy approximation algorithms for various NP hard problems were extensively studied [Joh74, Lov75, Chv79, Dob82, Wol82, KP94] The relaxed approach of multi objective approximation was studied in [RMR 93] for the similarly structured problem of constructing a spanning tree of minimum weight among those whose degree is bounded by d. Returning to the BDST problem, perhaps the simplest case of the BDST ....

L. A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2:385--393, 1982.

....SSC ) problem, and it is a common generalization of SC, IC, and the matroid base problems. In light of those facts mentioned above it is quite natural to consider the greedy approach for SSC as well, and Wolsey analyzed its performance and generalized the aforementioned results for SC and IC in [34], which will be presented in Sect. 3.1. Paralleling the development in analysis of the greedy algorithm for SC, IC, and SSC, another natural heuristic for SSC has been designed based on the LP duality, and its performance generalizes the approximation bound of Hochbaum for SC (TU 3.2) he ....

....function fS defined on 2 N S s.t. fS( X) f( X # S) f( S) It is easy to verify that if f is nondecreasing and submodular on N then so is fS on N S; thus, we can define another submodular system( N S, fS ) for any S # N . Let fS( j) denote fS( j ) It was shown by Wolsey [34] that SSC can be formulated by the following integer linear programming: Min # j#Nc j x j s.t. t. # j#N S fS( j)x j # fS( N S) S # N x j # 0, 1 j # N o see it let x T # 0, 1 N be the characteristic vector of T # N.IfT is an SSC solution, i.e. f( T) f( N ) x ....

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L.A. Wolsey, "An analysis of the greedy algorithm for the submodular set covering problem," Combinatorica, vol.2, no.4, pp.385--393, 1982.

....can be formulated as the problem of finding a minimum spanning set in a certain polymatroid. The general concept of polymatroids provides a natural way to shape the arguments needed for the analysis of the greedy algorithm. An analysis based on linear programming duality was given by Wolsey [18]. Our new analysis relies on the fact that each polymatroid can be represented by a system of subsets of an appropriately chosen matroid. The above mentioned result of Feige [5] implies that the approximation ratio of the greedy algorithm for spanning sets in k polymatroids and for k MSS is best ....

....m X i=1 u i 1 (h) u i (h) u i 1 (h) H u 0 (h) H um (h) um (h) 0, and that u 0 (h) k by (1) this implies w(ALG) H(k) X h2OPT w(h) H(k) w(OPT) proving the theorem. 5 Acknowledgement We are grateful to Toshihiro Fujito for bringing the article of Laurence Wolsey [18] to our attention. ....

Laurence A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2(4):385--393, 1982.

....W denote the set of uncontrolled vertices. Initially M = and W = V . While W 6= pick a vertex v 2 V n M maximizing j Gamma(v) W j, add it to M and remove from W any vertex that is now controlled by M . Since the problem is submodular, it is easy to see, using an analysis similar to [D82, W82] that the resulting monopoly M is at most ln jEj 1 times greater than the minimum one. Hence we have Proposition 5.1 The greedy algorithm yields a ratio ln jEj 1 approximation for the minimum monopoly problem. A similar situation (in terms of hardness and approximability) holds for ....

L. A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2:385--393, 1982.

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