R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover algorithm. Journal of Algorithms, 1981, Vol 2, pp 198-210.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... j j folklore j j E2 LIN 2 1:1383 [20] 12 11 E3 SAT 8 7 [25] 80 77 [9] 8 7 E2 SAT 1:0741 [15] 220 217 [9] 22 21 E4 Set Splitting 8 7 folklore 1 [27] 8 7 Max Cut 1:1383 [20] 72 71 [9] 17 16 Max di Cut 1:164 [20] 72 71 [9] 12 11 Vertex cover 2 [19, 7, 22] 233 218 [9] 7 6 Our lower bounds using gadgets (E2 SAT, E2 LIN 2, Max Cut, Max di Cut) rely on the gadgets produced by Trevisan et al. 35] and since the prior published work in some cases depended on worse gadgets the improvement are not only due to our results. The 2 approximation ....

....and since the prior published work in some cases depended on worse gadgets the improvement are not only due to our results. The 2 approximation algorithm for vertex cover is an unpublished result due to Gavril that is given in [19] The case of weighted graphs was treated by Bar Yehuda and Even [7] and Hochbaum [22] The inapproximability result for linear systems of equations mod p of Amaldi and Kann [1] needed arbitrary systems of linear equations mod p and hence did not, strictly speaking, apply to Max E3 Lin p. An outline of the paper is as follows. In Section 2 we introduce notation, ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover algorithm. Journal of Algorithms, 1981, Vol 2, pp 198-210.

....M in G (for example, by picking edges greedily) and take all vertices in the edges of M as a cover. This idea can be easily generalized to the case of general r. Several algorithms having approximation ratio 2 for the general (weighted) case are known, the algorithm of Bar Yehuda and Even [3] being the simplest of them. Most (if not all) of the more sophisticated approximation algorithms rely heavily on the theorem of Nemhauser and Trotter [21] enabling to reduce a given problem to the problem restricted to a class of graphs G satisfying jV (G)j= G) 2. This reduction is based on ....

R. Bar-Yehuda and S. Even, A linear time approximation algorithm for the weighted vertex cover problem, J. Algorithms 2 (1981), 189-203.

....the example of minimum vertex cover and maximum independent set, which are very similar since the complement of an independent set is a vertex cover in the graph. However, the approximabilities of these two problems are vastly different. Vertex cover can be approximated within a factor of 2 [27, 7, 8] (see [28] for additional references) whereas maximum independent set cannot be approximated within a factor of Omega Gamma n 1 Gammaffl ) for some fixed ffl 0 assuming NP 6= ZPP [21] Ways of overcoming such drawbacks were suggested by Zemel [38] where an axiomatic approach is used to ....

....that for the latter inequality we use the assumption that the edge weights are non negative. This follows from the above inequalities for c(S max ) and c(S min ) 4 Vertex cover, max clique and cover by independent sets There is a simple greedy algorithm that gives an approximation factor of 2 [7, 27] for vertex cover and this is the best asymptotic factor that is known. See [8] for an improvement to 2 Gamma log log jV j 2 log jV j . As an optimization problem, vertex cover is equivalent to maximum independent set since the complement of a vertex cover is an independent set. However, for ....

R. Bar-Yehuda and S. Even, "A linear time approximation algorithm for the weighted vertex cover problem", Journal of Algorithms, 2:198--203 (1981).

....in which the edges correspond to elements and vertices correspond to sets; in this set cover instance, each element is in exactly two sets. Both these problems are NP hard and polynomial time approximation algorithms for both are well studied. For set cover see [12, 26, 29] For vertex cover see [6, 7, 13, 21, 22, 30]. In this paper we study the generalization of covering to partial covering [27, 31] Specifically, in k set cover, we wish to find a minimum number (or, in the weighted version, a minimum weight collection) of sets that cover at least k elements. When k is the total number of elements, we ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. J. of Algorithms 2:198-203, 1981.

....through the example of minimum vertex cover and maximum independent set, which are very similar since the complement of an independent set is a vertex cover in the graph. However, the approximabilities of these two problems are vastly di erent. Vertex cover can be approximated within a factor of 2 [27, 7, 8] (see [28] for additional references) whereas maximum independent set cannot be approximated within a factor of n 1 ) for some xed 0 assuming NP 6= ZPP [21] Ways of overcoming such drawbacks were suggested by Zemel [38] where an axiomatic approach is used to construct di erent ....

....that for the latter inequality we use the assumption that the edge weights are non negative. This follows from the above inequalities for c(S max ) and c(S min ) 4 Vertex cover, max clique and cover by independent sets There is a simple greedy algorithm that gives an approximation factor of 2 [7, 27] for vertex cover and this is the best asymptotic factor that is known. See [8] for an improvement to 2 log log jV j 2 log jV j . As an optimization problem, vertex cover is equivalent to maximum independent set since the complement of a vertex cover is an independent set. However, for ....

R. Bar-Yehuda and S. Even, \A linear time approximation algorithm for the weighted vertex cover problem", Journal of Algorithms, 2:198-203 (1981).

....been considered [13, 24] Unfortunately, there is no longer a one to one relationship between 3HS and Minimum Set Cover for size three subsets, so approximation results do not transfer in this case. The known approximation algorithms for 3HS so far only achieve an approximation factor of three [8, 13, 28]. Hence, it is particularly interesting to develop ecient ( xed parameter) algorithms providing optimal solutions. Finally, we only mention in passing that 3HS is also mentioned in [25] and that an average case analysis of a greedy algorithm for d Hitting Set for constant d has been done in [16] ....

R. Bar-Yehuda and S. Even. A linear-time approximation algorithm for the Weighted Vertex Cover problem. Journal of Algorithms, 2:198-203, 1981.

.... in M a cover for T of weight at most h (i.e. is there a M 0 M such that T [ M 0 and X M i 2M 0 w(M i ) h ) This weighted variant of the minimum set cover problem is well studied, and we can therefore use known techniques developed for the WMSC problem in solving the WOPC problem [5] [9] Clearly, an exact solution to WOPC can clearly be obtained by performing an exhaustive search of all subset combinations. As we did in Section 2.4, we can decrease the computation time of this exponential algorithm by resorting to branch and bound techniques: keeping track of the weights of ....

....partial solutions will enable the pruning of numerous branches of the search tree. Given the analysis in Section 2. 5 of the greedy heuristic for the MSC problem, it is not surprising that a greedy heuristic for the WMSC problem also has a worst case performance bound of (log e jT j 1) DeltaOPT [5] [9] The only difference between the unweighted greedy heuristic (from Figure 2.3) and the weighted variant of the heuristic is the selection criteria. At each step, we now select the subset that covers the maximum number of yetuncovered elements in T at the lowest cost per element (i.e. we ....

R. Bar-Yehuda and S. Even, A Linear-Time Approximation Algorithm for the Weighted Vertex Cover Problem, J. Algorithms, 2 (1981), pp. 199--203.

.... vertices have weight 1, one speaks of Unweighted Vertex Cover (UVC for short) Even when restricted to planar graphs with maximum vertex degree 3, UVC is NP complete [16] There are linear time algorithms giving approximation factor 2 for the unweighted case [16] as well as for the weighted case [6]. Both results can be improved to an approximation factor that is asymptotically better: 2 log log V 2 log V [7, 20] Until now, no further improvements of these bounds have been obtained. The parameterized complexity [11] of UVC recently has received considerable interest [4, 8, ....

R. Bar-Yehuda and S. Even. A linear-time approximation algorithm for the Weighted Vertex Cover problem. Journal of Algorithms, 2:198--203, 1981.

.... vertices have weight 1, one speaks of Unweighted Vertex Cover (UVC for short) Even when restricted to planar graphs with maximum vertex degree 3, UVC is NP complete [16] There are linear time algorithms giving approximation factor 2 for the unweighted case [16] as well as for the weighted case [6]. Both results can be improved to an approximation factor that is asymptotically better: 2 log log jV j=2 log jV j [7, 21] Until now, no further improvements of these bounds have been obtained. The parameterized complexity [12] of UVC recently has received considerable interest [4, 8, 12, 14, ....

R. Bar-Yehuda and S. Even. A linear-time approximation algorithm for the Weighted Vertex Cover problem. Journal of Algorithms, 2:198-203, 1981.

.... research efforts have been aimed to the design of efficient off line approximation algorithms for such problems ( 4] 5] 7] 8] 13] 14] 17] 20] 22] 23] and in order to design efficient on line algorithms we will start from the approach introduced in [13] and then exploited in [3]. We will use the algorithm presented in [3] because, while obtaining the same approximation ratio found in [13] it achieves a better time complexity. In this paper, we first study Min Weighted Node Cover and present a dynamic algorithm which, for any sequence of edge insertions and edge ....

.... design of efficient off line approximation algorithms for such problems ( 4] 5] 7] 8] 13] 14] 17] 20] 22] 23] and in order to design efficient on line algorithms we will start from the approach introduced in [13] and then exploited in [3] We will use the algorithm presented in [3] because, while obtaining the same approximation ratio found in [13] it achieves a better time complexity. In this paper, we first study Min Weighted Node Cover and present a dynamic algorithm which, for any sequence of edge insertions and edge deletions, maintains an approximate solution whose ....

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R. Bar-Yehuda, S. Even. A Linear--Time Approximation Algorithm for the Weighted Vertex Cover Problem. Journal of Algorithms, 2 (1981), 198--203.

....k n, w i 0, a; b; c 2 ZZ m and ; u 2 IN n . Obviously this problem is a generalization of the well known minimum weight vertex cover problem (VC) and the minimum weight 2 satisfiability problem (2SAT) Both problems are known to be NP hard [6] and the best known approximation ratio for VC [3, 9, 11] and 2SAT [8] is 2. Both results are best viewed via the local ratio technique (see [2, 4] Hochbaum et al. 10] presented a 2 approximation algorithm for the 2VIP problem. Their algorithm uses a maximum flow algorithm, therefore the time complexity of their algorithm is relatively high, i.e. ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2:198--203, 1981.

....1.1383 [19] 12 11 # E3 SAT 8 7 [24] 80 77 # [8] 8 7 # E2 SAT 1.0741 [14] 220 217 # [8] 22 21 # E4 Set Splitting 8 7 folklore 1 # [26] 8 7 # Max Cut 1.1383 [19] 72 71 # [8] 17 16 # Max di Cut 1. 164 [19] 72 71 # [8] 12 11 # Vertex cover 2 [18, 6, 21] 233 218 # [8] 7 6 # Our lower bounds using gadgets (E2 SAT, E2 LIN 2, Max Cut, Max di Cut) rely on the gadgets produced by Trevisan et al. 34] and since the prior published work in some cases depended on worse gadgets the improvement are not only due to our results. The ....

....and since the prior published work in some cases depended on worse gadgets the improvement are not only due to our results. The 2 approximation algorithm for vertex cover is an unpublished result due to Gavril that is given in [18] The case of weighted graphs was treated by Bar Yehuda and Even [6] and Hochbaum [21] The inapproximability result for linear systems of equations mod p of Amaldi and Kann [1] needed arbitrary systems of linear equations mod p and hence did not, strictly speaking, apply to Max E3 Lin p. An outline of the paper is as follows. In Section 2 we introduce notation, ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover algorithm. Journal of Algorithms, 1981, Vol 2, pp 198--210.

....problems, there is no longer a one to one relationship between 3HS and Minimum Set Cover for size three subsets. However, 3HS is applied in computational biology for computing evolutionary phylogenies [11, 12] The known approximation algorithms for 3HS so far only achieve approximation factor 3 [3, 7, 17]. Hence, it is particularily interesting if we gain ecient exact algorithms for the problem. This is also emphasized by the growing importance of taking the parameterized complexity of problems into consideration [11, 12] The central concern here is whether or not one can obtain an ecient ....

R. Bar-Yehuda and S. Even. A linear-time approximation algorithm for the Weighted Vertex Cover problem. Journal of Algorithms, 2:198-203, 1981.

.... V . In [7] H astad shows that for VCP there is no polynomial ffi approximation algorithm if ffi 7=6, i.e. no algorithm finds a cover of weight less than 7 6 of optimum in polynomial time, unless P = NP . Bar Yehuda and Even proposed a linear time 2 approximation algorithm for this problem in [2]. No polynomial algorithms with better constant ratio approximation bound are known for the general case of VCP, although various polynomial algorithms with stronger performance guarantees are found for different restricted cases (see e.g. 8] Let us start with a definition of everywhere ....

....algorithm A ff . Algorithm DVC ff For all v 2 V do V 0 V n(N(v) fvg) Find a vertex cover C v for G(V 0 ) using the algorithm A ff . C(v) N(v) C v : Return C = argmin v2V w(C(v) The DVC algorithm [9] may be considered as DVC 2 , where the 2 approximation algorithm [2] is used as A ff algorithm. In case of weighted VCP on the everywhere dense graph the DVC ff scheme does not yield a substantial improvement to the performance guarantee of A ff . Indeed suppose that L 1 ; L 2 ; are the copies of the graph for which the approximation factor ff is attained ....

Bar-Yehuda, R.; Even, S. (1981): A Linear Time Approximation Algorithm for the Weighted Vertex Cover Problem. Journal of Algorithms, Vol. 2. pp. 198-203.

....sufficient to guarantee r approximation because (ii) is approximately satisfied in that every edge has at most r vertices in the cover. Since an optimal dual solution can be found in polynomial time by solving the linear program, Hochbaum obtained a polynomial time algorithm. Bar Yehuda and Even [BE81] observed that sequentially raising the edge packing weights as much as possible yields a maximal edge packing, thus obtaining a linear time algorithm. For our algorithm, we relax (i) further, insisting only that every vertex in the cover nearly have its constraint met, and we show how to ....

....number of sets in which any element occurs. The size M is the sum of the set sizes. The dual problems are also equivalent. 2 Reduction of Vertex Cover to ffl Maximal Packing We first reduce our problem to the problem of finding what we call an ffl maximal packing. This reduction generalizes [Ho82, BE81], who considered ffl = 0. Lemma 1 (Duality) Let C be an arbitrary vertex cover and p an arbitrary edge packing. Then p(E) w(C) Proof: p(E) X e2E p(e) X e2E je Cj p(e) X v2C p(E(v) X v2C w(v) w(C) Lemma 2 (Approximate Complimentary Slackness) Let C be a vertex cover and ....

R. Bar-Yehuda and S. Even. A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2:198--203, 1981.

....the row sum heuristic. Generally speaking, the column sum outperforms with an error bound O(log k) which is due to Johnson [8] Chvatal [4] et al. where k is the largest number of nonzeroes in each column. On the other hand, the bound of row sum heuristics, due to Hochbaum [6] Yehuda and Even [19], Balas [2] et al. is r max which is inferior in general, but becomes a small constant in some cases, where r max is the largest number of nonzeroes in each row. Therefore for the approximation of FVS one question is whether there exists such heuristics similar to that of SC. Also another ....

....Typically, there are two kinds of schemes. One approach is first proposed by Johnson [8] then studied by Chvatal [4] et al. which achieves a bound of log(d) where d is the maximum number of nonzeroes in one column. Another kind of algorithm is the so called row sum heuristic, which are studied by [19] [6] 2] and achieve an error bound of r max where r max is the maximum number of nonzeroes in one row. To apply the approximation algorithm from set covering to FVS, we note that there is one major difficulty to overcome: the detection of all essential cycles in a digraph, which will be a ....

Yehuda, B. and Even, S., A Linear Time Approximation Algorithm For The Weighted Vertex Cover Problem, Journal Of Algorithms, 2, pp. 198-203, (1981).

....and SET COVER is log approximable (see [KT95] it follows that all the problems in MIN F Pi 1 are constant approximable and all the problems in MIN F Pi 2 (1) are log approximable. It can be shown that weight( k HYPERVERTEX COVER continues to be constant approximable (the results in [BYE81] and [Hoc82] concerning weight( VERTEX COVER can be extended to weight( k HYPERVERTEX COVER) Chv atal [Chv79] has proved that weight( SET COVER also continues to be log approximable. From the L completeness of these problems, we obtain the following theorem. Theorem 3.6 (1) All problems ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. J. of Algorithms, 2(2):198--203, 1981.

.... empirically, but not within any provable simultaneous bounds) 7 Experimental Results We implemented the exact algorithm and the approximation algorithms discussed above using the C pro 2 This weighting criterion and its performance with respect to the weighted MSC problem are discussed in [2]; a heuristic for the unweighted MSC problem is analyzed in [3] gramming language in the UNIX environment (code is available from the authors upon request) In this section we compare the performance and running times of three algorithms: the efficient branch and bound optimal (BBOPT) algorithm ....

R. Bar-Yehuda and S. Even, A Linear-Time Approximation Algorithm for the Weighted Vertex Cover Problem, J. Algorithms, 2 (1981), pp. 199-- 203.

....connection is the following Proposition 3.3 [10] Let c; f; s be constants. Then FPCP c;s [ log; f ] D Gap Min VC c 0 ;s 0 with gap c 0 =s 0 = 1 (c Gamma s) 2 f Gamma c) The best known approximation algorithm for Min VC achieves a factor of 2 Gamma o(1) 9, 42] improving [8, 34] and Gavril [29] Putting together Theorem 3.2, Proposition 3.3 and Proposition 2.1 says that on the other hand we can t get within 6:8 of the optimum Corollary 3.4 [10] Assuming P 6= NP, Min VC has no factor 1:068 approximation algorithm. Obviously this is not a tight result. But it is not ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. In Jour. of Algorithms Vol. 2, 1981, pages 198--201.

....problems. The first question is whether a .878 approximation algorithm for MAX CUT can be obtained without explicitly solving the semidefinite program. For example, the first 2approximation algorithms for weighted vertex cover involved solving a linear program [32] but later Bar Yehuda and Even [3] devised a primal dual algorithm in which linear programming was used only in the analysis of the algorithm. Perhaps a semidefinite analog is possible for MAX CUT. The second question is whether adding additional constraints to the semidefinite program leads to a better worst case bound. There is ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2:198--203, 1981.

.... in M a cover for T of weight at most h (i.e. is there a M 0 M such that T [ M 0 and X M i 2M 0 w(M i ) h ) This weighted variant of the minimum set cover problem is well studied, and we can therefore use known techniques developed for the WMSC problem in solving the WOPC problem [2] [3] Clearly, an exact solution to WOPC can clearly be obtained by performing an exhaustive search of all subset combinations. As we did in Section 4, we can decrease the computation time of this exponential algorithm by resorting to branch and bound techniques: keeping track of the weights of ....

....of partial solutions will enable the pruning of numerous branches of the search tree. Given the analysis in Section 5 of the greedy heuristic for the MSC problem, it is not surprising that a greedy heuristic for the WMSC problem also has a worst case performance bound of (log e jT j 1) DeltaOPT [2] [3] The only difference between the unweighted greedy heuristic (from Figure 5) and the weighted variant of the heuristic is the selection criteria. At each step, we now select the subset that covers the maximum number of yet uncovered elements in T at the lowest cost per element (i.e. we select ....

R. Bar-Yehuda and S. Even, A Linear-Time Approximation Algorithm for the Weighted Vertex Cover Problem, J. Algorithms, 2 (1981), pp. 199--203.

....SET COVER is log approximable (see [KT91] it follows that all the problems in MIN F Pi 1 are constant approximable and all the problems in MIN F Pi 2 (1) are log approximable. It can be shown that weight( k HYPERVERTEX COVER continues to be constant approximable (the results in [BYE81] and [Hoc82] concerning weight( VERTEX COVER can be extended to weight( k HYPERVERTEX COVER) Chv atal [Chv79] has proved that weight( SET COVER also continues to be log approximable. From the L completeness of these problems, we obtain: Theorem 3.4 (1) All problems in weight( MIN F ....

....cannot be approximated with approximation ratio 2k 1. Proof : Assume that weight VERTEX COVER can be approximated with ratio k 1 2k Gamma 1. Let G be an instance of weight( VERTEX COVER and let b the weight of the minimum vertex cover of G. By the approximation algorithms from [Hoc82] and [BYE81] we can find a value c such that b c 2b. Since we can consider k k 1 2k Gamma 1, it follows that b 2(k 1 Gammak) k 1 Gamma1 b. Let a = k 1 Gammak k Gamma1 Delta c and consider the graph G 1 which consists of one vertex v having weight Gammaa. Let G 2 be the disjoint union of G ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. J. of Algorithms, 2(2):198--203, 1981.

....Each of these m vertex cover problems is 2 approximable in polynomial time. The 2 approximation for the general node biclique problem is the minimum of P j2V nV s;t w j z s;t for all pairs s and t. It is possible to use Bar Yehuda and Even s 2 approximation algorithm for vertex cover, [BYE81], that runs in time linear in the number of elements edges that need to be covered. Here this number is O(n 2 ) The procedure has to be run for each selected edge (s; t) and thus the overall complexity is O(mn 2 ) The appropriate network is depicted in Figure 4. Therefore we have a ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. J. of Algorithms 2, 198--203, 1981.

....weight VC. Clearly U is a vertex cover, because for (u; v) 2 E we have x (u) x (v) 1, which implies x (u) 1=2 or x (v) 1=2. Also X v2U w(v) X v2V w(v)2x (v) 2LB since 2x (v) 1 for all v 2 U . 3.4. 2 Primal Dual applied to VC This is due to Bar Yehuda and Even [4]. First formulate the dual problem. Let y 2 R jEj ; the elements of y are y(e) for e = u; v) 2 E. The dual is: max X e2E y(e) X u:e= v;u)2E y(e) w(v) 8v 2 V (3) y(e) 0 8e 2 E: 4) Initialize C (the vertex cover) to the empty set, y = 0 and F = E. The algorithm proceeds by repeating ....

R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2:198--203, 1981. Approx-47

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R. Bar-Yehuda and S. Even (1981). A linear time approximationalgorithm for the weighted vertex cover problem. J. Algorithms 2, 198--203.

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