L. Lovasz, On the ratio of optimal integral and fractional covers, Discrete Mathematics, 13, 383-390, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....k Dominating Set may exist for any particular given graph. The problem of nding the optimal size set is NP Complete, so it would be interesting to nd a distributed algorithm that constructs a good approximation. As this problem can be formulated as a special case of the set cover problem of [Lo], the greedy sequential algorithm described therein provides an approximation ratio of log n 1. The problem was studied further in the realm of sequential algorithms in [BKP] where its load balanced case is given an approximate solution. As an application of the k Dominating Set al..gorithm we ....

L. Lovasz, On the ratio of optimal integral and fractional covers, Discrete Mathematics, 13:383-390, 1975.

....of sets of equal size. The first technique is by using a greedy algorithm that starts with P = and iteratively adds to the set P an element in B occurring at the most uncovered sets. The algorithm stops when P becomes a cover. The set P is termed a greedy cover for H. Lemma 2. 2 (Lov asz [24]) Let P be a greedy cover for H. Then jP j jBj(ln jHj 1) t. The second method is randomized, and takes each element of B to the set P with probability (c ln jHj) t, for some constant c 1. Lemma 2.3 (Awerbuch et al. 2] Let P be the set constructed by the randomized cover algorithm under ....

L. Lov' asz, On the ratio of optimal integral and fractional covers, Discrete Mathematics, 13 (1975), pp. 383--390.

....explaining why no ecient algorithm with performance guarantee much better than k has been found. Previous Hardness Results The vertex cover problem on hypergraphs where the size of the hyperedges is unbounded is no other than Set Cover. For this problem there is a ln n approximation algorithm [20, 18], and a matching (1 o(1) ln n hardness result due to Feige [8] The rst explicit hardness result shown for Ek Vertex Cover was due to Trevisan [23] who considered the approximability of bounded degree instances of several combinatorial problems, and speci cally showed an inapproximability ....

L. Lovasz. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383-390, 1975.

....Rather, the major cost is the number of servers needed to service all the clients and the access bandwidth required at each server s network interface. Our model of server placement resembles more closely to the set cover problem. The classic greedy algorithm for solving set cover problem [10, 12] achieves an O(log n) performance ratio. In geometric spaces, the problem is easier. In [7] Hochbaum proposed a shifting strategy that gives an (1 #) performance ratio. Unfortunately, the interconnections between networks dictate that the network propagation delay no longer exhibits the ....

....order of set size; Step 2: Starting from the smallest set, check if it can be removed without leaving any of its nodes uncovered; If so, remove the set. Step 3: Repeat Step 2 until all sets are checked. 4.2 Greedy Heuristics A greedy algorithm is usually attractive due to its simplicity. In [10, 12], Johnson and Lov asz introduced a greedy algorithm for the set cover problem with an O(log n) approximation ratio. The basic greedy attribute of the algorithm is to select a set at every step that contains the maximum number of uncovered elements. For the backup problem variant, we extend the ....

L. Lov asa. On the Ratio of Optimal Integral and Fractional Covers. Discrete Mathematics, 13:383--390, 1975.

....sets in increasing order of set size; Step 2: Starting from the smallest set, check if it can be removed without leaving any of its nodes uncovered. Step 3: Repeat Step 2 until all sets are checked. 5.1.2 Greedy Heuristics A greedy algorithm is usually attractive due to its simplicity. In [41, 48], Johnson and Lov asz introduced a greedy algorithm for the set cover problem with an O(log n) approximation ratio. The basic greedy attribute of the algorithm is to select a set at every step that contains the maximum number of uncovered elements. For the backup problem variant, we extend the ....

....providers. Rather, the major cost is the number of servers needed to service all clients and the access bandwidth required at each server s network interface. Our model of server placement more closely resembles the set cover problem. The classic greedy algorithm for solving the set cover problem [41, 48] achieves an O(log n) performance ratio. In geometric spaces, the problem is easier. In [35] Hochbaum proposed a shifting strategy that gives an O(1 #) performance ratio. Unfortunately, the interconnections between networks dictate that the network propagation delay no longer exhibits the ....

L. Lov asa. On the Ratio of Optimal Integral and Fractional Covers. Discrete Mathematics, 13:383--390, 1975.

....all the sets U ij for which jU ij j s into a collection of sets called C. It then calls algorithm hitting set to nd a set that hits all the sets in this collection. Algorithm hitting set uses the greedy heuristic to nd a set B that hits all the sets in the collection C. As shown by Lov asz [Lov75] and Chv atal [Chv79] the size of the hitting set returned by hitting set is at most (ln ) 1 times the size of the optimal fractional hitting set, where is the maximal number of sets that a single element can hit. As each set in the collection C contains at least s elements, there is a ....

L. Lovasz. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383{ 390, 1975.

....greedy algorithm which repeatedly adds the node (resp. set) that covers the most number of uncovered nodes (resp. elements) The greedy algorithm achieves an H approximation where is the maximum degree of a node (resp. maximum number of elements in a set) and H i is the ith harmonic number [9, 17, 22]. Furthermore, Feige has shown that the approximation ratio achieved by the greedy algorithm for either problem is the best possible (to within a lower order additive term) unless NP has n O(log log n) time deterministic algorithms [11] Dominating sets and their variants (e.g. k dominating ....

....of v is immaterial. Furthermore, u is covered in exactly one round; thus, cost(u) is well de ned. The sum of cost(u) taken over all the nodes u, is closely tied to the performance of the greedy algorithm, and the following lemma, which is based on the well known analysis of the greedy algorithm [17, 22], places an upper bound on this sum. Lemma 3.5. P u2V cost(u) H jOPTj: Proof: Consider any v 2 OPT, and let denote jC(v)j. We sort all u 2 C(v) to obtain the sequence u1 , u2 , u such that for any 1 i j , u i is assigned a cost before u j . Then we have cost(u i ) 1 ....

L. Lovasz. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383-390, 1975.

....on A; b, and c in the above definition are without loss of generality: an arbitrary packing problem can be reduced to the above form (see [29] We are interested in PIPs where b i = B for 1 i d. When A 2 f0; 1g d Thetan this problem is known as the simple B matching in hypergraphs [24]: given a hypergraph with non negative edge weights, find a maximum weight collection of edges such that no vertex occurs in more than B of them. When B = 1 this is the usual hypergraph matching problem. We note that the maximum independent set problem in graphs is a special case of the hypergraph ....

L. Lov'asz. On the ratio of the optimal integral and fractional covers. Discrete Math., 13:383-- 390, 1975.

....is a special case of set cover 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. ....

....S of subsets of T , S = fS 1 ; S 2 ; Sm g, a cost function c : S Q , and an integer k, find a minimum cost subcollection of S that covers at least k elements of T . Previous Results: For the full coverage version, a ln n 1 approximation was proposed by Johnson [26] and Lov asz [29]. This analysis of the greedy algorithm can be improved to H ( Delta) see the proof in [14] where Delta is the size of the largest set 1 . Chv atal [12] generalized this to the case when sets have costs. Slav ik [32] shows the same bound for the partial cover problem. When Delta = 3, Duh ....

L. Lov'asz. On the ratio of optimal integral and fractional covers. Discrete Math. 13:383-390, 1975.

.... communication, but using myopic bidders, by adapting the algorithm of [10] Somewhat surprisingly, the situation for procurement auctions is di erent: an approximation to within a factor of ln n is possible (with myopic bidders) in polynomial communication, adapting the greedy method of [11], but a better approximation ratio requires exponential communication. The proofs use communication complexity types of reasoning, and are presented in a companion paper aimed at the computer science community [14] 4.3 Revenue If one considers cases where the k players are partitioned into ....

L. Lov'asz. The ratio of optimal integral and fractional covers. Discrete Mathematics, 13, 1975.

.... within n 1=2 [2] It seems that this similarity is rooted in the gap between the fractional solution and the integral one: The gap between the fractional cover number and the integral cover number is O(log n) and the gap between the fractional packing number and the integral one is O( p n) [4, 1]. Indeed, it is interesting to observe that the fractional cover number and fractional packing number may be both computed exactly in a polynomial amount of communication. The linear programs describing the fractional cover number and the fractional packing number, each have a variable for each ....

....all players is chosen. The elements that re contained in this set are removed, and a new stage starts, now considering only the remaining elements. The protocol ends when all items have been covered after at most n stages. Thus the total amount of communication is O(kn 2 ) It is well known [4] that this procedure produces a (ln n) approximation to the minimum cover. Note: The weighted case (where each player has a valuation function v i : P (f1: ng) R and the aim is to minimize P i v i (S i ) over all covers S 1 : S k ) can also be approximated this way to within the same ....

L. Lov'asz. The ratio of optimal integral and fractional covers. Discrete Mathematics, 13, 1975.

....by discs, which he shows gives a near optimal solution with high probability when p is (log n) and o(n= log n) and the points are randomly distributed in the unit square. The problem can be modeled as a capacitated set cover problem. The well known greedy algorithm of Johnson [15] and Lov asz [17], as modi ed for the capacitated case by Bar Ilan, Kortsarz, and Peleg [2] would yield a ln n approximate solution, where n is the number of galaxies. This algorithm is not good enough in practice. For this particular set cover problem the dual of the set system has bounded VC dimension; in this ....

Laszlo Lovasz. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383-390, 1975.

....techniques and Lagrangian relaxation. Our algorithms are also relatively simple, although they are not as effective for some problems of large width. Flavor of Oblivious Rounding Algorithms: For the (integer) set cover problem, oblivious rounding yields the greedy set cover algorithm [10, 14]. For the fractional set cover problem, it yields an algorithm that repeatedly chooses a set whose elements have the largest net weight, where the weight of an element is initially 1 and is multiplied by 1 Gamma ffl each time a set containing it is chosen. To obtain the final cover, each set is ....

....chooses each set to minimize the number of elements remaining uncovered. Nonetheless, it is guaranteed to keep up with the random experiment, finding a cover within djC j ln ne steps. This is the greedy set cover algorithm, originally analyzed non probabilistically by Johnson [10] and Lov asz [14]. Versus fractional cover: If the cover C is a fractional cover, the analyses of both algorithms carry over directly to show a ln n performance guarantee. What enables oblivious rounding We call such algorithms oblivious rounding algorithms. What kinds of randomized rounding schemes admit ....

L'aszl'o Lov'asz. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383--390, 1975.

....there exists a fractional solution of assignment cost d and facility cost k, produces a solution for which the assignment cost plus the facility cost is at most d 2k. 2 2 Background In the mid 1970 s Johnson and Lovasz gave a greedy [H k] approximation algorithm for unweighted set cover [8, 10]. In 1979 Chvatal generalized it to a [H k] approximation algorithm for weighted set cover [3] In 1982 Hochbaum gave a greedy [H (d k) approximation algorithm for the uncapacitated facilitylocation problem by an implicit reduction to the weighted set cover problem [7] 1 Above is at ....

Laszlo Lovasz. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383-390, 1975.

....for fractional coloring that achieves ratio jV j d , unless P = NP. Feige and Kilian [6] proved that the chromatic number cannot be approximated within jV j 1 e for any e 0, unless NP ZPP. This also holds for the fractional coloring problem, due to the logarithmic relationship (see also [11]) of the fractional chromatic number and the chromatic number (i.e. c f (G) c(G) c f (G) 1 lna(G) 2 The Conversion Technique Using the definitions from Section 1.1, we can state our main results as follows. Theorem 1 For every class of graphs that is (a;b) tractable or ....

LOV ASZ, L. On the ratio of the optimal integral and fractional covers. Discrete Mathematics 13 (1975), 383--390.

....subsets E (edges) of a set V (vertices) A set M E is a matching in H if no vertex occurs in more than one edge in M ; a basic and well known NP hard problem is to find a maximumcardinality matching in a given finite hypergraph. A generalization of this is matching, for integral 1 (Lov asz [21]) each vertex can be in at most elements of M . The matching problem is naturally written as a PIP with 0 1 variables x i , and with B = Similarly, CIPs model, e.g. the classical set cover problem covering V using the smallest number of edges in E (such problems have natural weighted ....

L. Lov'asz, On the ratio of optimal integral and fractional covers, Discrete Math., 13 (1975), pp. 383--390.

....covering number may be used as an estimate for the covering number. It is natural to ask how good this estimate is, or, in other words, how large the ratio = can be for certain types of hypergraphs. A very useful upper bound on the ratio = was obtained independently by Lov asz([3]) Sapozhenko( 6] and Stein( 7] this bound asserts that = 1 log D, where D = max v2V j e : v 2 e j the maximum degree in the hypergraph H. In this paper we focus on bounds based on the rank r = r(H) Since the union of a maximum matching forms a cover, we have r , so ....

L. Lovasz, On the ratio of optimal integral and fractional covers, Discrete Math. 13 (1975), 383-390.

No context found.

L. Lovasz, On the ratio of optimal integral and fractional covers, Discrete Mathematics, 13, 383-390, 1975.

No context found.

L. Lov' asz, On the ratio of optimal integral and fractional covers, Discrete Mathematics, 13 (1975), pp. 383--390.

No context found.

L. Lovasz. On the ratio of optimal integral and fractional covers. Discrete Math., 13:383-390, 1975.

No context found.

L. Lov asz, On the ratio of optimal integral and fractional covers, Discrete Mathematics, 13 (1975), pp. 383-390.

No context found.

L. Lovasz. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383-390, 1975.

No context found.

L. Lovasz, On the ratio of optimal integral and fractional covers. Discrete Math., 13, pages 383-390, 1975.

No context found.

Lov'asz, L. (1975) \The ratio of optimal integral and fractional covers." Discrete Mathematics, 13.

No context found.

Lov'asz, L. (#975) "The ratio of optimal integral and fractional covers." Discrete Mathematics, #3.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC