LOV ASZ, L. 1973. Coverings and colorings in hypergraphs. In Proceedings of 4th South Eastern Conference on Combinatorics, Graph Theory and Computing. Utilitas Math., Winnipeg, B.C., Canada, 3--12.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for about eighty years (e.g. 4, 10, 11, 15, 16, 19, 20] For k = 2, i.e. for graphs, the problem is well understood since a graph is 2 colorable if and only if it has no odd cycle. For k 3, though, much less is known and deciding the 2 colorability of k uniform hypergraphs is NP complete [17]. In this paper we discuss the 2 colorability of random k uniform hypergraphs for k 3. For the evolution of odd cycles in random graphs see [12] Let H k (n; m) be a random k uniform hypergraph on n vertices, where the edge set is formed by selecting uniformly, independently and with ....

Lov asz, L. Coverings and coloring of hypergraphs. Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Florida, 1973), 3-12.

....of dominated and non dominated coteries are proved. Alternative protocols based on quorum systems (rather than on voting) appear in [28] using finite projective planes) 1] the Tree system) 5, 25] using a grid) and [23, 24, 38] hierarchical systems) The triangular system is due to [26, 9]. The Wheel system appears in [29] The CWlog system appears in [37, 36] The motivation for several of these alternative systems was to find constructions with high availability and low load (which is referred to in most of these papers as quorum systems with small quorums) In [19] the ....

L. Lov' asz, Coverings and colorings of hypergraphs, in Proc. 4th Southeastern Conf. Combinatorics, Graph Theory and Computing, 1973, pp. 3--12.

....U = U 1 [ U 2 if S j U 1 6= and S j U 2 6= This problem is also called Max Hypergraph Cut. Max E k Set Splitting is a special case of Max Set Splitting in which all subsets in C have cardinality exactly k. For any xed k 2, Max E k Set Splitting was shown to be NP hard by Lov asz [12]. Petrank [15] proved that there exists some constant 0 such that the existence of a polynomial time (1 ) approximation algorithm would imply P=NP. Max E 2 Set Splitting is exactly the extensively studied MAX CUT problem, which is known to be NP hard to approximate within 16 17 for ....

L. Lovasz, \Coverings and Colorings of Hypergraphs," In Proc. 4th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica Publishing, Winnipeg, (1973) 3-12.

....eighty years (see, e.g. 8, 9, 2, 17] For k = 2, i.e. for graphs, the problem is well understood, since graph 2 colorability is equivalent to having no odd cycle. For k 3, though, much less is known and deciding the 2 colorability of k uniform hypergraphs is NP complete for every xed k 3 [15]. In this paper we discuss 2 colorability of random k uniform hypergraphs for k 3. For the evolution of odd cycles in random graphs see [10] Let H(k; n; p) be a random k uniform hypergraph on n labeled vertices V = f1; ng, where each k subset of V is chosen to be an edge of H ....

Lov asz, L. Coverings and coloring of hypergraphs. Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Florida, 1973), 3-12.

....have a non empty intersection. In this paper we develop a solution for grid based generation of quorums for mutual exclusion protocols which brings down the quorum size from 2 p N to p 2 p N . This construction results in the same suboptimal quorum size as the Triangular system of Lov asz [Lov73]. An advantage of our construction is that it extends Maekawa s grid based approach: once the method for generating quorums is given, it is geometrically obvious that they satisfy the properties required of quorum sets. Thus although suboptimal from the theoretical point of view, the p 2 p N ....

L. Lov'asz. Coverings and Colorings of Hypergraph. In Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 3--12, 1973.

....Theorem 2 The problem to decide for an arbitrary mixed hypergraph H whether Omega Gamma H) 0 is NP complete. Proof. We will prove that the recognition problem of colorable mixed hypergraphs is at least as hard as the problem of hypergraph 2 colorability. Since the latter is NP complete [10], the same will follow for the former, too. For an arbitrary hypergraph H = X; E) with vertex set X, construct the mixed hypergraph H 3 = X; i X 3 j ; E) 6 where i X 3 j denotes the collection of all 3 element subsets of X, i.e. each triple of vertices forms a co edge in H 3 ....

Lov'asz L.: Coverings and colorings of hypergraphs, In: Proc. 4th Southeastern Conf. on Combinatorics, Graph Theory, and Computing , Utilitas Math. Publ., Winnipeg, 1973, 3--12.

....that the grid construction would work for some non square grids as well. The conditions for the construction to work are that all rows contain at least 4 columns, and that there be sufficient elements to choose singletons on different columns in the non cross rows. For example, triangular grids [Lov73], or CWlog grids [PW95] that contain 2 i rows of length 2 i each) could be adapted to the grid construction. As the square grid already possesses a load within a constant factor of the optimal, we do not pursue such constructions further. Despite its asymptotically optimal load, grid is ....

L. Lov'asz. Coverings and colorings of hypergraphs. In Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 3--12, 1973.

....in general (including intersecting hypergraphs in particular) were studied extensively in recent years (cf. B86] The terms coterie and nondominated coterie (NDC) are defined in [GB85] and many properties of coteries and NDCs are presented. Some interesting properties of NDCs are derived in [L73]. In [IK90] a relationship is established between coteries and boolean functions. Properties of coteries and NDCs are derived from properties of the appropriate functions. 1.4. Contributions. This paper focuses on a number of questions related to the issue of balancing the load on processors ....

L. Lov asz, Coverings and colorings of hypergraphs, in Proc. 4th Southeastern Conf. on Combinatorics, Graph Theory and Computing, Florida Atlantic University, Utilitas Mathematica, Winnipeg, Canada, 1973, pp. 47--56.

....: ng. These examples play an important role in the results of this paper. 1. 2 Related Work Set systems have attracted attention as combinatorial objects since the 19 th century, but have matured into a broad field of study in the 60 s and 70 s (cf. Bol86] In particular, these studies [Lov73, EL75] have provided a number of constructions of quorum systems, further developed and studied in [Tuz85, MP92] The first distributed control protocols using quorum systems [Tho79, Gif79] use voting to define the quorums. Each processor has a number of votes, and a quorum is any set of processors ....

....2 p 1, F p (S) 1 as n 1) The Condorcet Jury Theorem [Con] translated to our terminology, states that the majority NDC has this property. We show that several other constructions also have Condorcet failure probabilities (e.g. the constructions of [EL75, AE89] while others still (e.g. [Lov73]) have a nonzero limit function when p 1 2 . Moreover, we show that the finite projective plane construction [Mae85] and the grid construction [CAA90, MP92] both of which are dominated) have failure probabilities tending to 1 for all values 0 p 1, so these constructions have poor ....

[Article contains additional citation context not shown here]

L. Lov'asz. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conf. Combinatorics, Graph Theory and Computing, pages 3--12, 1973.

....properties of dominated and non dominated coteries are proved. Alternative protocols based on quorum systems (rather than on voting) appear in [27] using finite projective planes) 1] the Tree system) 6] using a grid) and [23, 24, 36] hierarchical systems) The triangular system is due to [25, 10]. The Wheel system appears in [28] The CWlog system appears in [34] The motivation for several of these alternative systems was to find constructions with high availability and low load (which is referred to in most of these papers as quorum systems with small quorums) In [19] the question of ....

L. Lov'asz. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conf. Combinatorics, Graph Theory and Computing, pages 3--12, 1973.

....is polynomially equivalent to TRANSVERSAL HYPERGRAPH. A restricted version of the latter problem is the question whether a given hypergraph is equal to its own transversal hypergraph, i.e. T r(H) H. This problem, which we call SELFTRANSVERSALITY, has attracted much interest by mathematicians [Lov73, Ben80, Ber89] but no complexity results have been derived so far. We show that SELFTRANSVERSALITY is complete for TRANSVERSAL HYPERGRAPH, i.e. it has the same complexity as the general problem. In Section 5 we identify subclasses of the well known SATISFIABILITY problem which are polynomially ....

....TRANS HYP reduces to a very specific subproblem which is closely related to the issue where the boundary lies between the hypergraphs that are 2 colorable and those that are not. This issue is the more of interest as graph k colorability (NPcomplete for k 3 reduces to hypergraph 2 colorability [Lov73] Hypergraphs that are 2 uncolorable but become 2 colorable if any edge is removed (edge critical hypergraphs) have been studied extensively in the literature, cf. Lov68, Lov73, EL73, AL86, Tof74] Lov asz observed that such a critical hypergraph must have strict properties if it is intersecting ....

[Article contains additional citation context not shown here]

L. Lov'asz. Coverings and Colorings of Hypergraphs. In Proceedings of the 4th Conference on Combinatorics, Graph Theory, and Computing, pages 3--12, 1973.

....forty years (see, e.g. 8, 9, 2, 16] For k = 2, i.e. for graphs, the problem is well understood, since graph 2 colorability is equivalent to having no odd cycle. For k 3, though, much less is known and deciding the 2 colorability of k uniform hypergraphs is NP complete for every fixed k 3 [14]. In this paper we discuss 2 colorability of random k uniform hypergraphs for k 3. For the evolution of odd cycles in random graphs see [10] Let H(k;n; p) be a random k uniform hypergraph on n labeled vertices V = f1; ng, where each k subset of V is chosen to be an edge of H ....

LOV ASZ, L. Coverings and coloring of hypergraphs. 3--12. 12 Proceedings in Informatics

....for 2 coloring hypergraphs which have an appropriately small number of edges. This paper focuses on the algorithmic question of finding good 2 colorings for hypergraphs. Unfortunately, the general problem of 2 coloring hypergraphs is equivalent to set splitting and thus known to be NP complete [10]. We instead find 2 colorings of hypergraphs restricted by size and degree. Let f(k) be the largest s such that all k hypergraphs with s edges can be 2 colored on line. Our first result matches the original lower bound of Erdos constructively, showing that f(k) 2 k Gamma1 by giving an on line ....

L. Lov'asz. Coverings and colorings of hypergraphs. In Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 3--11. Utilitas Mathematica Publishing, 1973.

....f6; 7; 8gg 5.3 Disjoint Set Protocols As a generalization of the above grid protocols, nondominated coteries and bicoteries can be constructed by using any collection of disjoint sets. 5.3. 1 Coteries An interesting method to construct k uniform strange hypergraphs was proposed by Lov asz [39]. This method can be used to construct nondominated coteries. Let U be a non empty set of nodes. Let V = fV 1 ; V 2 ; Delta Delta Delta ; V k g be a collection of pairwise disjoint subsets of U such that jV i j = i. Construct a quorum, G, 45 by selecting all elements in V i and one element ....

L. Lov'asz. Coverings and colorings of hypergraphs. In Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 3--12, 1973. 103

.... in C is entirely contained in either S 1 or S 2 Both problems, the Partition Problem as well as the Set Splitting Problem are NP complete (in the weak and in the strong sense, respectively) The latter problem remains NP complete in the strong sense even if all C 2 C have at most 3 elements (see [4, 9]) The Graph K Colourability Problem Instance Given an undirected graph G = V; E) and a positive integer K jV j Question Is G K colourable, i.e. does there exist a function f : V f1; 2; Kg such that f(u) 6= f(v) whenever fu; vg 2 E The Graph K Colourability Problem is solvable ....

....I is given. Let (S; C = fC 1 ; C k g) be an instance of the Set Splitting Problem. Consider that j C i j 3 for all C i 2 C. This can be done without loss of generality since the Set Splitting Problem remains NP complete in the strong sense even if all C 2 C have at most 3 elements (see [4, 9]) We will find the solution for this instance running the approximation algorithm A at most k = jCj times, beginning with I 1 = S; F = fC 1 g) as an initial instance. For I 1 , a feasible schedule of length 2 can be constructed in the following way. Schedule all tasks contained in C 1 at time ....

L. Lov'asz. Covering and coloring of hypergraphs. In Proc. 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 3--12. Utilitas Mathematica Publishing, Winnipeg, 1973.

.... context of intersecting set systems is [GB85] Alternative protocols based on quorum systems (rather than on voting) appear in [Mae85] using finite projective planes) AE91] the Tree system) CAA90] using a grid) and [Kum91, KC91, RT91] hierarchical systems) The triangular system is due to [Lov73, EL75] The Wheel system appears in [MP92] The CWlog system appears in [PW94] The motivation for several of these alternative systems was to find constructions with high availability and low load (which is referred to in most of these papers as quorum systems with small quorums) In [HMP93] ....

L. Lov'asz. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conf. Combinatorics,

....of dominated and non dominated coteries are proved. Alternative protocols based on quorum systems (rather than on voting) appear in [28] using finite projective planes) 1] the Tree system) 5, 25] using a grid) and [23, 24, 38] hierarchical systems) The triangular system is due to [26, 9]. The Wheel system appears in [29] The CWlog system appears in [37, 36] The motivation for several of these alternative systems was to find constructions with high availability and low load (which is referred to in most of these papers as quorum systems with small quorums) In [19] the question ....

L. Lov' asz, Coverings and colorings of hypergraphs, in Proc. 4th Southeastern Conf. Combinatorics, Graph Theory and Computing, 1973, pp. 3--12.

....system s total number of votes. The simple majority system is the most obvious voting system. Alternative protocols based on quorum systems appear in [Mae85] using finite projective planes) AE89, AE91] the Tree system) and [CAA90] using a grid construction) The triangular system is due to [Lov73, EL75]. The Wheel system appears in [MP92, PW93] The first paper to explicitly consider mutual exclusion protocols in the context of intersecting set systems is [GB85] In this work the term coterie and the concept of domination are introduced. Several basic properties of dominated and nondominated ....

....[HMP93] Figure 1: The crumbling wall CWh1; 5; 4; 4; 6; 5; 3; 4i, with one quorum shaded. 1. 3 New Results This paper introduces a new class of quorum system constructions, which we call Crumbling Walls (or simply walls) The crumbling walls are a generalization of the triangular construction of [Lov73, EL75], and of the Wheel [MP92, PW93] The elements are arranged in rows, and a quorum is the union of one full row and a representative from every row below the full row. However, unlike the triangular system, we do not require that row i have exactly i elements, and allow the wall to crumble at its ....

[Article contains additional citation context not shown here]

L. Lov'asz. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conf. Combinatorics, Graph Theory and Computing, pages 3-- 12, 1973.

....k Set Splitting problem, the instance consists of subsets of size k of a finite set of elements. The problem is to partition the elements into two parts, such that as many subsets as possible will be split, i.e. contain elements from both parts. The problem was shown to be NP complete by Lov asz [11]. Recently it was shown to be Apx complete [15] We also consider another generalization of Max 3 Set Splitting, namely Max NAE 3Sat, that was shown to be Apx complete in [14] First, we give an approximation algorithm for Max 3 Set Splitting using semi definite programming. Our algorithm is ....

L. Lov'asz. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, pages 3--12. Utilitas Mathematica Publishing, Winnipeg, 1973.

....is a collection of sets of elements, where every two sets in the collection intersect. The foundations of quorum systems have been laid in [4] and [11] 12] contains a good survey on the topic. A dozen ways have been proposed of how to construct quorum systems, starting in the early seventies [10, 3] and continuing until today [8, 13, 9, 14] 2 Distributed Counting: The Model Consider an asynchronous, distributed system of n processors in a message passing network, where each processor is uniquely identified with one of the integers from 1 to n. Each processor has unbounded local memory; ....

Lov' asz, L. Coverings and colorings of hypergraphs. In Proc. 4th Southwestern Conf. Combinatorics, Graph Theory and Computing (1973), pp. 3--12.

....a description could potentially cause a high blowup. 6.3 The Crumbling Wall System 6.3. 1 The System The Crumbling Walls (CW) are a family of quorum systems due to [PW94, PW95] This family includes, among others, the CWlog system (see Figure 6) the grid of [CAA92] and the triangular wall of [Lov73]. Figure 6: A CWlog with n = 49 elements and d = 15 rows, with one quorum shaded. The elements of a wall are logically arranged in rows of varying widths. A quorum in a wall is the union of one full row and a representative from every row below the full row. In the CWlog system, the width of row i ....

L. Lov'asz. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conf. Combinatorics, Graph Theory and Computing, pages 3--12, 1973.

....meets the bounds given in Corollary 4.2. The grid construction above is dominated: e.g. all of the column elements above the top row in each quorum can be removed (and the quorum system remains dominated) The grid construction would work for some non square grids as well, e.g. triangular grids [Lov73], or CWlog grids [PW95] that contain 2 i rows of length 2 i each) As the square grid already possesses a load within a constant factor of the optimal, we do not pursue such constructions further. k k Figure 1: Grid construction, k Theta k = n. Example 4.3 (Partition) Suppose that B = fB 1 ; ....

L. Lov'asz. Coverings and colorings of hypergraphs. In Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 3--12, 1973.

No context found.

LOV ASZ, L. 1973. Coverings and colorings in hypergraphs. In Proceedings of 4th South Eastern Conference on Combinatorics, Graph Theory and Computing. Utilitas Math., Winnipeg, B.C., Canada, 3--12.

No context found.

L. Lovasz, Covering and colorings of hypergraphs, Utilitas Math. (1973), 3-12.

No context found.

L. Lov asz, Coverings and coloring of hypergraphs, in Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1973.

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