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22
On Contention Resolution Protocols and Associated Probabilistic Phenomena
 IN PROCEEDINGS OF THE 26TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1994
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Connected Domination and Spanning Trees with Many Leaves
 SIAM J. Discrete Math
, 2000
"... Abstract Let G = (V; E) be a connected graph. A connected dominating set S ae V is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted fl ..."
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Cited by 15 (2 self)
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Abstract Let G = (V; E) be a connected graph. A connected dominating set S ae V is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted fl
Signed domination in regular graphs and setsystems
 J. Combin. Theory Ser. B
, 1999
"... Abstract. Suppose G is a graph on n vertices with minimum degree r. Using standard random methods it is shown that there exists a twocoloring of the vertices of G with colors, +1 and −1, such that all closed neighborhoods contain more 1’s than −1’s, and altogether the number of 1’s does not exceed ..."
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Cited by 8 (0 self)
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Abstract. Suppose G is a graph on n vertices with minimum degree r. Using standard random methods it is shown that there exists a twocoloring of the vertices of G with colors, +1 and −1, such that all closed neighborhoods contain more 1’s than −1’s, and altogether the number of 1’s does not exceed the number of −1’s by more than (4 p log r/r + 1/r)n. For large r this greatly improves earlier results and is almost optimal, since starting with an Hadamard matrix of order r, a bipartite rregular graph is constructed on 4r vertices with signed domination number at least (1/2) √ r − O(1). The determination of limn→ ∞ γs(G)/n remains open, and is conjectured to be Θ(1 / √ r). 1. Discrepancy and domination of hypergraphs Discrepancy theory has originated from number theory and in the last few decades it has developed into an elaborate field related also to geometry, probability theory, ergodic theory, computer science, and combinatorics. The combinatorial setting of these problems proved to be a successful approach. See the monograph of Beck and Chen [4], the chapter from the Handbook of Combinatorics [6], or [17]. One of the basic problems in combinatorial discrepancy theory is the following: Suppose H is a hypergraph with vertex set S and edge set {A1,..., Am}. Our object is to color the elements of S by 2 colors such that all of the edges have almost the same number of elements in each color. A partition of S can be given by a function
Online and offline approximation algorithms for vector covering problems
 IN PROC. 4TH EUROPEAN SYMPOSIUM ON ALGORITHMS, LNCS
, 1998
"... This paper deals with vector covering problems in ddimensional space. The input to a vector covering problem consists of a set X of ddimensional vectors in [0, 1] d. The goal is to partition X into a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at ..."
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Cited by 7 (3 self)
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This paper deals with vector covering problems in ddimensional space. The input to a vector covering problem consists of a set X of ddimensional vectors in [0, 1] d. The goal is to partition X into a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NPcomplete, and we are mainly interested in its online and offline approximability. For the online version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d ≥ 2 dimensions. This result contradicts a statement of Csirik and Frenk (1990) in [5] where it is claimed that for d ≥ 2, no online algorithm can have a worst case ratio better than zero. Moreover, we prove that for d ≥ 2, no online algorithm can have worst case ratio better than 2/(2d + 1). For the offline version, we derive polynomial time approximation algorithms with worst case guarantee Θ(1 / log d). For d = 2, we present a very fast and very simple offline approximation algorithm that has worst case ratio 1/2. Moreover, we show that a method from the area of compact vector summation can be used to construct offline approximation algorithms with worst case ratio 1/d for every d ≥ 2.
Spanning directed trees with many leaves
 SIAM J. Discrete Math
"... Abstract. The Directed Maximum Leaf OutBranching problem is to find an outbranching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in outbranchings. We show that – every strong ..."
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Cited by 6 (4 self)
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Abstract. The Directed Maximum Leaf OutBranching problem is to find an outbranching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in outbranchings. We show that – every strongly connected nvertex digraph D with minimum indegree at least 3 has an outbranching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an outbranching with k leaves, then the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an outbranching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1). 1
A Survey on Packing and Covering Problems in the Hamming Permutation Space
"... Consider the symmetric group Sn equipped with the Hamming metric dH. Packing and covering problems in the finite metric space (Sn,dH) are surveyed, including a combination of both. 1 ..."
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Consider the symmetric group Sn equipped with the Hamming metric dH. Packing and covering problems in the finite metric space (Sn,dH) are surveyed, including a combination of both. 1
Dominating a Family of Graphs With Small Connected Subgraphs
"... Let F = fG 1 ; : : : ; G t g be a family of nvertex graphs defined on the same vertexset V , and let k be a positive integer. A subset of vertices D ae V is called an (F; k)core if for each v 2 V and for each i = 1; : : : ; t, there are at least k neighbors of v in G i which belong to D. The s ..."
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Cited by 2 (1 self)
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Let F = fG 1 ; : : : ; G t g be a family of nvertex graphs defined on the same vertexset V , and let k be a positive integer. A subset of vertices D ae V is called an (F; k)core if for each v 2 V and for each i = 1; : : : ; t, there are at least k neighbors of v in G i which belong to D. The subset D is called a connected (F; k)core, if the subgraph induced by D in each G i is connected. Let ffi i be the minimum degree of G i and let ffi (F ) = min t i=1 ffi i . Clearly, an (F; k)core exists if and only if ffi (F ) k, and a connected (F; k)core exists if and only if ffi (F ) k and each G i is connected. Let c(k; F ) and c c (k; F ) be the minimum size of an (F; k)core and a connected (F; k)core, respectively. The following asymptotic results are proved for every t ! ln ln ffi and k ! p ln ffi : c(k; F ) n ln ffi ffi (1 + o ffi (1)) c c (k; F ) n ln ffi ffi (1 + o ffi (1)): The results are asymptotically tight for infinitely many families F . The results ...
Economical Covers With Geometric Applications
, 1999
"... A cover of a hypergraph is a collection of edges whose union contains all vertices. Let H = (V, E) be a kuniform, Dregular hypergraph on n vertices, in which no two vertices are contained in more than o(D/e 2k log D) edges as D tends to infinity. Our results include that if k = o(log D), then t ..."
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Cited by 1 (1 self)
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A cover of a hypergraph is a collection of edges whose union contains all vertices. Let H = (V, E) be a kuniform, Dregular hypergraph on n vertices, in which no two vertices are contained in more than o(D/e 2k log D) edges as D tends to infinity. Our results include that if k = o(log D), then there is a cover of (1 + o(1))n/k edges, extending the known result that this holds for fixed k. On the other hand, if k # 4 log D then there are kuniform, Dregular hypergraphs on n vertices in which no two vertices are contained in more than one edge, and yet the smallest cover has at least ## n k log( k log D )) edges. Several extensions and variants are also obtained, as well as the following geometric application. The minimum number of lines required to separate n random points in the unit square is, almost surely, #(n 2/3 /(log n) 1/3 )). # Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Research suppor...
Rainbow edgecoloring and rainbow domination
, 2012
"... Let G be an edgecolored graph with n vertices. A rainbow subgraph is a subgraph whose edges have distinct colors. The rainbow edgechromatic number of G, written ˆχ ′(G), is the minimum number of rainbow matchings needed to cover E(G). An edgecolored graph is ttolerant if it contains no monochroma ..."
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Let G be an edgecolored graph with n vertices. A rainbow subgraph is a subgraph whose edges have distinct colors. The rainbow edgechromatic number of G, written ˆχ ′(G), is the minimum number of rainbow matchings needed to cover E(G). An edgecolored graph is ttolerant if it contains no monochromatic star with t+1 edges. If G is ttolerant, then ˆχ ′(G) < t(t + 1)n ln n, and examples exist with ˆχ ′(G) ≥ t 2 (n − 1). The rainbow domination number, written ˆγ(G), is the minimum number of disjoint rainbow stars needed to cover V (G). For ttolerant edgecolored nvertex graphs, we generalize 1+ln k δ(G) classical bounds on the domination number: (1) ˆγ(G) ≤ k n (where k =