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47
Simple Constructions of Almost kwise Independent Random Variables
, 1992
"... We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the dist ..."
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Cited by 264 (40 self)
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We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ɛ < 1/(k log n)). An additional advantage of our constructions is their simplicity.
SmallBias Probability Spaces: Efficient Constructions and Applications
 SIAM J. Comput
, 1993
"... We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called fflbiased random variables. The number of random bits needed to generate the random variables is ..."
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Cited by 253 (14 self)
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We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called fflbiased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are fflbiased can be used to construct "almost" kwise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using fflbiased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...
On the complexity of the parity argument and other inefficient proofs of existence
 JCSS
, 1994
"... We define several new complexity classes of search problems, "between " the classes FP and FNP. These new classes are contained, along with factoring, and the class PLS, in the class TFNP of search problems in FNP that always have a witness. A problem in each of these new classes is defined in terms ..."
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Cited by 153 (8 self)
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We define several new complexity classes of search problems, "between " the classes FP and FNP. These new classes are contained, along with factoring, and the class PLS, in the class TFNP of search problems in FNP that always have a witness. A problem in each of these new classes is defined in terms of an implicitly given, exponentially large graph. The existence of the solution sought is established via a simple graphtheoretic argument with an inefficiently constructive proof; for example, PLS can be thought of as corresponding to the lemma "every dag has a sink. " The new classes are based on lemmata such as "every graph has an even number of odddegree nodes. " They contain several important problems for which no polynomial time algorithm is presently known, including the computational versions of Sperner's lemma, Brouwer's fixpoint theorem, Chfvalley's theorem, and the BorsukUlam theorem, the linear complementarity problem for Pmatrices, finding a mixed equilibrium in a nonzero sum game, finding a second Hamilton circuit in a Hamiltonian cubic graph, a second Hamiltonian decomposition in a quartic graph, and others. Some of these problems are shown to be complete. © 1994 Academic Press, Inc. 1.
Packet routing and jobshop scheduling in O(congestion+dilation) steps
 Combinatorica
, 1994
"... In this paper, we prove that there exists a schedule for routing any set of packets with edgesimple paths, on any network, in O(c+d) steps, where c is the congestion of the paths in the network, and d is the length of the longest path. The result has applications to packet routing in parallel machi ..."
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Cited by 102 (8 self)
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In this paper, we prove that there exists a schedule for routing any set of packets with edgesimple paths, on any network, in O(c+d) steps, where c is the congestion of the paths in the network, and d is the length of the longest path. The result has applications to packet routing in parallel machines, network emulations, and jobshop scheduling.
The algorithmic aspects of the Regularity Lemma
 J. Algorithms
, 1994
"... The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that ..."
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Cited by 93 (31 self)
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The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is coNPcomplete. However, we also prove that despite this difficulty the lemma can be made constructive; we show how to obtain, for any input graph, a partition with the properties guaranteed by the lemma, efficiently. The desired partition, for an nvertex graph, can be found in time O(M(n)), where M(n) = O(n 2.376) is the time needed to multiply two n by n matrices with 0, 1entries over the integers. The algorithm can be parallelized and implemented in NC 1. Besides the curious phenomenon of exhibiting a natural problem in which the search for a solution is easy whereas the decision if a given instance is a solution is difficult (if P and NP differ), our constructive version of the Regularity Lemma supplies efficient sequential and parallel algorithms for many problems, some of which are naturally motivated by the study of various graph embedding and graph coloring problems.
Probabilistic Methods in Combinatorics
, 1974
"... Computer Science these questions take on an algorithmic tone, having proven the existence of a graph or other structure can it be constructed in polynomial time. A recent success of J. Beck allows the Lov'asz Local Lemma to be derandomized. Sometimes. We close with two forays into a land dubbed Asym ..."
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Cited by 76 (2 self)
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Computer Science these questions take on an algorithmic tone, having proven the existence of a graph or other structure can it be constructed in polynomial time. A recent success of J. Beck allows the Lov'asz Local Lemma to be derandomized. Sometimes. We close with two forays into a land dubbed Asymptopia by David Aldous. There the asymptotic behavior of random objects are given by an infinite object, allowing powerful noncombinatorial tools to be used. 1 Chernoff, Azuma, Janson, Talagrand Let X = X 1 + . . . +Xm with the X i mutually independent and normalized so that E[X] = E[X i ] = 0. The socalled Chernoff bounds (Bernstein or antiquity might be more accurate attributions) bound the "large deviation" Pr[X ? a] ! e ] = e ] (See, e.g., the appendix of [2].) The power in the inequality is that it holds for all ? 0 and one chooses = (a) for optimal results. Suppose, for example, that jX i j 1. One can show E[e ] cosh() exp( =2), the extreme case when X i =
Percolation on finite graphs and isoperimetric inequalities
, 2002
"... Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c> 0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least cGn, with probability going to on ..."
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Cited by 45 (10 self)
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Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c> 0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least cGn, with probability going to one, uniformly in p. The method from Ajtai, Komlós and Szemerédi [Combinatorica 2 (1982) 1–7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented. 1. Introduction. In
A constructive proof of the general Lovász Local Lemma, preprint
"... The Lovász Local Lemma [EL75] is a powerful tool to nonconstructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Si ..."
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Cited by 33 (1 self)
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The Lovász Local Lemma [EL75] is a powerful tool to nonconstructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Simplifications of his procedure and relaxations of its restrictions were subsequently exhibited in several publications [Alo91, MR98, CS00, Mos06, Sri08, Mos08a]. In [Mos08b], a constructive proof was presented that works under negligible restrictions, formulated in terms of the Bounded Occurrence Satisfiability problem. In the present paper, we reformulate and improve upon these findings so as to directly apply to almost all known applications of the general Local Lemma. Key Words and Phrases. Lovász Local Lemma, constructive proof, parallelization. 1
Improved bounds and algorithms for hypergraph twocoloring
, 1998
"... We show that for all large n, every nuniform hypergraph with at most 0:7pn = ln n \Theta 2n edges can be 2colored. This makes progress on a problem of Erd""os (1963), improving the previousbest bound of n1=3 \Gamma o(1) \Theta 2n due to Beck (1978). We further generalize this to a ..."
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Cited by 31 (0 self)
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We show that for all large n, every nuniform hypergraph with at most 0:7pn = ln n \Theta 2n edges can be 2colored. This makes progress on a problem of Erd""os (1963), improving the previousbest bound of n1=3 \Gamma o(1) \Theta 2n due to Beck (1978). We further generalize this to a
Lifts, Discrepancy and Nearly Optimal Spectral Gaps
"... Let G be a graph on n vertices. A 2lift of G is a graph H on 2n vertices, with a covering map : H ! G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n \new" eigenvalues. We conjecture that every dregular graph has a 2lift such that all new eige ..."
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Cited by 30 (4 self)
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Let G be a graph on n vertices. A 2lift of G is a graph H on 2n vertices, with a covering map : H ! G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n \new" eigenvalues. We conjecture that every dregular graph has a 2lift such that all new eigenvalues are in the range [ 2 d 1; 2 d 1] (If true, this is tight , e.g. by the AlonBoppana bound). Here we show that every graph of maximal degree d has a 2lift such that all \new" eigenvalues are in the range [ c d; c d] for some constant c.