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Searching in metric spaces
, 2001
"... The problem of searching the elements of a set that are close to a given query element under some similarity criterion has a vast number of applications in many branches of computer science, from pattern recognition to textual and multimedia information retrieval. We are interested in the rather gen ..."
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Cited by 432 (38 self)
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The problem of searching the elements of a set that are close to a given query element under some similarity criterion has a vast number of applications in many branches of computer science, from pattern recognition to textual and multimedia information retrieval. We are interested in the rather
The Maximum Clique Problem
, 1999
"... Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computation ..."
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Cited by 195 (21 self)
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Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .
Ramsey numbers of cubes versus cliques
"... The cube graph Qn is the skeleton of the ndimensional cube. It is an nregular graph on 2n vertices. The Ramsey number r(Qn,Ks) is the minimum N such that every graph of order N contains the cube graph Qn or an independent set of order s. Burr and Erdős in 1983 asked whether the simple lower bound ..."
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The cube graph Qn is the skeleton of the ndimensional cube. It is an nregular graph on 2n vertices. The Ramsey number r(Qn,Ks) is the minimum N such that every graph of order N contains the cube graph Qn or an independent set of order s. Burr and Erdős in 1983 asked whether the simple lower
Minimum Clique Number, Chromatic Number, and Ramsey Numbers
"... Let Q(n, c) denote the minimum clique number over graphs with n vertices and chromatic number c. We investigate the asymptotics of Q(n, c) when n/c is held constant. We show that when n/c is an integer α, Q(n, c) has the same growth order as the inverse function of the Ramsey number R(α + 1, t) (as ..."
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Cited by 1 (0 self)
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Let Q(n, c) denote the minimum clique number over graphs with n vertices and chromatic number c. We investigate the asymptotics of Q(n, c) when n/c is held constant. We show that when n/c is an integer α, Q(n, c) has the same growth order as the inverse function of the Ramsey number R(α + 1, t) (as
Hypergraph Ramsey Numbers: Triangles versus Cliques
, 2012
"... A celebrated result in Ramsey Theory states that the order of magnitude of the graph Ramsey numbers R(3, t) is t 2 / log t. In this paper, we consider an analogue of this problem for uniform hypergraphs. A triangle is a hypergraph consisting of edges e, f, g such that e ∩ f  = f ∩ g  = g ∩ e  ..."
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Cited by 4 (2 self)
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A celebrated result in Ramsey Theory states that the order of magnitude of the graph Ramsey numbers R(3, t) is t 2 / log t. In this paper, we consider an analogue of this problem for uniform hypergraphs. A triangle is a hypergraph consisting of edges e, f, g such that e ∩ f  = f ∩ g  = g ∩ e
What is Ramseyequivalent to a clique?
, 2013
"... A graph G is Ramsey for H if every twocolouring of the edges of G contains a monochromatic copy of H. Two graphs H and H ′ are Ramseyequivalent if every graph G is Ramsey for H if and only if it is Ramsey for H ′. In this paper, we study the problem of determining which graphs are Ramseyequivalen ..."
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A graph G is Ramsey for H if every twocolouring of the edges of G contains a monochromatic copy of H. Two graphs H and H ′ are Ramseyequivalent if every graph G is Ramsey for H if and only if it is Ramsey for H ′. In this paper, we study the problem of determining which graphs are Ramsey
Tractable inference for complex stochastic processes
 In Proc. UAI
, 1998
"... The monitoring and control of any dynamic system depends crucially on the ability to reason about its current status and its future trajectory. In the case of a stochastic system, these tasks typically involve the use of a belief state—a probability distribution over the state of the process at a gi ..."
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Cited by 306 (15 self)
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The monitoring and control of any dynamic system depends crucially on the ability to reason about its current status and its future trajectory. In the case of a stochastic system, these tasks typically involve the use of a belief state—a probability distribution over the state of the process at a given point in time. Unfortunately, the state spaces of complex processes are very large, making an explicit representation of a belief state intractable. Even in dynamic Bayesian networks (DBNs), where the process itself can be represented compactly, the representation of the belief state is intractable. We investigate the idea of maintaining a compact approximation to the true belief state, and analyze the conditions under which the errors due to the approximations taken over the lifetime of the process do not accumulate to make our answers completely irrelevant. We show that the error in a belief state contracts exponentially as the process evolves. Thus, even with multiple approximations, the error in our process remains bounded indefinitely. We show how the additional structure of a DBN can be used to design our approximation scheme, improving its performance significantly. We demonstrate the applicability of our ideas in the context of a monitoring task, showing that orders of magnitude faster inference can be achieved with only a small degradation in accuracy. 1
Hypergraph Ramsey numbers: tight cycles versus cliques
, 2015
"... For s ≥ 4, the 3uniform tight cycle C3s has vertex set corresponding to s distinct points on a circle and edge set given by the s cyclic intervals of three consecutive points. For fixed s ≥ 4 and s 6 ≡ 0 (mod 3) we prove that there are positive constants a and b with 2at < r(C3s,K 3 t) < 2 bt ..."
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For s ≥ 4, the 3uniform tight cycle C3s has vertex set corresponding to s distinct points on a circle and edge set given by the s cyclic intervals of three consecutive points. For fixed s ≥ 4 and s 6 ≡ 0 (mod 3) we prove that there are positive constants a and b with 2at < r(C3s,K 3 t) < 2 bt2 log t. The lower bound is obtained via a probabilistic construction. The upper bound for s> 5 is proved by using supersaturation and the known upper bound for r(K34,K 3 t), while for s = 5 it follows from a new upper bound for r(K3−5,K 3 t) that we develop. 1
Results 11  20
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