Results 11  20
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166
On Markov chains for independent sets
 Journal of Algorithms
, 1997
"... Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. A new rapidly mixing Markov chain for independent sets is defined in this paper. We show that it is rapidly mixing for a wider range of values of the parameter than the LubyVigoda chain, ..."
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Cited by 66 (16 self)
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Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. A new rapidly mixing Markov chain for independent sets is defined in this paper. We show that it is rapidly mixing for a wider range of values of the parameter than the LubyVigoda chain, the best previously known. Moreover the new chain is apparently more rapidly mixing than the LubyVigoda chain for larger values of (unless the maximum degree of the graph is 4). An extension of the chain to independent sets in hypergraphs is described. This chain gives an efficient method for approximately counting the number of independent sets of hypergraphs with maximum degree two, or with maximum degree three and maximum edge size three. Finally, we describe a method which allows one, under certain circumstances, to deduce the rapid mixing of one Markov chain from the rapid mixing of another, with the same state space and stationary distribution. This method is applied to two Markov ch...
Sampling Plausible Solutions to Multibody Constraint Problems
, 2000
"... Traditional collision intensive multibody simulations are difficult to control due to extreme sensitivity to initial conditions or model parameters. Furthermore, there may be multiple ways to achieve any one goal, and it may be difficult to codify a user's preferences before they have seen the avai ..."
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Cited by 60 (2 self)
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Traditional collision intensive multibody simulations are difficult to control due to extreme sensitivity to initial conditions or model parameters. Furthermore, there may be multiple ways to achieve any one goal, and it may be difficult to codify a user's preferences before they have seen the available solutions. In this paper we extend simulation models to include plausible sources of uncertainty, and then use a Markov chain Monte Carlo algorithm to sample multiple animations that satisfy constraints. A user can choose the animation they prefer, or applications can take direct advantage of the multiple solutions. Our technique is applicable when a probability can be attached to each animation, with "good" animations having high probability, and for such cases we provide a definition of physical plausibility for animations. We demonstrate our approach with examples of multibody rigidbody simulations that satisfy constraints of various kinds, for each case presenting animations that are true to a physical model, are significantly different from each other, and yet still satisfy the constraints. CR Descriptors: I.3.7 [Computer Graphics]: ThreeDimensional Graphics and Realism  Animation; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling  Physically based modeling; I.6.5 [Simulation and Modeling]: Model Development  Modeling methodologies G.3 [Probability and Statistics]: Probabilistic algorithms; Keywords: plausible motion, Markov chain Monte Carlo, motion synthesis, spacetime constraints 1
A Combinatorial Auction with Multiple Winners for Universal Service
, 1998
"... We describe a discretetime auction procedure called PAUSE (Progressive Adaptive User Selection Environment) for use in assigning COLR (Carrier of Last Resort) responsibility for Universal Service. The auction incorporates synergies by permitting all combinatorial bids, allows for multiple winners, ..."
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Cited by 59 (0 self)
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We describe a discretetime auction procedure called PAUSE (Progressive Adaptive User Selection Environment) for use in assigning COLR (Carrier of Last Resort) responsibility for Universal Service. The auction incorporates synergies by permitting all combinatorial bids, allows for multiple winners, and minimizes the possibility of bidder collusion. The procedure is computationally manageable for the auctioneer and thus is very efficient to run. The inherent computational complexity of combinatorial bidding cannot be eliminated. However, in this auction the computational burden of evaluating synergies rests with the bidders claiming those synergies, while the auctioneer simply checks that a bid is valid.
On Counting Independent Sets in Sparse Graphs
, 1998
"... We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if ..."
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Cited by 58 (11 self)
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We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if \Delta 25, unless RP = NP. 1 Introduction Counting independent sets in graphs is one of several combinatorial counting problems which have received recent attention. The problem is known to be #Pcomplete, even for low degree graphs [3]. On the other hand, it has been shown that, for graphs of maximum degree \Delta = 4, randomized approximate counting is possible [7, 3]. This success has been achieved using the Monte Carlo Markov chain method to construct a fully polynomial randomized approximation scheme (fpras). This has led to a natural question as to how far this success might extend. Here we consider in more detail this question of counting independent sets in graphs with constant m...
Toward CaseBased Preference Elicitation: Similarity Measures on Preference Structures
 In Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence
, 1998
"... While decision theory provides an appealing normative framework for representing rich preference structures, eliciting utility or value functions typically incurs a large cost. For many applications involving interactive systems this overhead precludes the use of formal decisiontheoretic models of ..."
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Cited by 48 (6 self)
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While decision theory provides an appealing normative framework for representing rich preference structures, eliciting utility or value functions typically incurs a large cost. For many applications involving interactive systems this overhead precludes the use of formal decisiontheoretic models of preference. Instead of performing elicitation in a vacuum, it would be useful if we could augment directly elicited preferences with some appropriate default information. In this paper we propose a casebased approach to alleviating the preference elicitation bottleneck. Assuming the existence of a population of users from whom we have elicited complete or incomplete preference structures, we propose eliciting the preferences of a new user interactively and incrementally, using the closest existing preference structures as potential defaults. Since a notion of closeness demands a measure of distance among preference structures, this paper takes the first step of studying various distance mea...
Random Walks on Truncated Cubes and Sampling 01 Knapsack Solutions
 in Proc. 40th IEEE Symp. on Foundations of Computer Science
, 2002
"... We solve an open problem concerning the mixing time of symmetric random walk on the n dimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a fullypolynomial randomized approximation scheme for counting the feasible solutions of a 01 knapsa ..."
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Cited by 46 (1 self)
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We solve an open problem concerning the mixing time of symmetric random walk on the n dimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a fullypolynomial randomized approximation scheme for counting the feasible solutions of a 01 knapsack problem. The results extend to the case of any xed number of hyperplanes.
Adiabatic quantum state generation and statistical zeroknowledge
 in Proc. 35th STOC
, 2003
"... The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem. We systematically study ’quantum state generation’, namely, which superpositions can be efficiently generated. We first show that all problems in Statistical Z ..."
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Cited by 42 (3 self)
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The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem. We systematically study ’quantum state generation’, namely, which superpositions can be efficiently generated. We first show that all problems in Statistical Zero Knowledge (SZK), a class which contains many languages that are natural candidates for BQP, can be reduced to an instance of quantum state generation. This was known before for graph isomorphism, but we give a general recipe for all problems in SZK. We demonstrate the reduction from the problem to its quantum state generation version for three examples: Discrete log, quadratic residuosity and a gap version of closest vector in a lattice. We then develop tools for quantum state generation. For this task, we define the framework of ’adiabatic quantum state generation ’ which uses the language of ground states, spectral gaps and Hamiltonians instead of the standard unitary gate language. This language stems from the recently suggested adiabatic computation model [20] and seems to be especially tailored for the task of quantum state generation. After defining the paradigm, we provide two basic lemmas for adiabatic quantum state generation: • The Sparse Hamiltonian lemma, which gives a general technique for implementing sparse Hamiltonians efficiently, and, • The jagged adiabatic path lemma, which gives conditions for a sequence of Hamiltonians to allow efficient adiabatic state generation. We use our tools to prove that any quantum state which can be generated efficiently in the standard model can also be generated efficiently adiabatically, and vice versa. Finally we show how to apply our techniques to generate superpositions corresponding to limiting distributions of a large class of Markov chains, including the uniform distribution over all perfect
The SwendsenWang process does not always mix rapidly
 Proc. 29th ACM Symp. on Theory of Computing
, 1997
"... The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The l ..."
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Cited by 40 (3 self)
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The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The legitimacy of such simulations depends on the rate of convergence of the process to equilibrium, often known as the mixing rate. Empirical observations suggest that the SwendsenWang process mixes rapidly in many instances of practical interest. In spite of this, we show that there are occasions on which the SwendsenWang process requires exponential time (in the size of the system) to approach equilibrium.
A more rapidly mixing Markov chain for graph colourings
, 1997
"... We define a new Markov chain on (proper) kcolourings of graphs, and relate its convergence properties to the maximum degree \Delta of the graph. The chain is shown to have bounds on convergence time appreciably better than those for the wellknown Jerrum/SalasSokal chain in most circumstances. For ..."
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Cited by 39 (11 self)
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We define a new Markov chain on (proper) kcolourings of graphs, and relate its convergence properties to the maximum degree \Delta of the graph. The chain is shown to have bounds on convergence time appreciably better than those for the wellknown Jerrum/SalasSokal chain in most circumstances. For the case k = 2\Delta, we provide a dramatic decrease in running time. We also show improvements whenever the graph is regular, or fewer than 3\Delta colours are used. The results are established using the method of path coupling. We indicate that our analysis is tight by showing that the couplings used are optimal in a sense which we define. 1 Introduction Markov chains on the set of proper colourings of graphs have been studied in computer science [9] and statistical physics [13]. In both applications, the rapidity of convergence of the chain is the main focus of interest, though for somewhat different reasons. The papers [9, 13] introduced a simple Markov chain, which we shall refer to a...
On the Relative Complexity of Approximate Counting Problems
, 2000
"... Two natural classes of counting problems that are interreducible under approximationpreserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reducibili ..."
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Cited by 34 (12 self)
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Two natural classes of counting problems that are interreducible under approximationpreserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically dened subclass of #P. Research Report 370, Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK. This work was supported in part by the EPSRC Research Grant \Sharper Analysis of Randomised Algorithms: a Computational Approach" and by the ESPRIT Projects RANDAPX and ALCOMFT. y dyer@scs.leeds.ac.uk, School of Computer Studies, University of Leeds, Leeds LS2 9JT, United Kingdom. z leslie@dcs.warwick.ac.uk, http://www.dcs.warw...