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196
AND/OR Search Spaces for Graphical Models
, 2004
"... The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the gr ..."
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Cited by 119 (44 self)
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The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the graphical model explicitly and may sometime reduce the search space exponentially. Indeed, most
Decentralised Coordination of Low-Power Embedded Devices Using the Max-Sum Algorithm
- In: 7 th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS-08
, 2008
"... This paper considers the problem of performing decentralised coordination of low-power embedded devices (as is required within many environmental sensing and surveillance applications). Specifically, we address the generic problem of maximising social welfare within a group of interacting agents. We ..."
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Cited by 96 (30 self)
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This paper considers the problem of performing decentralised coordination of low-power embedded devices (as is required within many environmental sensing and surveillance applications). Specifically, we address the generic problem of maximising social welfare within a group of interacting agents. We propose a novel representation of the problem, as a cyclic bipartite factor graph, composed of variable and function nodes (representing the agents’ states and utilities respectively). We show that such representation allows us to use an extension of the max-sum algorithm to generate approximate solutions to this global optimisation problem through local decentralised message passing. We empirically evaluate this approach on a canonical coordination problem (graph colouring), and benchmark it against state of the art approximate and complete algorithms (DSA and DPOP). We show that our approach is robust to lossy communication, that it generates solutions closer to those of DPOP than DSA is able to, and that it does so with a communication cost (in terms of total messages size) that scales very well with the number of agents in the system (compared to the exponential increase of DPOP). Finally, we describe a hardware implementation of our algorithm operating on low-power Chipcon CC2431 System-on-Chip sensor nodes.
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
, 2002
"... Consider the following simple, greedy Davis-Putnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occu ..."
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Cited by 76 (6 self)
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Consider the following simple, greedy Davis-Putnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42.
A comparison of algorithms for inference and learning in probabilistic graphical models
- IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2005
"... Computer vision is currently one of the most exciting areas of artificial intelligence re-search, largely because it has recently become possible to record, store and process large amounts of visual data. While impressive achievements have been made in pattern clas-sification problems such as handwr ..."
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Cited by 70 (4 self)
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Computer vision is currently one of the most exciting areas of artificial intelligence re-search, largely because it has recently become possible to record, store and process large amounts of visual data. While impressive achievements have been made in pattern clas-sification problems such as handwritten character recognition and face detection, it is even more exciting that researchers may be on the verge of introducing computer vision systems that perform scene analysis, decomposing image input into its constituent objects, lighting conditions, motion patterns, and so on. Two of the main challenges in computer vision are finding efficient models of the physics of visual scenes and finding efficient algorithms for inference and learning in these models. In this paper, we advocate the use of graph-based probability models and their associated inference and learning algorithms for computer vision and scene analysis. We review exact techniques and various approximate, computationally efficient techniques, including iterative conditional modes, the expectation maximization (EM) algorithm, the mean field method, variational techniques, structured variational techniques, Gibbs sampling, the sum-product algorithm and “loopy ” belief propagation. We describe how each technique can be applied in a model of multiple, occluding objects, and contrast the behaviors and performances of the techniques using a unifying cost function, free energy.
A New Look at Survey Propagation and its Generalizations
"... We study the survey propagation algorithm [19, 5, 4], which is an iterative technique that appears to be very effective in solving random k-SAT problems even with densities close to threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), ..."
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Cited by 66 (11 self)
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We study the survey propagation algorithm [19, 5, 4], which is an iterative technique that appears to be very effective in solving random k-SAT problems even with densities close to threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), parameterized by a real number ρ. We then show that applying belief propagation— a well-known “message-passing” technique—to this family of MRFs recovers various algorithms, ranging from pure survey propagation at one extreme (ρ = 1) to standard belief propagation on the uniform distribution over SAT assignments at the other extreme (ρ = 0). Configurations in these MRFs have a natural interpretation as generalized satisfiability assignments, on which a partial order can be defined. We isolate cores as minimal elements in this partial
Maximum weight matching via max-product belief propagation
- in International Symposium of Information Theory
, 2005
"... Abstract — The max-product “belief propagation ” algorithm is an iterative, local, message passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many applicati ..."
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Cited by 63 (12 self)
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Abstract — The max-product “belief propagation ” algorithm is an iterative, local, message passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding and computer vision which involve graphs with many cycles, theoretical convergence results are only known for graphs which are tree-like or have a single cycle. In this paper, we consider a weighted complete bipartite graph and define a probability distribution on it whose MAP assignment corresponds to the maximum weight matching (MWM) in that graph. We analyze the fixed points of the max-product algorithm when run on this graph and prove the surprising result that even though the underlying graph has many short cycles, the maxproduct assignment converges to the correct MAP assignment. We also provide a bound on the number of iterations required by the algorithm. I.
Max-product for maximum weight matching: convergence, correctness and LP duality
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Algorithmic barriers from phase transitions. preprint
"... For many random Constraint Satisfaction Problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomialtime algorithms stop finding solutions at much smaller densities. For exa ..."
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Cited by 51 (4 self)
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For many random Constraint Satisfaction Problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomialtime algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 − ɛ)χ colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of {1,..., k} n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ≥ 2χ, but like an error-correcting code for k ≤ (2 − ɛ)χ. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step Replica-Symmetry-Breaking hypothesis of statistical physics for random CSPs.
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worst-case exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 50 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worst-case exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,
Towards efficient sampling: Exploiting random walk strategies
- In Proceedings of the AAAI Conference on Artificial Intelligence
, 2004
"... From a computational perspective, there is a close connec-tion between various probabilistic reasoning tasks and the problem of counting or sampling satisfying assignments of a propositional theory. We consider the question of whether state-of-the-art satisfiability procedures, based on random walk ..."
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Cited by 49 (6 self)
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From a computational perspective, there is a close connec-tion between various probabilistic reasoning tasks and the problem of counting or sampling satisfying assignments of a propositional theory. We consider the question of whether state-of-the-art satisfiability procedures, based on random walk strategies, can be used to sample uniformly or near-uniformly from the space of satisfying assignments. We first show that random walk SAT procedures often do reach the full set of solutions of complex logical theories. Moreover, by interleaving random walk steps with Metropolis transitions, we also show how the sampling becomes near-uniform.
