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197
A New Class of Upper Bounds on the Log Partition Function
 In Uncertainty in Artificial Intelligence
, 2002
"... Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis [11, 5, 4]. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distribution ..."
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Cited by 225 (32 self)
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Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis [11, 5, 4]. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distributions in the exponential domain, that is applicable to an arbitrary undirected graphical model. In the special case of convex combinations of treestructured distributions, we obtain a family of variational problems, similar to the Bethe free energy, but distinguished by the following desirable properties: (i) they are convex, and have a unique global minimum; and (ii) the global minimum gives an upper bound on the log partition function. The global minimum is defined by stationary conditions very similar to those defining xed points of belief propagation (BP) or treebased reparameterization [see 13, 14]. As with BP fixed points, the elements of the minimizing argument can be used as approximations to the marginals of the original model. The analysis described here can be extended to structures of higher treewidth (e.g., hypertrees), thereby making connections with more advanced approximations (e.g., Kikuchi and variants [15, 10]).
Topologically Sweeping Visibility Complexes via Pseudotriangulations
, 1996
"... This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal run ..."
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Cited by 78 (8 self)
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This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using redblack trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of th...
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 kSAT 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 kSAT 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 wellknown “messagepassing” 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
Tree Consistency and Bounds on the Performance of the MaxProduct Algorithm and Its Generalizations
, 2002
"... Finding the maximum a posteriori (MAP) assignment of a discretestate distribution specified by a graphical model requires solving an integer program. The maxproduct algorithm, also known as the maxplus or minsum algorithm, is an iterative method for (approximately) solving such a problem on gr ..."
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Cited by 65 (5 self)
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Finding the maximum a posteriori (MAP) assignment of a discretestate distribution specified by a graphical model requires solving an integer program. The maxproduct algorithm, also known as the maxplus or minsum algorithm, is an iterative method for (approximately) solving such a problem on graphs with cycles.
Enumeration of totally positive Grassmann cells
, 2005
"... (Article begins on next page) The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. ..."
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Cited by 63 (9 self)
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(Article begins on next page) The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters.
Faces of generalized permutohedra
, 2007
"... The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f, h and γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formul ..."
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Cited by 49 (3 self)
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The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f, h and γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for hvectors and γvectors involving descent statistics. This includes a combinatorial interpretation for γvectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal’s conjecture on nonnegativity of γvectors. We calculate explicit generating functions and formulae for hpolynomials of various families of graphassociahedra, including those corresponding to all Dynkin diagrams of finite and affine types. We also discuss relations with Narayana numbers and with Simon Newcomb’s problem. We give (and conjecture) upper and lower bounds for f, h, and γvectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deformations of simple polytopes.
A characterization of the quadrilateral meshes of a surface which admit a compatible hexahedral mesh of the enclosed volume
 Proceedings of the 13th Annual Symposium on Theoretical Aspects of Computer Science
, 1996
"... Abstract. A popular threedimensional mesh generation scheme is to start with a quadrilateral mesh of the surface of a volume, and then attempt to fill the interior of the volume with hexahedra, so that the hexahedra touch the surface in exactly the given quadrilaterals[24]. Folklore has maintained ..."
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Cited by 45 (9 self)
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Abstract. A popular threedimensional mesh generation scheme is to start with a quadrilateral mesh of the surface of a volume, and then attempt to fill the interior of the volume with hexahedra, so that the hexahedra touch the surface in exactly the given quadrilaterals[24]. Folklore has maintained that there are many quadrilateral meshes for which no such compatible hexahedral mesh exists. In this paper we give an existence proof which contradicts this folklore: A quadrilateral mesh need only satisfy some very weak conditions for there to exist a compatible hexahedral mesh. For a volume that is topologically a ball, any quadrilateral mesh composed of an even number of quadrilaterals admits a compatible hexahedral mesh. We extend this to certain nonball volumes: there is a construction to reduce to the ball case, and we give a necessary condition as well.
Counting integer points in parametric polytopes using Barvinok’s rational functions
 Algorithmica
, 2007
"... Abstract Many compiler optimization techniques depend on the ability to calculate the number of elements that satisfy certain conditions. If these conditions can be represented by linear constraints, then such problems are equivalent to counting the number of integer points in (possibly) parametric ..."
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Cited by 44 (9 self)
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Abstract Many compiler optimization techniques depend on the ability to calculate the number of elements that satisfy certain conditions. If these conditions can be represented by linear constraints, then such problems are equivalent to counting the number of integer points in (possibly) parametric polytopes. It is well known that the enumerator of such a set can be represented by an explicit function consisting of a set of quasipolynomials each associated with a chamber in the parameter space. Previously, interpolation was used to obtain these quasipolynomials, but this technique has several disadvantages. Its worstcase computation time for a single quasipolynomial is exponential in the input size, even for fixed dimensions. The worstcase size of such a quasipolynomial (measured in bits needed to represent the quasipolynomial) is also exponential in the input size. Under certain conditions this technique even fails to produce a solution. Our main contribution is a novel method for calculating the required quasipolynomials analytically. It extends an existing method, based on Barvinok’s decomposition,
A new proof of the density HalesJewett theorem
, 2009
"... Abstract The HalesJewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1, . . . , k} n contains a monochromatic combinatorial line. This result is a generalization of van der Waerden's theorem, and it is one of the fundament ..."
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Cited by 39 (2 self)
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Abstract The HalesJewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1, . . . , k} n contains a monochromatic combinatorial line. This result is a generalization of van der Waerden's theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdős and Turán in 1936, proved by Szemerédi in 1975, and given a different proof by Furstenberg in 1977. The HalesJewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemerédi's theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson and the first to provide a quantitative bound on how large n needs to be. In particular, we show that a subset of {1, 2, 3} n of density δ contains a combinatorial line if n is at least as big as a tower of 2s of height O(1/δ 2 ). Our proof is surprisingly simple: indeed, it gives arguably the simplest known proof of Szemerédi's theorem.
Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations
 In International Symposium on Symbolic and Algebraic Computation 2004 (ISSAC04
, 2004
"... This paper presents the algebraic foundation for an approach for generating polynomial loop invariants in imperative programs. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Using this connection, a procedure for finding loop invaria ..."
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Cited by 34 (6 self)
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This paper presents the algebraic foundation for an approach for generating polynomial loop invariants in imperative programs. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Using this connection, a procedure for finding loop invariants is given in terms of operations on ideals, for which Gröbner basis constructions can be employed. Most importantly, it is proved that if the assignment statements in a loop are solvable (in particular, affine) mappings with positive eigenvalues, then the procedure terminates in at most 2m + 1 iterations, where m is the number of variables in the loop. The proof is done by showing that the irreducible subvarieties of the variety associated with the polynomial ideal approximating the invariant polynomial ideal of the loop either stay the same or increase their dimension in every iteration. This yields a correct and complete algorithm for inferring conjunctions of polynomial equations as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover nontrivial invariants for several examples to illustrate the power of the techniques.