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58
SemiSupervised Learning Using Randomized Mincuts
 IN PROCEEDINGS OF THE 21ST INTERNATIONAL CONFERENCE ON MACHINE LEARNING
, 2004
"... In many application domains there is a large amount of unlabeled data but only a very limited amount of labeled training data. One general approach that has been explored for utilizing this unlabeled data is to construct a graph on all the data points based on distance relationships among exam ..."
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Cited by 78 (4 self)
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In many application domains there is a large amount of unlabeled data but only a very limited amount of labeled training data. One general approach that has been explored for utilizing this unlabeled data is to construct a graph on all the data points based on distance relationships among examples, and then to use the known labels to perform some type of graph partitioning. One natural
The complexity of the counting constraint satisfaction problem
 In ICALP (1
, 2008
"... The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structureH can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G toH. In this paper we characterize relational structuresH for which#CSP(H) can ..."
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Cited by 45 (7 self)
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The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structureH can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G toH. In this paper we characterize relational structuresH for which#CSP(H) can be solved in polynomial time and prove that for all other structures the problem is #Pcomplete. 1
Inapproximability of the Tutte polynomial
, 2008
"... The Tutte polynomial of a graph G is a twovariable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G; x, y). Jaeger, V ..."
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Cited by 29 (9 self)
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The Tutte polynomial of a graph G is a twovariable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G; x, y). Jaeger, Vertigan and Welsh have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #Phard, except along the hyperbola (x − 1)(y − 1) = 1 and at four special points. We are interested in determining for which points (x, y) there is a fully polynomial randomised approximation scheme (FPRAS) for T(G; x, y). Under the assumption RP = NP, we prove that there is no FPRAS at (x, y) if (x, y) is is in one of the halfplanes x < −1 or y < −1 (excluding the easytocompute cases mentioned above). Two exceptions to this result are the halfline x < −1, y = 1 (which is still open) and the portion of the hyperbola (x − 1)(y − 1) = 2 corresponding to y < −1 which we show
An approximation trichotomy for Boolean #CSP
, 2007
"... We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is ..."
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Cited by 25 (7 self)
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We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the coclone IM2 from Post’s lattice, then the problem of counting satisfying assignments is complete with respect to approximationpreserving reductions in the complexity class #RHΠ1. This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately counting the number of independent sets in a bipartite graph. For every other fixed constraint language, the problem is complete for #P with respect to approximationpreserving reductions, meaning that there is no fully polynomial randomised approximation scheme for counting satisfying assignments unless NP=RP. 1
The Trichotomy of HAVING Queries on a Probabilistic Database
 VLDBJ
"... We study the evaluation of positive conjunctive queries with Boolean aggregate tests (similar to HAVING in SQL) on probabilistic databases. More precisely, we study conjunctive queries with predicate aggregates on probabilistic databases where the aggregation function is one of MIN, MAX, EXISTS, C ..."
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Cited by 20 (3 self)
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We study the evaluation of positive conjunctive queries with Boolean aggregate tests (similar to HAVING in SQL) on probabilistic databases. More precisely, we study conjunctive queries with predicate aggregates on probabilistic databases where the aggregation function is one of MIN, MAX, EXISTS, COUNT, SUM, AVG, or COUNT(DISTINCT) and the comparison function is one of =, �, ≥,>, ≤, or <. The complexity of evaluating a HAVING query depends on the aggregation function, α, and the comparison function, θ. In this paper, we establish a set of trichotomy results for conjunctive queries with HAVING predicates parametrized by (α, θ). For such queries (without self joins), one of the following three statements is true: (1) The exact evaluation problem has Ptime data complexity. In this case, we call the query safe. (2) The exact evaluation problem is ♯Phard, but the approximate evaluation problem has (randomized) Ptime data complexity. More precisely, there exists an fptras for the query. In this case, we call the query apxsafe. (3) The exact evaluation problem is ♯Phard, and the approximate evaluation problem is also hard. We call these queries hazardous. The precise definition of each class depends on the aggregate considered and the comparison function. Thus, we have queries that are (MAX, ≥)safe, (COUNT, ≤)apxsafe, (SUM, =)hazardous, etc. Our trichotomy result is a signifi
The complexity of ferromagnetic Ising with local fields
, 2005
"... We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomised approximation scheme for the case in which the system is consistent in ..."
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Cited by 19 (10 self)
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We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomised approximation scheme for the case in which the system is consistent in the sense that the local external fields all favour the same spin. We characterise the complexity of the general problem by showing that it is equivalent in complexity to the problem of approximately counting independent sets in bipartite graphs, thus it is complete in a logicallydefined subclass of #P previously studied by Dyer, Goldberg, Greenhill and Jerrum. By contrast, we show that the corresponding computational task for the qstate Potts model with local external magnetic fields and q> 2 is complete for all of #P with respect to approximationpreserving reductions.
A fully polynomial time approximation scheme for singleitem stochastic inventory control with discrete demand
, 2006
"... We develop a framework for obtaining (deterministic) Fully Polynomial Time Approximation Schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone singleperiod cost functions. Using our framework, we give the first FPTASs for several NPhard problems in various fiel ..."
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Cited by 15 (0 self)
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We develop a framework for obtaining (deterministic) Fully Polynomial Time Approximation Schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone singleperiod cost functions. Using our framework, we give the first FPTASs for several NPhard problems in various fields of research such as knapsackrelated problems, logistics, operations management, economics, and mathematical finance. 1
Quantum computation and the evaluation of tensor networks
, 2008
"... We present a quantum algorithm that additively approximates the value of a tensor network to a certain scale. When combined with existing results, this provides a complete problem for quantum computation. The result is a simple new way of looking at quantum computation in which unitary gates are rep ..."
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Cited by 9 (0 self)
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We present a quantum algorithm that additively approximates the value of a tensor network to a certain scale. When combined with existing results, this provides a complete problem for quantum computation. The result is a simple new way of looking at quantum computation in which unitary gates are replaced by tensors and time is replaced by the order in which the tensornetwork is “swallowed”. We use this result to derive new quantum algorithms that approximate the partition function of a variety of classical statistical mechanics models, including the Potts model.
The expressibility of functions on the Boolean domain, with applications to Counting CSPs
"... An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post’s lattice gives a complete classification of ..."
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Cited by 8 (6 self)
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An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post’s lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of logsupermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial nonlsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any nontrivial functional clone (in a sense that will be made precise) contains the binary function “implies”. As a consequence, in the conservative case, all nontrivial counting CSPs are as hard to approximate as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexitytheoretic results, it is natural to ask whether the “implies” clone is equivalent to the clone of lsm functions. We use the Möbius transform and the Fourier transform to show that these