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81
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 156 (27 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]).
MAP estimation via agreement on trees: Messagepassing and linear programming
, 2002
"... We develop and analyze methods for computing provably optimal maximum a posteriori (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of treestructured distributions, we obtain an upper bound ..."
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Cited by 132 (8 self)
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We develop and analyze methods for computing provably optimal maximum a posteriori (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of treestructured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: (a) a treerelaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds; and (b) a treereweighted maxproduct messagepassing algorithm that is related to but distinct from the maxproduct algorithm. In this way, we establish a connection between a certain LP relaxation of the modefinding problem, and a reweighted form of the maxproduct (minsum) messagepassing algorithm.
MAP estimation via agreement on (hyper)trees: Messagepassing and linear programming approaches
 IEEE Transactions on Information Theory
, 2002
"... We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of treestructured problems, we obtain an upper bound on the optimal value of the original ..."
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Cited by 108 (11 self)
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We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of treestructured problems, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is met with equality if and only if the tree problems share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original problem. Next we present and analyze two methods for attempting to obtain tight upper bounds: (a) a treereweighted messagepassing algorithm that is related to but distinct from the maxproduct (minsum) algorithm; and (b) a treerelaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds. Finally, we discuss the conditions that govern when the relaxation is tight, in which case the MAP configuration can be obtained. The analysis described here generalizes naturally to convex combinations of hypertreestructured distributions.
Algorithm theories and design tactics
 Science of Computer Programming
, 1990
"... Algorithm theories represent the structure common to a class of algorithms, such as divideandconquer or backtrack. An algorithm theory for a class A provides the basis for design tactics – specialized methods for designing Aalgorithms from formal problem specifications. We illustrate this approac ..."
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Cited by 47 (15 self)
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Algorithm theories represent the structure common to a class of algorithms, such as divideandconquer or backtrack. An algorithm theory for a class A provides the basis for design tactics – specialized methods for designing Aalgorithms from formal problem specifications. We illustrate this approach with recent work on the theory of global search algorithms and briefly mention several others. Several design tactics have been implemented in the KIDS/CYPRESS system and have been used to semiautomatically derive many algorithms. 1.
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal
 In Proc. of ACM Symposium on Theory of computing (STOC
, 2007
"... ABSTRACT In the MINIMUM BOUNDED DEGREE SPANNING TREE problem,we are given an undirected graph with a degree upper bound Bv oneach vertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this ..."
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Cited by 46 (6 self)
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ABSTRACT In the MINIMUM BOUNDED DEGREE SPANNING TREE problem,we are given an undirected graph with a degree upper bound Bv oneach vertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this paper, we present a polynomial time algorithm which returns a spanning tree T ofcost at most OPT and dT (v) ^ Bv + 1 for all v, where dT (v)denotes the degree of v in T. This generalizes a result of Furerand Raghavachari [8] to weighted graphs, and settles a 15yearold conjecture of Goemans [10] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound Av and a degreeupper bound Bv, and returns a spanning tree with cost at most OPTand Av \Gamma 1 ^ dT (v) ^ Bv + 1 for all v. This is essentially thebest possible. The main technique used is an extension of the iterative rounding method introduced by Jain [12] for the design ofapproximation algorithms.
The convex hull of two core capacitated network design problems
 MATHEMATICAL PROGRAMMING
, 1993
"... The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed pointtopoint demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs ..."
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Cited by 43 (0 self)
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The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed pointtopoint demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost. This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.
Solving small TSPs with constraints
 PROCEEDINGS OF THE 14TH INTERNATIONAL CONFERENCE ON LOGIC PROGRAMMING
, 1997
"... This paper presents a set of techniques that makes constraint programming a technique of choice for solving small (up to 30 nodes) traveling salesman problems. These techniques include a propagation scheme to avoid intermediate cycles (a global constraint), a branching scheme and a redundant constra ..."
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Cited by 42 (0 self)
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This paper presents a set of techniques that makes constraint programming a technique of choice for solving small (up to 30 nodes) traveling salesman problems. These techniques include a propagation scheme to avoid intermediate cycles (a global constraint), a branching scheme and a redundant constraint that can be used as a bounding method. The resulting improvement is that we can solve problems twice larger than those solved previously with constraint programming tools. We evaluate the use of Lagrangean Relaxation to narrow the gap between constraint programming and other Operations Research techniques and we show that improved constraint propagation has now a place in the array of techniques that should be used to solve a traveling salesman problem.
Minimum Bounded Degree Spanning Trees
, 2006
"... We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. Thi ..."
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Cited by 31 (0 self)
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We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. This is almost best possible. The approach uses a sequence of simple algebraic, polyhedral and combinatorial arguments. It illustrates many techniques and ideas in combinatorial optimization as it involves polyhedral characterizations, uncrossing, matroid intersection, and graph orientations (or packing of spanning trees). The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree. It also gives a better understanding of the subtour relaxation for both the symmetric and asymmetric traveling salesman problems. The generalization to ledgeconnected subgraphs is briefly discussed.
Symmetry and approximability of submodular maximization problems
"... A number of recent results on optimization problems involving submodular functions have made use of the ”multilinear relaxation” of the problem [3], [8], [24], [14], [13]. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of ”symmetry ..."
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Cited by 24 (2 self)
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A number of recent results on optimization problems involving submodular functions have made use of the ”multilinear relaxation” of the problem [3], [8], [24], [14], [13]. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of ”symmetry gap”. Our main result is that for any fixed instance that exhibits a certain ”symmetry gap ” in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1 − 1/e)approximation for monotone submodular maximization under a cardinality constraint [20], [7], and the impossibility of ( 1 +ɛ)approximation for uncon2 strained (nonmonotone) submodular maximization [8]. It follows from our result that ( 1 + ɛ)approximation is also impossible for 2 nonmonotone submodular maximization subject to a (nontrivial) matroid constraint. On the algorithmic side, we present a 0.309approximation for this problem, improving the previously known factor of 1 − o(1) [14]. 4 As another application, we consider the problem of maximizing a nonmonotone submodular function over the bases of a matroid. A ( 1 − o(1))approximation has been developed for this problem, 6 assuming that the matroid contains two disjoint bases [14]. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k ≥ 2, there is a class of matroids of fractional base packing number k k−1 ν = , such that any algorithm achieving a better than (1 − 1)approximation for this class would require exponentially many