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19
On Dual Decomposition and Linear Programming Relaxations for Natural Language Processing
 In Proc. EMNLP
, 2010
"... This paper introduces dual decomposition as a framework for deriving inference algorithms for NLP problems. The approach relies on standard dynamicprogramming algorithms as oracle solvers for subproblems, together with a simple method for forcing agreement between the different oracles. The approa ..."
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Cited by 71 (3 self)
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This paper introduces dual decomposition as a framework for deriving inference algorithms for NLP problems. The approach relies on standard dynamicprogramming algorithms as oracle solvers for subproblems, together with a simple method for forcing agreement between the different oracles. The approach provably solves a linear programming (LP) relaxation of the global inference problem. It leads to algorithms that are simple, in that they use existing decoding algorithms; efficient, in that they avoid exact algorithms for the full model; and often exact, in that empirically they often recover the correct solution in spite of using an LP relaxation. We give experimental results on two problems: 1) the combination of two lexicalized parsing models; and 2) the combination of a lexicalized parsing model and a trigram partofspeech tagger. 1
The Steiner tree polytope and related polyhedra
, 1994
"... We consider the vertexweighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is seriesparallel. For ..."
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Cited by 30 (1 self)
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We consider the vertexweighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is seriesparallel. For general graphs, this formulation can be interpreted as a (partial) extended formulation for the Steiner tree problem. By projecting this formulation, we obtain some very large classes of facetdefining valid inequalities for the Steiner tree polytope.
A Catalog of Steiner Tree Formulations
, 1993
"... We present some existing and some new formulations for the Steiner tree and Steiner arborescence problems. We show the equivalence of many of these formulations. In particular, we establish the equivalence between the classical bidirected dicut relaxation and two vertex weighted undirected relaxatio ..."
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Cited by 29 (0 self)
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We present some existing and some new formulations for the Steiner tree and Steiner arborescence problems. We show the equivalence of many of these formulations. In particular, we establish the equivalence between the classical bidirected dicut relaxation and two vertex weighted undirected relaxations. The motivation behind this study is a characterization of the feasible region of the dicut relaxation in the natural space corresponding to the Steiner tree problem.
A Tutorial on Dual Decomposition and Lagrangian Relaxation for Inference in Natural Language Processing
"... Dual decomposition, and more generally Lagrangian relaxation, is a classical method for combinatorial optimization; it has recently been applied to several inference problems in natural language processing (NLP). This tutorial gives an overview of the technique. We describe example algorithms, descr ..."
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Cited by 21 (4 self)
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Dual decomposition, and more generally Lagrangian relaxation, is a classical method for combinatorial optimization; it has recently been applied to several inference problems in natural language processing (NLP). This tutorial gives an overview of the technique. We describe example algorithms, describe formal guarantees for the method, and describe practical issues in implementing the algorithms. While our examples are predominantly drawn from the NLP literature, the material should be of general relevance to inference problems in machine learning. A central theme of this tutorial is that Lagrangian relaxation is naturally applied in conjunction with a broad class of combinatorial algorithms, allowing inference in models that go significantly beyond previous work on Lagrangian relaxation for inference in graphical models.
Exact Decoding of Syntactic Translation Models through Lagrangian Relaxation
, 2011
"... We describe an exact decoding algorithm for syntaxbased statistical translation. The approach uses Lagrangian relaxation to decompose the decoding problem into tractable subproblems, thereby avoiding exhaustive dynamic programming. The method recovers exact solutions, with certificates of optimalit ..."
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Cited by 14 (3 self)
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We describe an exact decoding algorithm for syntaxbased statistical translation. The approach uses Lagrangian relaxation to decompose the decoding problem into tractable subproblems, thereby avoiding exhaustive dynamic programming. The method recovers exact solutions, with certificates of optimality, on over 97 % of test examples; it has comparable speed to stateoftheart decoders.
Extended Formulations for Packing and Partitioning Orbitopes
, 2008
"... We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in [6]. These polytopes are the convex hulls of all 0/1matrices with lexicographically sorted columns and at most, resp. exactly, one 1entry per row. ..."
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Cited by 5 (0 self)
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We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in [6]. These polytopes are the convex hulls of all 0/1matrices with lexicographically sorted columns and at most, resp. exactly, one 1entry per row. They are important objects for symmetry reduction in certain integer programs. Using the extended formulations, we also derive a rather simple proof of the fact [6] that basically shiftedcolumn inequalities suffice in order to describe those orbitopes linearly.
The mixing set with divisible capacities: a simple approach
, 2008
"... We give a simple algorithm for linear optimization over the mixing set with divisible capacities, and derive a compact extended formulation from such an algorithm. The main idea is to apply a suitable unimodular transformation to obtain an equivalent problem that is easier to analyze. ..."
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Cited by 3 (0 self)
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We give a simple algorithm for linear optimization over the mixing set with divisible capacities, and derive a compact extended formulation from such an algorithm. The main idea is to apply a suitable unimodular transformation to obtain an equivalent problem that is easier to analyze.
Constant Integrality Gap LP Formulations of Unsplittable Flow on a Path
, 2013
"... The Unsplittable Flow Problem on a Path (UFPP) isacore problem in many important settings such as network flows, bandwidth allocation, resource constraint scheduling, and interval packing. We are given a path with capacities on the edges and a set of tasks, each task having a demand, a profit, a sou ..."
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Cited by 3 (2 self)
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The Unsplittable Flow Problem on a Path (UFPP) isacore problem in many important settings such as network flows, bandwidth allocation, resource constraint scheduling, and interval packing. We are given a path with capacities on the edges and a set of tasks, each task having a demand, a profit, a source and a destination vertex on the path. The goal is to compute a subset of tasks of maximum profit that does not violate the edge capacities. In practical applications generic approaches such as integer programming (IP) methods are desirable. Unfortunately, no IPformulation is known for the problem whose LPrelaxation has an integrality gap that is provably constant. For the unweighted case, we show that adding a few constraints to the standard LP of the problem is sufficient to make the integrality gap drop from Ω(n) to O(1). This positively answers an open question in [Chekuri et al., APPROX 2009]. For the general (weighted) case, we present an extended formulation with integrality gap bounded by 7+ε. This matches the best known approximation factor for the problem [Bonsma et al., FOCS 2011]. This result exploits crucially a technique for embedding dynamic programs into linear programs. We believe that this method could be useful to strengthen LPformulations for other problems as well and might eventually speed up computations due to stronger problem formulations.
Column Generation for Extended Formulations
, 2013
"... Working in an extended variable space allows one to develop tighter reformulations for mixed integer programs. However, the size of the extended formulation grows rapidly too large for a direct treatment by a MIPsolver. Then, one can work with inner approximations defined and improved by generating ..."
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Working in an extended variable space allows one to develop tighter reformulations for mixed integer programs. However, the size of the extended formulation grows rapidly too large for a direct treatment by a MIPsolver. Then, one can work with inner approximations defined and improved by generating dynamically variables and constraints. When the extended formulation stems from subproblems’ reformulations, one can implement column generation for the extended formulation using a DantzigWolfe decomposition paradigm. Pricing subproblem solutions are expressed in the variables of the extended formulation and added to the current restricted version of the extended formulation along with the subproblem constraints that are active for the subproblem solutions. This socalled “columnandrow generation” procedure is revisited here in a unifying presentation that generalizes the column generation algorithm and extends to the case of working with an approximate extended formulation. The interest of the approach is evaluated numerically on machine scheduling, bin packing, generalized assignment, and multiechelon lotsizing problems. We compare a direct handling of the extended formulation, a standard column generation approach, and the “columnandrow generation ” procedure, highlighting a key benefit of the latter: lifting pricing problem solutions in the space of the extended formulation permits their recombination into new subproblem solutions and results in faster convergence.