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251
On graph kernels: Hardness results and efficient alternatives
 IN: CONFERENCE ON LEARNING THEORY
, 2003
"... As most ‘realworld’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances tha ..."
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Cited by 164 (5 self)
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As most ‘realworld’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances that are represented by graphs. So far, only very specific graphs such as trees and strings have been considered. This paper investigates kernels on labeled directed graphs with general structure. It is shown that computing a strictly positive definite graph kernel is at least as hard as solving the graph isomorphism problem. It is also shown that computing an inner product in a feature space indexed by all possible graphs, where each feature counts the number of subgraphs isomorphic to that graph, is NPhard. On the other hand, inner products in an alternative feature space, based on walks in the graph, can be computed in polynomial time. Such kernels are defined in this paper.
Algorithms in Discrete Convex Analysis
 Math. Programming
, 2000
"... this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects. ..."
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Cited by 142 (32 self)
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this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects.
Dual decomposition for parsing with nonprojective head automata
 In Proc. of EMNLP
, 2010
"... This paper introduces algorithms for nonprojective parsing based on dual decomposition. We focus on parsing algorithms for nonprojective head automata, a generalization of headautomata models to nonprojective structures. The dual decomposition algorithms are simple and efficient, relying on standa ..."
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Cited by 97 (15 self)
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This paper introduces algorithms for nonprojective parsing based on dual decomposition. We focus on parsing algorithms for nonprojective head automata, a generalization of headautomata models to nonprojective structures. The dual decomposition algorithms are simple and efficient, relying on standard dynamic programming and minimum spanning tree algorithms. They provably solve an LP relaxation of the nonprojective parsing problem. Empirically the LP relaxation is very often tight: for many languages, exact solutions are achieved on over 98 % of test sentences. The accuracy of our models is higher than previous work on a broad range of datasets. 1
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 73 (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
An Improved LPbased Approximation for Steiner Tree
, 2009
"... The Steiner tree problem is one of the most fundamentalhard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robin ..."
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Cited by 64 (7 self)
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The Steiner tree problem is one of the most fundamentalhard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robins,ZelikovskySIDMA’05]. All these algorithms are purely combinatorial. A longstanding open problem is whether there is an LPrelaxation for Steiner tree with integrality gap smaller than [Vazirani,RajagopalanSODA’99]. In this paper we improve the approximation factor for Steiner tree, developing an LPbased approximation a� algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directedcomponent cut relaxation for the�restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together
Expressivity versus efficiency of graph kernels
 Proceedings of the First International Workshop on Mining Graphs, Trees and Sequences
, 2003
"... Abstract. Recently, kernel methods have become a popular tool for machine learning and data mining. As most ‘realworld ’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured dat ..."
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Cited by 47 (0 self)
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Abstract. Recently, kernel methods have become a popular tool for machine learning and data mining. As most ‘realworld ’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. In this paper we study the tradeoff between expressivity and efficiency of graph kernels. First, we motivate the need for this discussion by showing that fully general graph kernels can not even be approximated efficiently. We also discuss generalizations of graph kernels defined in literature and show that they are either not positive definite or not very useful. Finally, we propose a new graph kernel based on subtree patterns. We argue that while a little more computationally expensive, this kernel is more expressive than kernels based on walks. 1
Bisubmodular Function Minimization
 Mathematical Programming
, 2000
"... This paper presents the rst combinatorial, polynomialtime algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. A bisubmodular function arises as a rank function of a deltamatroid. The scali ..."
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Cited by 47 (4 self)
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This paper presents the rst combinatorial, polynomialtime algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. A bisubmodular function arises as a rank function of a deltamatroid. The scaling algorithm naturally leads to the rst combinatorial polynomialtime algorithm for testing membership in deltamatroid polyhedra. Unlike the case of matroid polyhedra, it remains open to develop a combinatorial strongly polynomial algorithm for this problem. Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 5608531, Japan (fujishig@sys.es.osakau.ac.jp). Research partly carried out while at Forschungsinstut fur Diskrete Mathematik, Universitat Bonn. y Department of Mathematical Engineering and Information Physics, University of Tokyo, Tokyo 1138656, Japan (iwata@sr3.t.utokyo.ac.jp). 1 1 Introduction Let V be a nite none...
Halfintegrality based algorithms for cosegmentation of images
 In CVPR
, 2009
"... We study the cosegmentation problem where the objective is to segment the same object (i.e., region) from a pair of images. The segmentation for each image can be cast using a partitioning/segmentation function with an additional constraint that seeks to make the histograms of the segmented regions ..."
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Cited by 47 (4 self)
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We study the cosegmentation problem where the objective is to segment the same object (i.e., region) from a pair of images. The segmentation for each image can be cast using a partitioning/segmentation function with an additional constraint that seeks to make the histograms of the segmented regions (based on intensity and texture features) similar. Using Markov Random Field (MRF) energy terms for the simultaneous segmentation of the images together with histogram consistency requirements using the squared L2 (rather than L1) distance, after linearization and adjustments, yields an optimization model with some interesting combinatorial properties. We discuss these properties which are closely related to certain relaxation strategies recently introduced in computer vision. Finally, we show experimental results of the proposed approach. 1.
Graph Kernels and Gaussian Processes for Relational Reinforcement Learning
 Machine Learning
, 2003
"... Relational reinforcement learning is a Qlearning technique for relational stateaction spaces. It aims to enable agents to learn how to act in an environment that has no natural representation as a tuple of constants. In this case, the learning algorithm used to approximate the mapping between stat ..."
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Cited by 45 (9 self)
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Relational reinforcement learning is a Qlearning technique for relational stateaction spaces. It aims to enable agents to learn how to act in an environment that has no natural representation as a tuple of constants. In this case, the learning algorithm used to approximate the mapping between stateaction pairs and their so called Q(uality)value has to be not only very reliable, but it also has to be able to handle the relational representation of stateaction pairs. In this paper we investigate...
Combinatorial network optimization with unknown variables: Multiarmed bandits with linear rewards and individual observations
 IEEE/ACM Transactions on Networking (TON
"... Abstract—We formulate the following combinatorial multiarmed bandit (MAB) problem: There are random variables with unknown mean that are each instantiated in an i.i.d. fashion over time. At each time multiple random variables can be selected, subject to an arbitrary constraint on weights associated ..."
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Cited by 36 (4 self)
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Abstract—We formulate the following combinatorial multiarmed bandit (MAB) problem: There are random variables with unknown mean that are each instantiated in an i.i.d. fashion over time. At each time multiple random variables can be selected, subject to an arbitrary constraint on weights associated with the selected variables. All of the selected individual random variables are observed at that time, and a linearly weighted combination of these selected variables is yielded as the reward. The goal is to find a policy that minimizes regret, defined as the difference between the reward obtained by a genie that knows the mean of each random variable, and that obtained by the given policy. This formulation is broadly applicable and useful for stochastic online versions of many interesting tasks in networks that can be formulated as tractable combinatorial optimization problems with linear objective functions, such as maximum weighted matching,