TY - CONF
AB - A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. We study which classes of finite-valued languages can be solved exactly by the basic linear programming relaxation (BLP). Thapper and Živný showed [20] that if BLP solves the language then the language admits a binary commutative fractional polymorphism. We prove that the converse is also true. This leads to a necessary and a sufficient condition which can be checked in polynomial time for a given language. In contrast, the previous necessary and sufficient condition due to [20] involved infinitely many inequalities. More recently, Thapper and Živný [21] showed (using, in particular, a technique introduced in this paper) that core languages that do not satisfy our condition are NP-hard. Taken together, these results imply that a finite-valued language can either be solved using Linear Programming or is NP-hard.
AU - Kolmogorov, Vladimir
ID - 2518
IS - 1
TI - The power of linear programming for finite-valued CSPs: A constructive characterization
VL - 7965
ER -
TY - JOUR
AB - We study the complexity of valued constraint satisfaction problems (VCSPs) parametrized by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimize the sum. Under the unique games conjecture, the approximability of finite-valued VCSPs is well understood, see Raghavendra [2008]. However, there is no characterization of finite-valued VCSPs, let alone general-valued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimization perspective. We consider the case of languages containing all possible unary cost functions. In the case of languages consisting of only {0, ∞}-valued cost functions (i.e., relations), such languages have been called conservative and studied by Bulatov [2003, 2011] and recently by Barto [2011]. Since we study valued languages, we call a language conservative if it contains all finite-valued unary cost functions. The computational complexity of conservative valued languages has been studied by Cohen et al. [2006] for languages over Boolean domains, by Deineko et al. [2008] for {0, 1}-valued languages (a.k.a Max-CSP), and by Takhanov [2010a] for {0, ∞}-valued languages containing all finite-valued unary cost functions (a.k.a. Min-Cost-Hom). We prove a Schaefer-like dichotomy theorem for conservative valued languages: if all cost functions in the language satisfy a certain condition (specified by a complementary combination of STP and MJN multimor-phisms), then any instance can be solved in polynomial time (via a new algorithm developed in this article), otherwise the language is NP-hard. This is the first complete complexity classification of general-valued constraint languages over non-Boolean domains. It is a common phenomenon that complexity classifications of problems over non-Boolean domains are significantly harder than the Boolean cases. The polynomial-time algorithm we present for the tractable cases is a generalization of the submodular minimization problem and a result of Cohen et al. [2008]. Our results generalize previous results by Takhanov [2010a] and (a subset of results) by Cohen et al. [2006] and Deineko et al. [2008]. Moreover, our results do not rely on any computer-assisted search as in Deineko et al. [2008], and provide a powerful tool for proving hardness of finite-valued and general-valued languages.
AU - Kolmogorov, Vladimir
AU - Živný, Stanislav
ID - 2828
IS - 2
JF - Journal of the ACM
TI - The complexity of conservative valued CSPs
VL - 60
ER -
TY - CONF
AB - We introduce the M-modes problem for graphical models: predicting the M label configurations of highest probability that are at the same time local maxima of the probability landscape. M-modes have multiple possible applications: because they are intrinsically diverse, they provide a principled alternative to non-maximum suppression techniques for structured prediction, they can act as codebook vectors for quantizing the configuration space, or they can form component centers for mixture model approximation. We present two algorithms for solving the M-modes problem. The first algorithm solves the problem in polynomial time when the underlying graphical model is a simple chain. The second algorithm solves the problem for junction chains. In synthetic and real dataset, we demonstrate how M-modes can improve the performance of prediction. We also use the generated modes as a tool to understand the topography of the probability distribution of configurations, for example with relation to the training set size and amount of noise in the data.
AU - Chen, Chao
AU - Kolmogorov, Vladimir
AU - Yan, Zhu
AU - Metaxas, Dimitris
AU - Lampert, Christoph
ID - 2901
TI - Computing the M most probable modes of a graphical model
VL - 31
ER -
TY - GEN
AB - We consider the problem of inference in agraphical model with binary variables. While in theory it is arguably preferable to compute marginal probabilities, in practice researchers often use MAP inference due to the availability of efficient discrete optimization algorithms. We bridge the gap between the two approaches by introducing the Discrete Marginals technique in which approximate marginals are obtained by minimizing an objective function with unary and pair-wise terms over a discretized domain. This allows the use of techniques originally devel-oped for MAP-MRF inference and learning. We explore two ways to set up the objective function - by discretizing the Bethe free energy and by learning it from training data. Experimental results show that for certain types of graphs a learned function can out-perform the Bethe approximation. We also establish a link between the Bethe free energy and submodular functions.
AU - Korc, Filip
AU - Kolmogorov, Vladimir
AU - Lampert, Christoph
ID - 5396
SN - 2664-1690
TI - Approximating marginals using discrete energy minimization
ER -
TY - GEN
AB - This paper addresses the problem of approximate MAP-MRF inference in general graphical models. Following [36], we consider a family of linear programming relaxations of the problem where each relaxation is specified by a set of nested pairs of factors for which the marginalization constraint needs to be enforced. We develop a generalization of the TRW-S algorithm [9] for this problem, where we use a decomposition into junction chains, monotonic w.r.t. some ordering on the nodes. This generalizes the monotonic chains in [9] in a natural way. We also show how to deal with nested factors in an efficient way. Experiments show an improvement over min-sum diffusion, MPLP and subgradient ascent algorithms on a number of computer vision and natural language processing problems.
AU - Kolmogorov, Vladimir
AU - Schoenemann, Thomas
ID - 2928
T2 - arXiv
TI - Generalized sequential tree-reweighted message passing
ER -
TY - CONF
AB - In this paper we investigate k-submodular functions. This natural family of discrete functions includes submodular and bisubmodular functions as the special cases k = 1 and k = 2 respectively.
In particular we generalize the known Min-Max-Theorem for submodular and bisubmodular functions. This theorem asserts that the minimum of the (bi)submodular function can be found by solving a maximization problem over a (bi)submodular polyhedron. We define a k-submodular polyhedron, prove a Min-Max-Theorem for k-submodular functions, and give a greedy algorithm to construct the vertices of the polyhedron.
AU - Huber, Anna
AU - Kolmogorov, Vladimir
ID - 2930
TI - Towards minimizing k-submodular functions
VL - 7422
ER -
TY - JOUR
AB - In this paper, we present a new approach for establishing correspondences between sparse image features related by an unknown nonrigid mapping and corrupted by clutter and occlusion, such as points extracted from images of different instances of the same object category. We formulate this matching task as an energy minimization problem by defining an elaborate objective function of the appearance and the spatial arrangement of the features. Optimization of this energy is an instance of graph matching, which is in general an NP-hard problem. We describe a novel graph matching optimization technique, which we refer to as dual decomposition (DD), and demonstrate on a variety of examples that this method outperforms existing graph matching algorithms. In the majority of our examples, DD is able to find the global minimum within a minute. The ability to globally optimize the objective allows us to accurately learn the parameters of our matching model from training examples. We show on several matching tasks that our learned model yields results superior to those of state-of-the-art methods.
AU - Torresani, Lorenzo
AU - Kolmogorov, Vladimir
AU - Rother, Carsten
ID - 2931
IS - 2
JF - IEEE Transactions on Pattern Analysis and Machine Intelligence
TI - A dual decomposition approach to feature correspondence
VL - 35
ER -
TY - JOUR
AB - We consider the problem of minimizing a function represented as a sum of submodular terms. We assume each term allows an efficient computation of exchange capacities. This holds, for example, for terms depending on a small number of variables, or for certain cardinality-dependent terms. A naive application of submodular minimization algorithms would not exploit the existence of specialized exchange capacity subroutines for individual terms. To overcome this, we cast the problem as a submodular flow (SF) problem in an auxiliary graph in such a way that applying most existing SF algorithms would rely only on these subroutines. We then explore in more detail Iwata's capacity scaling approach for submodular flows (Iwata 1997 [19]). In particular, we show how to improve its complexity in the case when the function contains cardinality-dependent terms.
AU - Kolmogorov, Vladimir
ID - 3117
IS - 15
JF - Discrete Applied Mathematics
TI - Minimizing a sum of submodular functions
VL - 160
ER -
TY - CONF
AB - We consider the problem of inference in a graphical model with binary variables. While in theory it is arguably preferable to compute marginal probabilities, in practice researchers often use MAP inference due to the availability of efficient discrete optimization algorithms. We bridge the gap between the two approaches by introducing the Discrete Marginals technique in which approximate marginals are obtained by minimizing an objective function with unary and pairwise terms over a discretized domain. This allows the use of techniques originally developed for MAP-MRF inference and learning. We explore two ways to set up the objective function - by discretizing the Bethe free energy and by learning it from training data. Experimental results show that for certain types of graphs a learned function can outperform the Bethe approximation. We also establish a link between the Bethe free energy and submodular functions.
AU - Korc, Filip
AU - Kolmogorov, Vladimir
AU - Lampert, Christoph
ID - 3124
TI - Approximating marginals using discrete energy minimization
ER -
TY - JOUR
AB - Consider a convex relaxation f̂ of a pseudo-Boolean function f. We say that the relaxation is totally half-integral if f̂(x) is a polyhedral function with half-integral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form x i=x j, x i=1-x j, and x i=γ where γ∈{0,1,1/2} is a constant. A well-known example is the roof duality relaxation for quadratic pseudo-Boolean functions f. We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-Boolean functions. Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations f̂ by establishing a one-to-one correspondence with bisubmodular functions. Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality. On the conceptual level, our results show that bisubmodular functions provide a natural generalization of the roof duality approach to higher-order terms. This can be viewed as a non-submodular analogue of the fact that submodular functions generalize the s-t minimum cut problem with non-negative weights to higher-order terms.
AU - Kolmogorov, Vladimir
ID - 3257
IS - 4-5
JF - Discrete Applied Mathematics
TI - Generalized roof duality and bisubmodular functions
VL - 160
ER -