@inproceedings{6725,
abstract = {A Valued Constraint Satisfaction Problem (VCSP) provides a common framework that can express a wide range of discrete optimization problems. A VCSP instance is given by a finite set of variables, a finite domain of labels, and an objective function to be minimized. This function is represented as a sum of terms where each term depends on a subset of the variables. To obtain different classes of optimization problems, one can restrict all terms to come from a fixed set Γ of cost functions, called a language.
Recent breakthrough results have established a complete complexity classification of such classes with respect to language Γ: if all cost functions in Γ satisfy a certain algebraic condition then all Γ-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language Γ is known to be NP-hard. We thus study exponential algorithms for this meta-problem. We show that the tractability condition of a finite-valued language Γ can be tested in O(3‾√3|D|⋅poly(size(Γ))) time, where D is the domain of Γ and poly(⋅) is some fixed polynomial. We also obtain a matching lower bound under the Strong Exponential Time Hypothesis (SETH). More precisely, we prove that for any constant δ<1 there is no O(3‾√3δ|D|) algorithm, assuming that SETH holds.},
author = {Kolmogorov, Vladimir},
booktitle = {46th International Colloquium on Automata, Languages and Programming},
isbn = {978-3-95977-109-2},
issn = {1868-8969},
location = {Patras, Greece},
pages = {77:1--77:12},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Testing the complexity of a valued CSP language}},
doi = {10.4230/LIPICS.ICALP.2019.77},
volume = {132},
year = {2019},
}
@article{6032,
abstract = {The main result of this article is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are even Δ-matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvorak and Kupec. Using a reduction to even Δ-matroids, we then extend the tractability result to larger classes of Δ-matroids that we call efficiently coverable. It properly includes classes that were known to be tractable before, namely, co-independent, compact, local, linear, and binary, with the following caveat:We represent Δ-matroids by lists of tuples, while the last two use a representation by matrices. Since an n ×n matrix can represent exponentially many tuples, our tractability result is not strictly stronger than the known algorithm for linear and binary Δ-matroids.},
author = {Kazda, Alexandr and Kolmogorov, Vladimir and Rolinek, Michal},
journal = {ACM Transactions on Algorithms},
number = {2},
publisher = {ACM},
title = {{Even delta-matroids and the complexity of planar boolean CSPs}},
doi = {10.1145/3230649},
volume = {15},
year = {2019},
}
@article{18,
abstract = {An N-superconcentrator is a directed, acyclic graph with N input nodes and N output nodes such that every subset of the inputs and every subset of the outputs of same cardinality can be connected by node-disjoint paths. It is known that linear-size and bounded-degree superconcentrators exist. We prove the existence of such superconcentrators with asymptotic density 25.3 (where the density is the number of edges divided by N). The previously best known densities were 28 [12] and 27.4136 [17].},
author = {Kolmogorov, Vladimir and Rolinek, Michal},
issn = {0381-7032},
journal = {Ars Combinatoria},
number = {10},
pages = {269 -- 304},
publisher = {Charles Babbage Research Centre},
title = {{Superconcentrators of density 25.3}},
volume = {141},
year = {2018},
}
@article{5975,
abstract = {We consider the recent formulation of the algorithmic Lov ́asz Local Lemma [N. Har-vey and J. Vondr ́ak, inProceedings of FOCS, 2015, pp. 1327–1345; D. Achlioptas and F. Iliopoulos,inProceedings of SODA, 2016, pp. 2024–2038; D. Achlioptas, F. Iliopoulos, and V. Kolmogorov,ALocal Lemma for Focused Stochastic Algorithms, arXiv preprint, 2018] for finding objects that avoid“bad features,” or “flaws.” It extends the Moser–Tardos resampling algorithm [R. A. Moser andG. Tardos,J. ACM, 57 (2010), 11] to more general discrete spaces. At each step the method picks aflaw present in the current state and goes to a new state according to some prespecified probabilitydistribution (which depends on the current state and the selected flaw). However, the recent formu-lation is less flexible than the Moser–Tardos method since it requires a specific flaw selection rule,whereas the algorithm of Moser and Tardos allows an arbitrary rule (and thus can potentially beimplemented more efficiently). We formulate a new “commutativity” condition and prove that it issufficient for an arbitrary rule to work. It also enables an efficient parallelization under an additionalassumption. We then show that existing resampling oracles for perfect matchings and permutationsdo satisfy this condition.},
author = {Kolmogorov, Vladimir},
issn = {0097-5397},
journal = {SIAM Journal on Computing},
number = {6},
pages = {2029--2056},
publisher = {Society for Industrial & Applied Mathematics (SIAM)},
title = {{Commutativity in the algorithmic Lovász local lemma}},
doi = {10.1137/16m1093306},
volume = {47},
year = {2018},
}
@inproceedings{273,
abstract = {The accuracy of information retrieval systems is often measured using complex loss functions such as the average precision (AP) or the normalized discounted cumulative gain (NDCG). Given a set of positive and negative samples, the parameters of a retrieval system can be estimated by minimizing these loss functions. However, the non-differentiability and non-decomposability of these loss functions does not allow for simple gradient based optimization algorithms. This issue is generally circumvented by either optimizing a structured hinge-loss upper bound to the loss function or by using asymptotic methods like the direct-loss minimization framework. Yet, the high computational complexity of loss-augmented inference, which is necessary for both the frameworks, prohibits its use in large training data sets. To alleviate this deficiency, we present a novel quicksort flavored algorithm for a large class of non-decomposable loss functions. We provide a complete characterization of the loss functions that are amenable to our algorithm, and show that it includes both AP and NDCG based loss functions. Furthermore, we prove that no comparison based algorithm can improve upon the computational complexity of our approach asymptotically. We demonstrate the effectiveness of our approach in the context of optimizing the structured hinge loss upper bound of AP and NDCG loss for learning models for a variety of vision tasks. We show that our approach provides significantly better results than simpler decomposable loss functions, while requiring a comparable training time.},
author = {Mohapatra, Pritish and Rolinek, Michal and Jawahar, C V and Kolmogorov, Vladimir and Kumar, M Pawan},
booktitle = {2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition},
isbn = {9781538664209},
location = {Salt Lake City, UT, USA},
pages = {3693--3701},
publisher = {IEEE},
title = {{Efficient optimization for rank-based loss functions}},
doi = {10.1109/cvpr.2018.00389},
year = {2018},
}
@inproceedings{1192,
abstract = {The main result of this paper is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are even Δ-matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvorak and Kupec. Knowing that edge CSP is tractable for even Δ-matroid constraints allows us to extend the tractability result to a larger class of Δ-matroids that includes many classes that were known to be tractable before, namely co-independent, compact, local and binary.},
author = {Kazda, Alexandr and Kolmogorov, Vladimir and Rolinek, Michal},
isbn = {978-161197478-2},
location = {Barcelona, Spain},
pages = {307 -- 326},
publisher = {SIAM},
title = {{Even delta-matroids and the complexity of planar Boolean CSPs}},
doi = {10.1137/1.9781611974782.20},
year = {2017},
}
@article{644,
abstract = {An instance of the valued constraint satisfaction problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. We study, assuming that P 6= NP, how the complexity of this very general problem depends on the set of functions allowed in the instances, the so-called constraint language. The case when all allowed functions take values in f0;1g corresponds to ordinary CSPs, where one deals only with the feasibility issue, and there is no optimization. This case is the subject of the algebraic CSP dichotomy conjecture predicting for which constraint languages CSPs are tractable (i.e., solvable in polynomial time) and for which they are NP-hard. The case when all allowed functions take only finite values corresponds to a finitevalued CSP, where the feasibility aspect is trivial and one deals only with the optimization issue. The complexity of finite-valued CSPs was fully classified by Thapper and Živný. An algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language was recently given by Kozik and Ochremiak. As our main result, we prove that if a constraint language satisfies this algebraic necessary condition, and the feasibility CSP (i.e., the problem of deciding whether a given instance has a feasible solution) corresponding to the VCSP with this language is tractable, then the VCSP is tractable. The algorithm is a simple combination of the assumed algorithm for the feasibility CSP and the standard LP relaxation. As a corollary, we obtain that a dichotomy for ordinary CSPs would imply a dichotomy for general-valued CSPs.},
author = {Kolmogorov, Vladimir and Krokhin, Andrei and Rolinek, Michal},
journal = {SIAM Journal on Computing},
number = {3},
pages = {1087 -- 1110},
publisher = {SIAM},
title = {{The complexity of general-valued CSPs}},
doi = {10.1137/16M1091836},
volume = {46},
year = {2017},
}
@inproceedings{274,
abstract = {We consider the problem of estimating the partition function Z(β)=∑xexp(−β(H(x)) of a Gibbs distribution with a Hamilton H(⋅), or more precisely the logarithm of the ratio q=lnZ(0)/Z(β). It has been recently shown how to approximate q with high probability assuming the existence of an oracle that produces samples from the Gibbs distribution for a given parameter value in [0,β]. The current best known approach due to Huber [9] uses O(qlnn⋅[lnq+lnlnn+ε−2]) oracle calls on average where ε is the desired accuracy of approximation and H(⋅) is assumed to lie in {0}∪[1,n]. We improve the complexity to O(qlnn⋅ε−2) oracle calls. We also show that the same complexity can be achieved if exact oracles are replaced with approximate sampling oracles that are within O(ε2qlnn) variation distance from exact oracles. Finally, we prove a lower bound of Ω(q⋅ε−2) oracle calls under a natural model of computation.},
author = {Kolmogorov, Vladimir},
pages = {1 -- 17},
publisher = {Unknown},
title = {{A faster approximation algorithm for the Gibbs partition function}},
year = {2017},
}
@inproceedings{1231,
abstract = {We study the time-and memory-complexities of the problem of computing labels of (multiple) randomly selected challenge-nodes in a directed acyclic graph. The w-bit label of a node is the hash of the labels of its parents, and the hash function is modeled as a random oracle. Specific instances of this problem underlie both proofs of space [Dziembowski et al. CRYPTO’15] as well as popular memory-hard functions like scrypt. As our main tool, we introduce the new notion of a probabilistic parallel entangled pebbling game, a new type of combinatorial pebbling game on a graph, which is closely related to the labeling game on the same graph. As a first application of our framework, we prove that for scrypt, when the underlying hash function is invoked n times, the cumulative memory complexity (CMC) (a notion recently introduced by Alwen and Serbinenko (STOC’15) to capture amortized memory-hardness for parallel adversaries) is at least Ω(w · (n/ log(n))2). This bound holds for adversaries that can store many natural functions of the labels (e.g., linear combinations), but still not arbitrary functions thereof. We then introduce and study a combinatorial quantity, and show how a sufficiently small upper bound on it (which we conjecture) extends our CMC bound for scrypt to hold against arbitrary adversaries. We also show that such an upper bound solves the main open problem for proofs-of-space protocols: namely, establishing that the time complexity of computing the label of a random node in a graph on n nodes (given an initial kw-bit state) reduces tightly to the time complexity for black pebbling on the same graph (given an initial k-node pebbling).},
author = {Alwen, Joel F and Chen, Binyi and Kamath Hosdurg, Chethan and Kolmogorov, Vladimir and Pietrzak, Krzysztof Z and Tessaro, Stefano},
location = {Vienna, Austria},
pages = {358 -- 387},
publisher = {Springer},
title = {{On the complexity of scrypt and proofs of space in the parallel random oracle model}},
doi = {10.1007/978-3-662-49896-5_13},
volume = {9666},
year = {2016},
}
@article{1794,
abstract = {We consider Conditional random fields (CRFs) with pattern-based potentials defined on a chain. In this model the energy of a string (labeling) (Formula presented.) is the sum of terms over intervals [i, j] where each term is non-zero only if the substring (Formula presented.) equals a prespecified pattern w. Such CRFs can be naturally applied to many sequence tagging problems. We present efficient algorithms for the three standard inference tasks in a CRF, namely computing (i) the partition function, (ii) marginals, and (iii) computing the MAP. Their complexities are respectively (Formula presented.), (Formula presented.) and (Formula presented.) where L is the combined length of input patterns, (Formula presented.) is the maximum length of a pattern, and D is the input alphabet. This improves on the previous algorithms of Ye et al. (NIPS, 2009) whose complexities are respectively (Formula presented.), (Formula presented.) and (Formula presented.), where (Formula presented.) is the number of input patterns. In addition, we give an efficient algorithm for sampling, and revisit the case of MAP with non-positive weights.},
author = {Kolmogorov, Vladimir and Takhanov, Rustem},
journal = {Algorithmica},
number = {1},
pages = {17 -- 46},
publisher = {Springer},
title = {{Inference algorithms for pattern-based CRFs on sequence data}},
doi = {10.1007/s00453-015-0017-7},
volume = {76},
year = {2016},
}
@inproceedings{1193,
abstract = {We consider the recent formulation of the Algorithmic Lovász Local Lemma [1], [2] for finding objects that avoid "bad features", or "flaws". It extends the Moser-Tardos resampling algorithm [3] to more general discrete spaces. At each step the method picks a flaw present in the current state and "resamples" it using a "resampling oracle" provided by the user. However, it is less flexible than the Moser-Tardos method since [1], [2] require a specific flaw selection rule, whereas [3] allows an arbitrary rule (and thus can potentially be implemented more efficiently). We formulate a new "commutativity" condition, and prove that it is sufficient for an arbitrary rule to work. It also enables an efficient parallelization under an additional assumption. We then show that existing resampling oracles for perfect matchings and permutations do satisfy this condition. Finally, we generalize the precondition in [2] (in the case of symmetric potential causality graphs). This unifies special cases that previously were treated separately.},
author = {Kolmogorov, Vladimir},
booktitle = {Proceedings - Annual IEEE Symposium on Foundations of Computer Science},
location = {New Brunswick, NJ, USA },
publisher = {IEEE},
title = {{Commutativity in the algorithmic Lovasz local lemma}},
doi = {10.1109/FOCS.2016.88},
volume = {2016-December},
year = {2016},
}
@article{1377,
abstract = {We consider the problem of minimizing the continuous valued total variation subject to different unary terms on trees and propose fast direct algorithms based on dynamic programming to solve these problems. We treat both the convex and the nonconvex case and derive worst-case complexities that are equal to or better than existing methods. We show applications to total variation based two dimensional image processing and computer vision problems based on a Lagrangian decomposition approach. The resulting algorithms are very effcient, offer a high degree of parallelism, and come along with memory requirements which are only in the order of the number of image pixels.},
author = {Kolmogorov, Vladimir and Pock, Thomas and Rolinek, Michal},
journal = {SIAM Journal on Imaging Sciences},
number = {2},
pages = {605 -- 636},
publisher = {Society for Industrial and Applied Mathematics },
title = {{Total variation on a tree}},
doi = {10.1137/15M1010257},
volume = {9},
year = {2016},
}
@inproceedings{1636,
abstract = {Constraint Satisfaction Problem (CSP) is a fundamental algorithmic problem that appears in many areas of Computer Science. It can be equivalently stated as computing a homomorphism R→ΓΓ between two relational structures, e.g. between two directed graphs. Analyzing its complexity has been a prominent research direction, especially for the fixed template CSPs where the right side ΓΓ is fixed and the left side R is unconstrained.
Far fewer results are known for the hybrid setting that restricts both sides simultaneously. It assumes that R belongs to a certain class of relational structures (called a structural restriction in this paper). We study which structural restrictions are effective, i.e. there exists a fixed template ΓΓ (from a certain class of languages) for which the problem is tractable when R is restricted, and NP-hard otherwise. We provide a characterization for structural restrictions that are closed under inverse homomorphisms. The criterion is based on the chromatic number of a relational structure defined in this paper; it generalizes the standard chromatic number of a graph.
As our main tool, we use the algebraic machinery developed for fixed template CSPs. To apply it to our case, we introduce a new construction called a “lifted language”. We also give a characterization for structural restrictions corresponding to minor-closed families of graphs, extend results to certain Valued CSPs (namely conservative valued languages), and state implications for (valued) CSPs with ordered variables and for the maximum weight independent set problem on some restricted families of graphs.},
author = {Kolmogorov, Vladimir and Rolinek, Michal and Takhanov, Rustem},
location = {Nagoya, Japan},
pages = {566 -- 577},
publisher = {Springer},
title = {{Effectiveness of structural restrictions for hybrid CSPs}},
doi = {10.1007/978-3-662-48971-0_48},
volume = {9472},
year = {2015},
}
@article{1841,
abstract = {We propose a new family of message passing techniques for MAP estimation in graphical models which we call Sequential Reweighted Message Passing (SRMP). Special cases include well-known techniques such as Min-Sum Diffusion (MSD) and a faster Sequential Tree-Reweighted Message Passing (TRW-S). Importantly, our derivation is simpler than the original derivation of TRW-S, and does not involve a decomposition into trees. This allows easy generalizations. The new family of algorithms can be viewed as a generalization of TRW-S from pairwise to higher-order graphical models. We test SRMP on several real-world problems with promising results.},
author = {Kolmogorov, Vladimir},
journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
number = {5},
pages = {919 -- 930},
publisher = {IEEE},
title = {{A new look at reweighted message passing}},
doi = {10.1109/TPAMI.2014.2363465},
volume = {37},
year = {2015},
}
@inproceedings{1675,
abstract = {Proofs of work (PoW) have been suggested by Dwork and Naor (Crypto’92) as protection to a shared resource. The basic idea is to ask the service requestor to dedicate some non-trivial amount of computational work to every request. The original applications included prevention of spam and protection against denial of service attacks. More recently, PoWs have been used to prevent double spending in the Bitcoin digital currency system. In this work, we put forward an alternative concept for PoWs - so-called proofs of space (PoS), where a service requestor must dedicate a significant amount of disk space as opposed to computation. We construct secure PoS schemes in the random oracle model (with one additional mild assumption required for the proof to go through), using graphs with high “pebbling complexity” and Merkle hash-trees. We discuss some applications, including follow-up work where a decentralized digital currency scheme called Spacecoin is constructed that uses PoS (instead of wasteful PoW like in Bitcoin) to prevent double spending. The main technical contribution of this work is the construction of (directed, loop-free) graphs on N vertices with in-degree O(log logN) such that even if one places Θ(N) pebbles on the nodes of the graph, there’s a constant fraction of nodes that needs Θ(N) steps to be pebbled (where in every step one can put a pebble on a node if all its parents have a pebble).},
author = {Dziembowski, Stefan and Faust, Sebastian and Kolmogorov, Vladimir and Pietrzak, Krzysztof Z},
location = {Santa Barbara, CA, United States},
pages = {585 -- 605},
publisher = {Springer},
title = {{Proofs of space}},
doi = {10.1007/978-3-662-48000-7_29},
volume = {9216},
year = {2015},
}
@inproceedings{1637,
abstract = {An instance of the Valued Constraint Satisfaction Problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. We study, assuming that P ≠ NP, how the complexity of this very general problem depends on the set of functions allowed in the instances, the so-called constraint language. The case when all allowed functions take values in {0, ∞} corresponds to ordinary CSPs, where one deals only with the feasibility issue and there is no optimization. This case is the subject of the Algebraic CSP Dichotomy Conjecture predicting for which constraint languages CSPs are tractable (i.e. solvable in polynomial time) and for which NP-hard. The case when all allowed functions take only finite values corresponds to finite-valued CSP, where the feasibility aspect is trivial and one deals only with the optimization issue. The complexity of finite-valued CSPs was fully classified by Thapper and Zivny. An algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language was recently given by Kozik and Ochremiak. As our main result, we prove that if a constraint language satisfies this algebraic necessary condition, and the feasibility CSP (i.e. the problem of deciding whether a given instance has a feasible solution) corresponding to the VCSP with this language is tractable, then the VCSP is tractable. The algorithm is a simple combination of the assumed algorithm for the feasibility CSP and the standard LP relaxation. As a corollary, we obtain that a dichotomy for ordinary CSPs would imply a dichotomy for general-valued CSPs.},
author = {Kolmogorov, Vladimir and Krokhin, Andrei and Rolinek, Michal},
location = {Berkeley, CA, United States},
pages = {1246 -- 1258},
publisher = {IEEE},
title = {{The complexity of general-valued CSPs}},
doi = {10.1109/FOCS.2015.80},
year = {2015},
}
@inproceedings{1859,
abstract = {Structural support vector machines (SSVMs) are amongst the best performing models for structured computer vision tasks, such as semantic image segmentation or human pose estimation. Training SSVMs, however, is computationally costly, because it requires repeated calls to a structured prediction subroutine (called \emph{max-oracle}), which has to solve an optimization problem itself, e.g. a graph cut.
In this work, we introduce a new algorithm for SSVM training that is more efficient than earlier techniques when the max-oracle is computationally expensive, as it is frequently the case in computer vision tasks. The main idea is to (i) combine the recent stochastic Block-Coordinate Frank-Wolfe algorithm with efficient hyperplane caching, and (ii) use an automatic selection rule for deciding whether to call the exact max-oracle or to rely on an approximate one based on the cached hyperplanes.
We show experimentally that this strategy leads to faster convergence to the optimum with respect to the number of requires oracle calls, and that this translates into faster convergence with respect to the total runtime when the max-oracle is slow compared to the other steps of the algorithm. },
author = {Shah, Neel and Kolmogorov, Vladimir and Lampert, Christoph},
location = {Boston, MA, USA},
pages = {2737 -- 2745},
publisher = {IEEE},
title = {{A multi-plane block-coordinate Frank-Wolfe algorithm for training structural SVMs with a costly max-oracle}},
doi = {10.1109/CVPR.2015.7298890},
year = {2015},
}
@article{2271,
abstract = {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. Finite-valued constraint languages contain functions that take on rational costs and general-valued constraint languages contain functions that take on rational or infinite costs. An instance of the problem is specified by a sum of functions from the language with the goal to minimise the sum. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs).
Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a general-valued constraint language Γ, BLP is a decision procedure for Γ if and only if Γ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language Γ, BLP is a decision procedure if and only if Γ admits a symmetric fractional polymorphism of some arity, or equivalently, if Γ admits a symmetric fractional polymorphism of arity 2.
Using these results, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees. },
author = {Kolmogorov, Vladimir and Thapper, Johan and Živný, Stanislav},
journal = {SIAM Journal on Computing},
number = {1},
pages = {1 -- 36},
publisher = {SIAM},
title = {{The power of linear programming for general-valued CSPs}},
doi = {10.1137/130945648},
volume = {44},
year = {2015},
}
@inproceedings{2275,
abstract = {Energies with high-order non-submodular interactions have been shown to be very useful in vision due to their high modeling power. Optimization of such energies, however, is generally NP-hard. A naive approach that works for small problem instances is exhaustive search, that is, enumeration of all possible labelings of the underlying graph. We propose a general minimization approach for large graphs based on enumeration of labelings of certain small patches.
This partial enumeration technique reduces complex high-order energy formulations to pairwise Constraint Satisfaction Problems with unary costs (uCSP), which can be efficiently solved using standard methods like TRW-S. Our approach outperforms a number of existing state-of-the-art algorithms on well known difficult problems (e.g. curvature regularization, stereo, deconvolution); it gives near global minimum and better speed.
Our main application of interest is curvature regularization. In the context of segmentation, our partial enumeration technique allows to evaluate curvature directly on small patches using a novel integral geometry approach.
},
author = {Olsson, Carl and Ulen, Johannes and Boykov, Yuri and Kolmogorov, Vladimir},
location = {Sydney, Australia},
pages = {2936 -- 2943},
publisher = {IEEE},
title = {{Partial enumeration and curvature regularization}},
doi = {10.1109/ICCV.2013.365},
year = {2014},
}
@inproceedings{2272,
abstract = {We consider Conditional Random Fields (CRFs) with pattern-based potentials defined on a chain. In this model the energy of a string (labeling) x1...xn is the sum of terms over intervals [i,j] where each term is non-zero only if the substring xi...xj equals a prespecified pattern α. Such CRFs can be naturally applied to many sequence tagging problems.
We present efficient algorithms for the three standard inference tasks in a CRF, namely computing (i) the partition function, (ii) marginals, and (iii) computing the MAP. Their complexities are respectively O(nL), O(nLℓmax) and O(nLmin{|D|,log(ℓmax+1)}) where L is the combined length of input patterns, ℓmax is the maximum length of a pattern, and D is the input alphabet. This improves on the previous algorithms of (Ye et al., 2009) whose complexities are respectively O(nL|D|), O(n|Γ|L2ℓ2max) and O(nL|D|), where |Γ| is the number of input patterns.
In addition, we give an efficient algorithm for sampling. Finally, we consider the case of non-positive weights. (Komodakis & Paragios, 2009) gave an O(nL) algorithm for computing the MAP. We present a modification that has the same worst-case complexity but can beat it in the best case. },
author = {Takhanov, Rustem and Kolmogorov, Vladimir},
booktitle = {ICML'13 Proceedings of the 30th International Conference on International},
location = {Atlanta, GA, USA},
number = {3},
pages = {145 -- 153},
publisher = {International Machine Learning Society},
title = {{Inference algorithms for pattern-based CRFs on sequence data}},
volume = {28},
year = {2013},
}