TY - CONF
AB - Representation languages for coalitional games are a key research area in algorithmic game theory. There is an inher-
ent tradeoff between how general a language is, allowing it to capture more elaborate games, and how hard it is computationally to optimize and solve such games. One prominent such language is the simple yet expressive
Weighted Graph Games (WGGs) representation (Deng and Papadimitriou 1994), which maintains knowledge about synergies between agents in the form of an edge weighted graph. We consider the problem of finding the optimal coalition structure in WGGs. The agents in such games are vertices in a graph, and the value of a coalition is the sum of the weights of the edges present between coalition members. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that finding the optimal coalition structure is not only hard for general graphs, but is also intractable for restricted families such as planar graphs which are amenable for many other combinatorial problems. We then provide algorithms with constant factor approximations for planar, minorfree and bounded degree graphs.
AU - Bachrach, Yoram
AU - Kohli, Pushmeet
AU - Kolmogorov, Vladimir
AU - Zadimoghaddam, Morteza
ID - 2270
TI - Optimal Coalition Structures in Cooperative Graph Games
ER -
TY - CONF
AB - 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.
AU - Takhanov, Rustem
AU - Kolmogorov, Vladimir
ID - 2272
IS - 3
T2 - ICML'13 Proceedings of the 30th International Conference on International
TI - Inference algorithms for pattern-based CRFs on sequence data
VL - 28
ER -
TY - GEN
AB - 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 Diusion (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. We present such a generalization for the case of higher-order graphical models, and test it on several real-world problems with promising results.
AU - Vladimir Kolmogorov
ID - 2273
TI - Reweighted message passing revisited
ER -
TY - GEN
AB - 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, using graphs with high "pebbling complexity" and Merkle hash-trees.
AU - Dziembowski, Stefan
AU - Faust, Sebastian
AU - Kolmogorov, Vladimir
AU - Pietrzak, Krzysztof Z
ID - 2274
TI - Proofs of Space
ER -
TY - CONF
AB - The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NP-hard problem was proposed by Kovtun [19, 20]. It identifies a part of an optimal solution by running k maxflow computations, where k is the number of labels. The number of “labeled” pixels can be significant in some applications, e.g. 50-93% in our tests for stereo. We show how to reduce the runtime to O (log k) maxflow computations (or one parametric maxflow computation). Furthermore, the output of our algorithm allows to speed-up the subsequent alpha expansion for the unlabeled part, or can be used as it is for time-critical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for Tree Metrics . We also show a connection to k-submodular functions from combinatorial optimization, and discuss k-submodular relaxations for general energy functions.
AU - Gridchyn, Igor
AU - Kolmogorov, Vladimir
ID - 2276
TI - Potts model, parametric maxflow and k-submodular functions
ER -
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 - 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 -