TY - JOUR AB - Given a fixed finite metric space (V,μ), the {\em minimum 0-extension problem}, denoted as 0-Ext[μ], is equivalent to the following optimization problem: minimize function of the form minx∈Vn∑ifi(xi)+∑ijcijμ(xi,xj) where cij,cvi are given nonnegative costs and fi:V→R are functions given by fi(xi)=∑v∈Vcviμ(xi,v). The computational complexity of 0-Ext[μ] has been recently established by Karzanov and by Hirai: if metric μ is {\em orientable modular} then 0-Ext[μ] can be solved in polynomial time, otherwise 0-Ext[μ] is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as L♮-convex functions. We consider a more general version of the problem in which unary functions fi(xi) can additionally have terms of the form cuv;iμ(xi,{u,v}) for {u,v}∈F, where set F⊆(V2) is fixed. We extend the complexity classification above by providing an explicit condition on (μ,F) for the problem to be tractable. In order to prove the tractability part, we generalize Hirai's theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai's algorithm for solving 0-Ext on orientable modular graphs. AU - Dvorak, Martin AU - Kolmogorov, Vladimir ID - 10045 JF - Mathematical Programming KW - minimum 0-extension problem KW - metric labeling problem KW - discrete metric spaces KW - metric extensions KW - computational complexity KW - valued constraint satisfaction problems KW - discrete convex analysis KW - L-convex functions SN - 0025-5610 TI - Generalized minimum 0-extension problem and discrete convexity ER - TY - CONF AB - A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We will consider sampling problems which come from Gibbs distributions, which are families of probability distributions over a discrete space Ω with probability mass function of the form μ^Ω_β(ω) ∝ e^{β H(ω)} for β in an interval [β_min, β_max] and H(ω) ∈ {0} ∪ [1, n]. The partition function is the normalization factor Z(β) = ∑_{ω ∈ Ω} e^{β H(ω)}, and the log partition ratio is defined as q = (log Z(β_max))/Z(β_min) We develop a number of algorithms to estimate the counts c_x using roughly Õ(q/ε²) samples for general Gibbs distributions and Õ(n²/ε²) samples for integer-valued distributions (ignoring some second-order terms and parameters), We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs and perfect matchings in a graph. AU - Harris, David G. AU - Kolmogorov, Vladimir ID - 14084 SN - 1868-8969 T2 - 50th International Colloquium on Automata, Languages, and Programming TI - Parameter estimation for Gibbs distributions VL - 261 ER - TY - CONF AB - We formalized general (i.e., type-0) grammars using the Lean 3 proof assistant. We defined basic notions of rewrite rules and of words derived by a grammar, and used grammars to show closure of the class of type-0 languages under four operations: union, reversal, concatenation, and the Kleene star. The literature mostly focuses on Turing machine arguments, which are possibly more difficult to formalize. For the Kleene star, we could not follow the literature and came up with our own grammar-based construction. AU - Dvorak, Martin AU - Blanchette, Jasmin ID - 13120 SN - 9783959772846 T2 - 14th International Conference on Interactive Theorem Proving TI - Closure properties of general grammars - formally verified VL - 268 ER - TY - CONF AB - We consider the problem of solving LP relaxations of MAP-MRF inference problems, and in particular the method proposed recently in [16], [35]. As a key computational subroutine, it uses a variant of the Frank-Wolfe (FW) method to minimize a smooth convex function over a combinatorial polytope. We propose an efficient implementation of this subroutine based on in-face Frank-Wolfe directions, introduced in [4] in a different context. More generally, we define an abstract data structure for a combinatorial subproblem that enables in-face FW directions, and describe its specialization for tree-structured MAP-MRF inference subproblems. Experimental results indicate that the resulting method is the current state-of-art LP solver for some classes of problems. Our code is available at pub.ist.ac.at/~vnk/papers/IN-FACE-FW.html. AU - Kolmogorov, Vladimir ID - 14448 SN - 1063-6919 T2 - Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition TI - Solving relaxations of MAP-MRF problems: Combinatorial in-face Frank-Wolfe directions VL - 2023 ER - TY - JOUR AB - We consider two models for the sequence labeling (tagging) problem. The first one is a Pattern-Based Conditional Random Field (PB), in which the energy of a string (chain labeling) x=x1⁢…⁢xn∈Dn is a sum of terms over intervals [i,j] where each term is non-zero only if the substring xi⁢…⁢xj equals a prespecified word w∈Λ. The second model is a Weighted Context-Free Grammar (WCFG) frequently used for natural language processing. PB and WCFG encode local and non-local interactions respectively, and thus can be viewed as complementary. We propose a Grammatical Pattern-Based CRF model (GPB) that combines the two in a natural way. We argue that it has certain advantages over existing approaches such as the Hybrid model of Benedí and Sanchez that combines N-grams and WCFGs. The focus of this paper is to analyze the complexity of inference tasks in a GPB such as computing MAP. We present a polynomial-time algorithm for general GPBs and a faster version for a special case that we call Interaction Grammars. AU - Takhanov, Rustem AU - Kolmogorov, Vladimir ID - 10737 IS - 1 JF - Intelligent Data Analysis SN - 1088-467X TI - Combining pattern-based CRFs and weighted context-free grammars VL - 26 ER - TY - JOUR AB - Weak convergence of inertial iterative method for solving variational inequalities is the focus of this paper. The cost function is assumed to be non-Lipschitz and monotone. We propose a projection-type method with inertial terms and give weak convergence analysis under appropriate conditions. Some test results are performed and compared with relevant methods in the literature to show the efficiency and advantages given by our proposed methods. AU - Shehu, Yekini AU - Iyiola, Olaniyi S. ID - 7577 IS - 1 JF - Applicable Analysis SN - 0003-6811 TI - Weak convergence for variational inequalities with inertial-type method VL - 101 ER - TY - CONF AB - The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs. AU - Harris, David G. AU - Iliopoulos, Fotis AU - Kolmogorov, Vladimir ID - 10072 SN - 1868-8969 T2 - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques TI - A new notion of commutativity for the algorithmic Lovász Local Lemma VL - 207 ER - TY - CONF AB - We study a class of convex-concave saddle-point problems of the form minxmaxy⟨Kx,y⟩+fP(x)−h∗(y) where K is a linear operator, fP is the sum of a convex function f with a Lipschitz-continuous gradient and the indicator function of a bounded convex polytope P, and h∗ is a convex (possibly nonsmooth) function. Such problem arises, for example, as a Lagrangian relaxation of various discrete optimization problems. Our main assumptions are the existence of an efficient linear minimization oracle (lmo) for fP and an efficient proximal map for h∗ which motivate the solution via a blend of proximal primal-dual algorithms and Frank-Wolfe algorithms. In case h∗ is the indicator function of a linear constraint and function f is quadratic, we show a O(1/n2) convergence rate on the dual objective, requiring O(nlogn) calls of lmo. If the problem comes from the constrained optimization problem minx∈Rd{fP(x)|Ax−b=0} then we additionally get bound O(1/n2) both on the primal gap and on the infeasibility gap. In the most general case, we show a O(1/n) convergence rate of the primal-dual gap again requiring O(nlogn) calls of lmo. To the best of our knowledge, this improves on the known convergence rates for the considered class of saddle-point problems. We show applications to labeling problems frequently appearing in machine learning and computer vision. AU - Kolmogorov, Vladimir AU - Pock, Thomas ID - 10552 T2 - 38th International Conference on Machine Learning TI - One-sided Frank-Wolfe algorithms for saddle problems ER - TY - CONF AB - The convex grabbing game is a game where two players, Alice and Bob, alternate taking extremal points from the convex hull of a point set on the plane. Rational weights are given to the points. The goal of each player is to maximize the total weight over all points that they obtain. We restrict the setting to the case of binary weights. We show a construction of an arbitrarily large odd-sized point set that allows Bob to obtain almost 3/4 of the total weight. This construction answers a question asked by Matsumoto, Nakamigawa, and Sakuma in [Graphs and Combinatorics, 36/1 (2020)]. We also present an arbitrarily large even-sized point set where Bob can obtain the entirety of the total weight. Finally, we discuss conjectures about optimum moves in the convex grabbing game for both players in general. AU - Dvorak, Martin AU - Nicholson, Sara ID - 9592 KW - convex grabbing game KW - graph grabbing game KW - combinatorial game KW - convex geometry T2 - Proceedings of the 33rd Canadian Conference on Computational Geometry TI - Massively winning configurations in the convex grabbing game on the plane ER - TY - JOUR AB - In this paper, we consider reflected three-operator splitting methods for monotone inclusion problems in real Hilbert spaces. To do this, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the reflected Krasnosel'skiĭ-Mann iteration for finding a fixed point of nonexpansive mapping in real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. We then apply our results to three-operator splitting for the monotone inclusion problem and consequently obtain the corresponding convergence analysis. Furthermore, we derive reflected primal-dual algorithms for highly structured monotone inclusion problems. Some numerical implementations are drawn from splitting methods to support the theoretical analysis. AU - Iyiola, Olaniyi S. AU - Enyi, Cyril D. AU - Shehu, Yekini ID - 9469 JF - Optimization Methods and Software SN - 1055-6788 TI - Reflected three-operator splitting method for monotone inclusion problem ER -