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 -