Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Hinge-Loss Markov Random Fields and Probabilistic Soft Logic: A Scalable Approach to Structured Prediction

A fundamental challenge in developing impactful artificial intelligence technologies is balancing the ability to model rich, structured domains with the ability to scale to big data. Many important problem areas are both richly structured and large scale, from social and biological networks, to knowledge graphs and the Web, to images, video, and natural language. In this thesis I introduce two new formalisms for modeling structured data, distinguished from previous approaches by their ability to both capture rich structure and scale to big data. The first, hinge-loss Markov random fields (HL-MRFs), is a new kind of probabilistic graphical model that generalizes different approaches to convex inference. I unite three views of inference from the randomized algorithms, probabilistic graphical models, and fuzzy logic communities, showing that all three views lead to the same inference objective. I then derive HL-MRFs by generalizing this unified objective. The second new formalism, probabilistic soft logic (PSL), is a probabilistic programming language that makes HL-MRFs easy to define, refine, and reuse for relational data. PSL uses a syntax based on first-order logic to compactly specify complex models. I next introduce an algorithm for inferring most-probable variable assignments (MAP inference) for HL-MRFs that is extremely scalable, much more so than commercially available software, because it uses message passing to leverage the sparse dependency structures common in inference tasks. I then show how to learn the parameters of HL-MRFs using a number of learning objectives. The learned HL-MRFs are as accurate as traditional, discrete models, but much more scalable. To enable HL-MRFs and PSL to capture even richer dependencies, I then extend learning to support latent variables, i.e., variables without training labels. To overcome the bottleneck of repeated inferences required during learning, I introduce paired-dual learning, which interleaves inference and parameter updates. Paired-dual learning learns accurate models and is also scalable, often completing before traditional methods make even one parameter update. Together, these algorithms enable HL-MRFs and PSL to model rich, structured data at scales not previously possible

Domain-independent local search for linear integer optimization

Integer and combinatorial optimization problems constitute a major challenge for algorithmics. They arise when a large number of discrete organizational decisions have to be made, subject to constraints and optimization criteria. This thesis describes and investigates new domain-independent local search strategies for linear integer optimization. We introduce WSAT(OIP), an integer local search method which operates on an algebraic problem representation. WSAT(OIP) generalizes Walksat, a successful local search procedure for propositional satisfiability (SAT), to more expressive constraint systems. For this purpose, we introduce over-constrained integer programs (OIPs), a constraint class which is closely related to integer programs. OIP allows for a natural generalization of the principles of SAT local search to integer optimization. Further, it will be shown that OIPs are a special case of integer linear programs and permit combinations with linear programming for bound computation, initialization by rounding, search space reduction, and feasibility testing. The representation is similar enough to integer programs to make use of existing algebraic modeling languages as front-end to a local search solver. To improve performance on realistic problems, WSAT(OIP) incorporates strategies from Tabu Search. We experimentally investigate WSAT(OIP) for a variety of realistic integer optimization problems from the domains of time tabling, sports scheduling, radar surveillance, course assignment, and capacitated production planning. The experimental design examines efficiency, scaling (with increasing problem size and constrainedness), and robustness. The results demonstrate that integer local search can outperform or compete with state-of-the-art integer programming (IP) branch-and-bound and constraint programming (CP) approaches to these problems in finding near-optimal solutions. Key findings of our empirical study include that integer local search is able to solve difficult constraint problems from time-tabling and sports scheduling when cast into a 0-1 representation, which are beyond the scope of IP branch-and-bound strategies and for which devising robust constraint programs is a non-trivial task. For several realistic optimization problems (0-1 integer and finite domain) we show that integer local search exhibits graceful runtime scaling with increasing problem size and constrainedness. It can therefore significantly outperform IP branch-and-bound strategies on large or tightly constrained problems in finding near-optimal solutions. The problems under consideration are mostly beyond the limitations of a previous general-purpose simulated annealing strategy for 0-1 integer programs.Ganzzahlige und kombinatorische Optimierungsprobleme stellen eine schwierige Herausforderung im Gebiet der Algorithmen dar. Sie treten auf, wenn eine große Anzahl diskreter organisatorischer Entscheidungen unter Berücksichtigung von Constraints und Optimierungskriterien zu treffen sind. Diese Arbeit beschreibt und untersucht neue, domänenunabhängige Strategien der lokalen Suche zur ganzzahligen linearen Optimierung. Wir beschreiben WSAT(OIP), eine Strategie "ganzzahliger lokaler Suche\u27;, die auf einer algebraischen Problemrepräsentation operiert. WSAT(OIP) verallgemeinert Walksat, eine erfolgreiche Prozedur lokaler Suche für das Erfüllbarkeitsproblem der Aussagenlogik (SAT), auf ausdrucksstärkere Constraint-Systeme. Für diesen Zweck führen wir die Klasse der "Over-constrained Integer Programs\u27;(OIPs) ein, eine Constraint-Klasse, die eng mit ganzzahligen Programmen verwandt ist. OIPs erlauben einerseits eine natürliche Verallgemeinerung der Prinzipien von lokaler Suche für SAT. Andererseits sind sie ein Spezialfall der ganzzahligen linearen Programme und ermöglichen die Kombination mit linearer Programmierung zur Berechnung von Schranken, Initialisierung durch Rundung, Suchraum-Reduktion und für Gültigkeits-Tests. OIPs sind ganzzahligen Programmen ähnlich, so daß existierende algebraische Modellierungssprachen als Eingabeschnittstelle für einen Problemlöser benutzt werden können, der auf lokaler Suche basiert. Um die Performanz auf realistischen Problemen zu verbessern, ist WSAT(OIP) mit Strategien der Tabu-Suche ausgestattet. Wir führen eine experimentelle Untersuchung von WSAT(OIP) auf einer Reihe von realistischen ganzzahligen Constraint- und Optimierungsproblemen durch. Die Probleme stammen aus den Domänen Zeitplan-Erstellung, Sport-Ablaufplanung, Radar- Überwachung, Kurs-Zuteilung und Produktions-Planung. Das experimentelle Design untersucht Effizienz, Skalierung mit zunehmender Problemgröße und stärkeren Constraints sowie Robustheit. Die Ergebnisse zeigen, daß ganzzahlige lokale Suche bezüglich Performanz auf diesen Problemklassen zeitgemäße Ansätze der ganzzahligen Programmierung und der Constraint-Programmierung beim Finden nahe-optimaler Lösungen schlägt oder mit ihnen konkurriert. Kernergebnisse der empirischen Untersuchung sind, daß ganzzahlige lokale Suche in der Lage ist, schwierige Constraint-Probleme der Zeitplan-Erstellung und Sport-Ablaufplanung in einer 0-1 Repräsentation zu lösen, die außerhalb der Grenzen der ganzzahligen linearen Programmierung liegen, und für die die Entwicklung eines robustes Constraint-Programms eine nicht-triviale Aufgabe darstellt. Für mehrere realistische Optimierungsprobleme (ganzzahlig 0-1 und endliche Bereiche)zeigen wir, daß ganzzahlige lokale Suche eine günstige Skalierung der Laufzeit mit zunehmender Problemgröße und Constrainedness aufweist. Dadurch zeigt das Verfahren auf großen Problemen und auf Problemen mit starken Constraints deutlich bessere Performanz für das Finden nahe-Lösungen als die Branch-and-Bound Strategie der ganzzahligen Programmierung. Die untersuchten Probleme liegen zumeist außerhalb der Grenzen einer existierenden Simulated Annealing Strategie für allgemeine lineare 0-1 Programme

Computer Aided Verification

The open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Computer Aided Verification

The open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Discrete Optimization in Early Vision - Model Tractability Versus Fidelity

Early vision is the process occurring before any semantic interpretation of an image takes place. Motion estimation, object segmentation and detection are all parts of early vision, but recognition is not. Some models in early vision are easy to perform inference with---they are tractable. Others describe the reality well---they have high fidelity. This thesis improves the tractability-fidelity trade-off of the current state of the art by introducing new discrete methods for image segmentation and other problems of early vision. The first part studies pseudo-boolean optimization, both from a theoretical perspective as well as a practical one by introducing new algorithms. The main result is the generalization of the roof duality concept to polynomials of higher degree than two. Another focus is parallelization; discrete optimization methods for multi-core processors, computer clusters, and graphical processing units are presented. Remaining in an image segmentation context, the second part studies parametric problems where a set of model parameters and a segmentation are estimated simultaneously. For a small number of parameters these problems can still be optimally solved. One application is an optimal method for solving the two-phase Mumford-Shah functional. The third part shifts the focus to curvature regularization---where the commonly used length and area penalization is replaced by curvature in two and three dimensions. These problems can be discretized over a mesh and special attention is given to the mesh geometry. Specifically, hexagonal meshes in the plane are compared to square ones and a method for generating adaptive meshes is introduced and evaluated. The framework is then extended to curvature regularization of surfaces. Finally, the thesis is concluded by three applications to early vision problems: cardiac MRI segmentation, image registration, and cell classification

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing