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Discrete Optimization in Computer Vision: Theory and Practice

Mid-Term Report Summary - DOICV (Discrete Optimization in Computer Vision: Theory and Practice)

Below I highlight outcomes of the DOiCV project which I view as the most significant.

A substantial progress has been made on WP4 ("Theory of discrete optimization"):
- In [Kolmogorov, Thapper, Zivny SICOMP'15] we provided a complete characterization of languages that can be solved exactly by a natural Linear Programming (LP) relaxation of the problem.
- In [Kolmogorov, Krokhin, Rolinek FOCS'15] we settled the complexity of general-valued CSPs modulo the complexity of ordinary CSPs, essentially closing one line of research.
- In [Kazda, Kolmogorov, Rolinek SODA'17] we developed a polynomial time algorithm for "edge CSPs" with a certain class of Delta-matroid constraints. It extends Edmonds's blossom-shrinking algorithm for computing perfect matchings in a graph. We can handle all previously known tractable classes of Delta-matroids, as well as new classes (such as even Delta-matroids). One implication of our result is resolving the complexity classification of Boolean CSPs restricted to planar instances.

Another goal of project was to design and implement efficient optimization algorithms for real-world applications (see WP5). Selected achievements on this objective are as follows:
- [Kolmogorov PAMI'15] developed an efficient message passing algorithm for arbitrary graphical models called "SRMP". For some classes of problems it showed a state-of-the-art performance.
- [Shah, Kolmogorov, Lampert CVPR'15] tackled the problem of training Structural Support Vector Machines (SSVMs), which amounts to minimizing a certain convex objective. In practice this is a computationally costly task, since it requires repeated calls to a structured prediction subroutine (called max-oracle) that can be quite slow. The technique that we developed improved on the previous state-of-the-art by several orders of magnitude on some real applications.

The codes for both algorithms have been made publicly available under the GPL license.

Three side projects were not directly related to DOiCV:
- In [Kolmogorov FOCS'16] I provided a deeper understanding of the algorithmic version of the Lovasz Local Lemma (LLL). LLL can be loosely connected to the topic of DOiCV project, since it can be applied to certain CSPs. However, the involved techniques (the probabilistic method) are entirely different from those envisaged in the proposal.
- In [Dziembowski et al. CRYPTO'15] and [Alwen et al. EUROCRYPT'16] we studied pebbling games on graphs and their application in cryptography.