A Study of Lagrangean Decompositions and Dual Ascent Solvers for Graph Matching

We study the quadratic assignment problem, in computer vision also known as graph matching. Two leading solvers for this problem optimize the Lagrange decomposition duals with sub-gradient and dual ascent (also known as message passing) updates. We explore s direction further and propose several additional Lagrangean relaxations of the graph matching problem along with corresponding algorithms, which are all based on a common dual ascent framework. Our extensive empirical evaluation gives several theoretical insights and suggests a new state-of-the-art any-time solver for the considered problem. Our improvement over state-of-the-art is particularly visible on a new dataset with large-scale sparse problem instances containing more than 500 graph nodes each.Comment: Added acknowledgment

A study of lagrangean decompositions and dual ascent solvers for graph matching

We study the quadratic assignment problem, in computer vision also known as graph matching. Two leading solvers for this problem optimize the Lagrange decomposition duals with sub-gradient and dual ascent (also known as message passing) updates. We explore this direction further and propose several additional Lagrangean relaxations of the graph matching problem along with corresponding algorithms, which are all based on a common dual ascent framework. Our extensive empirical evaluation gives several theoretical insights and suggests a new state-of-the-art anytime solver for the considered problem. Our improvement over state-of-the-art is particularly visible on a new dataset with large-scale sparse problem instances containing more than 500 graph nodes each

MAP inference via Block-Coordinate Frank-Wolfe Algorithm

We present a new proximal bundle method for Maximum-A-Posteriori (MAP) inference in structured energy minimization problems. The method optimizes a Lagrangean relaxation of the original energy minimization problem using a multi plane block-coordinate Frank-Wolfe method that takes advantage of the specific structure of the Lagrangean decomposition. We show empirically that our method outperforms state-of-the-art Lagrangean decomposition based algorithms on some challenging Markov Random Field, multi-label discrete tomography and graph matching problems

A dual ascent framework for Lagrangean decomposition of combinatorial problems

We propose a general dual ascent framework for Lagrangean decomposition of combinatorial problems. Although methods of this type have shown their efficiency for a number of problems, so far there was no general algorithm applicable to multiple problem types. In this work, we propose such a general algorithm. It depends on several parameters, which can be used to optimize its performance in each particular setting. We demonstrate efficacy of our method on graph matching and multicut problems, where it outperforms state-of-the-art solvers including those based on subgradient optimization and off-the-shelf linear programming solvers

A dual ascent framework for Lagrangean decomposition of combinatorial problems

We propose a general dual ascent framework for Lagrangean decomposition of combinatorial problems. Although methods of this type have shown their efficiency for a number of problems, so far there was no general algorithm applicable to multiple problem types. In this work, we propose such a general algorithm. It depends on several parameters, which can be used to optimize its performance in each particular setting. We demonstrate efficacy of our method on graph matching and multicut problems, where it outperforms state-of-the-art solvers including those based on subgradient optimization and off-the-shelf linear programming solvers

FastDOG: Fast Discrete Optimization on GPU

We present a massively parallel Lagrange decomposition method for solving 0--1 integer linear programs occurring in structured prediction. We propose a new iterative update scheme for solving the Lagrangean dual and a perturbation technique for decoding primal solutions. For representing subproblems we follow Lange et al. (2021) and use binary decision diagrams (BDDs). Our primal and dual algorithms require little synchronization between subproblems and optimization over BDDs needs only elementary operations without complicated control flow. This allows us to exploit the parallelism offered by GPUs for all components of our method. We present experimental results on combinatorial problems from MAP inference for Markov Random Fields, quadratic assignment and cell tracking for developmental biology. Our highly parallel GPU implementation improves upon the running times of the algorithms from Lange et al. (2021) by up to an order of magnitude. In particular, we come close to or outperform some state-of-the-art specialized heuristics while being problem agnostic. Our implementation is available at https://github.com/LPMP/BDD.Comment: Published at CVPR 2022. Alert before printing: last 10 pages just contains detailed results tabl

Deep Graph Matching via Blackbox Differentiation of Combinatorial Solvers

Building on recent progress at the intersection of combinatorial optimization and deep learning, we propose an end-to-end trainable architecture for deep graph matching that contains unmodified combinatorial solvers. Using the presence of heavily optimized combinatorial solvers together with some improvements in architecture design, we advance state-of-the-art on deep graph matching benchmarks for keypoint correspondence. In addition, we highlight the conceptual advantages of incorporating solvers into deep learning architectures, such as the possibility of post-processing with a strong multi-graph matching solver or the indifference to changes in the training setting. Finally, we propose two new challenging experimental setups. The code is available at https://github.com/martius-lab/blackbox-deep-graph-matchingComment: ECCV 2020 conference pape

Analysis of large scale linear programming problems with embedded network structures: Detection and solution algorithms

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Linear programming (LP) models that contain a (substantial) network structure frequently arise in many real life applications. In this thesis, we investigate two main questions; i) how an embedded network structure can be detected, ii) how the network structure can be exploited to create improved sparse simplex solution algorithms. In order to extract an embedded pure network structure from a general LP problem we develop two new heuristics. The first heuristic is an alternative multi-stage generalised upper bounds (GUB) based approach which finds as many GUB subsets as possible. In order to identify a GUB subset two different approaches are introduced; the first is based on the notion of Markowitz merit count and the second exploits an independent set in the corresponding graph. The second heuristic is based on the generalised signed graph of the coefficient matrix. This heuristic determines whether the given LP problem is an entirely pure network; this is in contrast to all previously known heuristics. Using generalised signed graphs, we prove that the problem of detecting the maximum size embedded network structure within an LP problem is NP-hard. The two detection algorithms perform very well computationally and make positive contributions to the known body of results for the embedded network detection. For computational solution a decomposition based approach is presented which solves a network problem with side constraints. In this approach, the original coefficient matrix is partitioned into the network and the non-network parts. For the partitioned problem, we investigate two alternative decomposition techniques namely, Lagrangean relaxation and Benders decomposition. Active variables identified by these procedures are then used to create an advanced basis for the original problem. The computational results of applying these techniques to a selection of Netlib models are encouraging. The development and computational investigation of this solution algorithm constitute further contribution made by the research reported in this thesis.This study is funded by the Turkish Educational Council and Mugla University