Solving the Set Packing Problem via a Maximum Weighted Independent Set Heuristic

The set packing problem (SPP) is a significant NP-hard combinatorial optimization problem with extensive applications. In this paper, we encode the set packing problem as the maximum weighted independent set (MWIS) problem and solve the encoded problem with an efficient algorithm designed to the MWIS problem. We compare the independent set-based method with the state-of-the-art algorithms for the set packing problem on the 64 standard benchmark instances. The experimental results show that the independent set-based method is superior to the existing algorithms in terms of the quality of the solutions and running time obtained the solutions.

Two-row and two-column mixed-integer presolve using hashing-based pairing methods

In state-of-the-art mixed-integer programming solvers, a large array of reduction techniques are applied to simplify the problem and strengthen the model formulation before starting the actual branch-and-cut phase. Despite their mathematical simplicity, these methods can have significant impact on the solvability of a given problem. However, a crucial property for employing presolve techniques successfully is their speed. Hence, most methods inspect constraints or variables individually in order to guarantee linear complexity. In this paper, we present new hashing-based pairing mechanisms that help to overcome known performance limitations of more powerful presolve techniques that consider pairs of rows or columns. Additionally, we develop an enhancement to one of these presolve techniques by exploiting the presence of set-packing structures on binary variables in order to strengthen the resulting reductions without increasing runtime. We analyze the impact of these methods on the MIPLIB 2017 benchmark set based on an implementation in the MIP solver SCIP.

Data Association via Set Packing for Computer Vision Applications

Significant progress has been made in the field of computer vision because of the development of supervised machine learning algorithms, which efficiently extract information from high-dimensional data such as images and videos. Such techniques are particularly effective at recognizing the presence or absence of entities in the domains where labeled data are abundant. However, supervised learning is not sufficient in applications where one needs to annotate each unique entity in crowded scenes respecting known domain-specific structures of those entities. This problem, known as data association, provides fertile ground for the application of combinatorial optimization. In this review paper, we present a unified framework based on column generation for some computer vision applications, namely multiperson tracking, multiperson pose estimation, and multicell segmentation, which can be formulated as set packing problems with a massive number of variables. To solve them, column generation algorithms are applied to circumvent the need to enumerate all variables explicitly. To enhance the solution process, we provide a general approach for applying subset-row inequalities to tighten the formulations and introduce novel dual-optimal inequalities to reduce the dual search space. The proposed algorithms and their enhancements are successfully applied to solve the three aforementioned computer vision problems and achieve superior performance over benchmark approaches. The common framework presented allows us to leverage operations research methodologies to efficiently tackle computer vision problems.

Accelerating Column Generation via Flexible Dual Optimal Inequalities with Application to Entity Resolution

In this paper, we introduce a new optimization approach to Entity Resolution. Traditional approaches tackle entity resolution with hierarchical clustering, which does not benefit from a formal optimization formulation. In contrast, we model entity resolution as correlation-clustering, which we treat as a weighted set-packing problem and write as an integer linear program (ILP). In this case, sources in the input data correspond to elements and entities in output data correspond to sets/clusters. We tackle optimization of weighted set packing by relaxing integrality in our ILP formulation. The set of potential sets/clusters can not be explicitly enumerated, thus motivating optimization via column generation. In addition to the novel formulation, we also introduce new dual optimal inequalities (DOI), that we call flexible dual optimal inequalities, which tightly lower-bound dual variables during optimization and accelerate column generation. We apply our formulation to entity resolution (also called de-duplication of records), and achieve state-of-the-art accuracy on two popular benchmark datasets. Our F-DOI can be extended to other weighted set-packing problems.