Models and algorithms for grouping disjunctive binary vectors in digital printing optimization tasks

Abstract

The paper investigates models and algorithms for the optimal grouping of disjunctive binary vectors arising in multilayer printing planning tasks. In modern printing technologies, each printing pass consists of a set of mutually compatible operations, such as the application of inks, varnishes, protective coatings, and special effects. The chemical, technological, and hardware compatibility constraints make the problem of minimizing the number of printing passes non-trivial, since any two operations that conflict in their properties cannot be executed simultaneously. A binary vector representation is used to formally describe the operations, where each vector component corresponds to a particular functional or technological attribute. Incompatibility between elements is determined based on the intersection of unit components in the corresponding vectors. Experimental studies demonstrated that the dynamic programming method provides optimal results for problems with no more than 25 vectors, but becomes impractical for larger instances due to the exponential growth of computational cost. For medium-sized problems, an integer linear programming model was employed, implemented in Python using the Gurobi solver. The experiments showed that for matrices up to 100×100 the solution can be obtained within an acceptable time, from a few seconds to several tens of minutes, whereas a 150 × 150 matrix may require up to two hours of computation. For large-scale problems, a specialized parallel algorithm is proposed. It constructs groups of vectors across multiple threads without the need to build the full incompatibility graph. The method is based on independent matrix scanning by each thread with strict row-selection rules, which minimizes synchronization conflicts and ensures high performance on sparse data. Experiments showed that for a 10⁴ × 10⁵ matrix with approximately 3% sparsity, the proposed algorithm finds a solution within 20 minutes, which is significantly faster than classical DSATUR-based approaches. While DSATUR may yield up to 15% better optimization results, it requires substantially more time to construct the graph and perform the coloring. The obtained results demonstrate the effectiveness of combining rigorous mathematical modeling, optimization solvers, and specially designed parallel methods for solving large-scale applied problems in printing and other domains where similar conflict-resolution challenges arise.