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

Abstract

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.

Efficient Pre-Solve Algorithms for the Schwerin and Falkenauer_U Bin Packing Benchmark Problems for Getting Optimal Solutions with High Probability

“…Reference [55] considers pre-solve for Mixed Integer Programming problems. Some further publications on the same topic are Reference [56,57], from the same authors, and Reference [58].…”

Special issue on: Computational discrete optimization

“…Besides branch and bound, cutting planes, and primal heuristics, presolve techniques have a significant impact on MILP solver performance. In Gemander et al (2020), Gemander et al consider a new hashing-based pairing mechanism to consider two rows or columns simultaneously. By this, formulations can be strengthened further and the solution times of larger instances can be decreased.…”