MILP-StuDio: MILP Instance Generation via Block Structure Decomposition

TL;DR

A novel MILP generation framework, called Block Structure Decomposition (MILP-StuDio), to generate high-quality instances by preserving the block structures and is able to significantly reduce over 10% of the solving time for learning-based solvers.

Abstract

Mixed-integer linear programming (MILP) is one of the most popular mathematical formulations with numerous applications. In practice, improving the performance of MILP solvers often requires a large amount of high-quality data, which can be challenging to collect. Researchers thus turn to generation techniques to generate additional MILP instances. However, existing approaches do not take into account specific block structures -- which are closely related to the problem formulations -- in the constraint coefficient matrices (CCMs) of MILPs. Consequently, they are prone to generate computationally trivial or infeasible instances due to the disruptions of block structures and thus problem formulations. To address this challenge, we propose a novel MILP generation framework, called Block Structure Decomposition (MILP-StuDio), to generate high-quality instances by preserving the block structures. Specifically, MILP-StuDio begins by identifying the blocks in CCMs and decomposing the instances into block units, which serve as the building blocks of MILP instances. We then design three operators to construct new instances by removing, substituting, and appending block units in the original instances, enabling us to generate instances with flexible sizes. An appealing feature of MILP-StuDio is its strong ability to preserve the feasibility and computational hardness of the generated instances. Experiments on the commonly-used benchmarks demonstrate that using instances generated by MILP-StuDio is able to significantly reduce over 10% of the solving time for learning-based solvers.

Figures (10)

• Figure 1: Figure \ref{['fig:intro1']} visualizes the CCMs of four instances from the FA problem, where the white points represent the nonzero entries in CCMs. As we can see, the CCMs exhibit similar block structures across instances, with the patterns in red boxes being the block units. Figure \ref{['fig:intro2']} illustrates the block decomposition process, advanced features, and applications of our proposed MILP-StuDio.
• Figure 2: Visualization of the CCMs of instances in four widely recognized benchmarks. The block structures can be commonly seen in MILP problems.
• Figure 3: Visualization of CCMs from original instances (left), instances generated by G2MILP (middle), and instances generated by MILP-StuDio (right).
• Figure 4: An overview of MILP-StuDio. (1) We detect the block structures in the original instances and decompose the CCMs into sub-matrices of block units. (2) The sub-matrices are transferred into the corresponding sub-graphs of instances' bipartite graph representations. These sub-graphs are used to construct the structure library. (3) We sample instances and sub-graphs of block units and perform block manipulations, including block reduction, mix-up and expansion.
• Figure 5: Mean solving time during iterations.