Determining Sharp Proximity Bounds For Low Row Rank and Delta-Modularity
Presenter Status
Undergraduate, Mathematics
Second Presenter Status
Professor, Mathematics
Third Presenter Status
Krener Assistant Professor, Mathematics
Preferred Session
Poster Session
Start Date
20-10-2023 2:00 PM
End Date
20-10-2023 3:00 PM
Presentation Abstract
A question of much interest in integer programming is that of proximity bounding: how big can the distance be between a vertex optimal solution of the corresponding linear program relaxation and the closest integer optimal solution in the integer linear program? The proximity bound has many applications, for example, it can be used to bound the integrality gap and estimate nearby integer solutions in a dynamic programming framework. The maximum absolute subdeterminant of the coefficient matrix is known to be an important parameter to measure the proximity bound. Here, we consider the integer program in the standard form max{cx : Ax = b, x >= 0} with a bounded feasible region, where the coefficient matrix A is of full row rank m and has a maximum order-m absolute subdeterminant of Delta (where we call the rank-m matrix A Delta-modular). Using Wolsey's b-hull results and the polyhedral fan theory, we propose a method to determine the proximity bound for a fixed A and parametric b,c. Furthermore, applying a modified algorithm by Averkov and Schymura for classifying such rank-m Delta-modular matrices up to unimodular transformation, we calculate sharp proximity bounds for small m and Delta using SageMath.
Determining Sharp Proximity Bounds For Low Row Rank and Delta-Modularity
A question of much interest in integer programming is that of proximity bounding: how big can the distance be between a vertex optimal solution of the corresponding linear program relaxation and the closest integer optimal solution in the integer linear program? The proximity bound has many applications, for example, it can be used to bound the integrality gap and estimate nearby integer solutions in a dynamic programming framework. The maximum absolute subdeterminant of the coefficient matrix is known to be an important parameter to measure the proximity bound. Here, we consider the integer program in the standard form max{cx : Ax = b, x >= 0} with a bounded feasible region, where the coefficient matrix A is of full row rank m and has a maximum order-m absolute subdeterminant of Delta (where we call the rank-m matrix A Delta-modular). Using Wolsey's b-hull results and the polyhedral fan theory, we propose a method to determine the proximity bound for a fixed A and parametric b,c. Furthermore, applying a modified algorithm by Averkov and Schymura for classifying such rank-m Delta-modular matrices up to unimodular transformation, we calculate sharp proximity bounds for small m and Delta using SageMath.