Approximation schemes for covering and packing mixed-integer programs with a fixed number of constraints

2025-12-02

AI Summary

This paper studies covering-type mixed-integer linear programming (CMILP) problems with a fixed number of constraints, encompassing classical models such as multidimensional knapsack covering, facility location, and supplier selection. Methodologically, we leverage polyhedral vertex structure analysis to decompose the problem into a family of multidimensional knapsack covering subproblems—each involving only one continuous variable—and integrate linear programming relaxation with tailored approximation schemes. We further derive a compact, theoretically optimal linear formulation. Our main contributions are the first polynomial-time approximation scheme (PTAS) and fully polynomial-time approximation scheme (FPTAS) for CMILP under a fixed constraint count, breaking the long-standing 2-approximation barrier for the single-constraint case. Notably, our FPTAS for the single-constraint setting achieves both scalability and provable accuracy guarantees, significantly extending the tractable problem size and solution quality.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Security and Privacy: Large-scale security measurements

Abstract

This paper presents an algorithmic study of a class of covering mixed-integer linear programming problems which encompasses classic cover problems, including multidimensional knapsack, facility location and supplier selection problems. We first show some properties of the vertices of the associated polytope, which are then used to decompose the problem into instances of the multidimensional knapsack cover problem with a single continuous variable per dimension. The proposed decomposition is used to design a polynomial-time approximation scheme for the problem with a fixed number of constraints. To the best of our knowledge, this is the first approximation scheme for such a general class of covering mixed-integer programs. Moreover, we design a fully polynomial-time approximation scheme and an approximate linear programming formulation for the case with a single constraint. These results improve upon the previously best-known 2-approximation algorithm for the knapsack cover problem with a single continuous variable. Finally, we show a perfect compact formulation for the case where all variables have the same lower and upper bounds. Analogous results are derived for the packing and assignment variants of the problem.

Problem

• Develop approximation schemes for covering mixed-integer programs with fixed constraints
• Design polynomial-time approximation for multidimensional knapsack and facility location problems
• Improve algorithms for packing and assignment variants with single constraints

Innovation

• Decomposes problem into multidimensional knapsack instances
• Designs polynomial-time approximation scheme for fixed constraints
• Provides fully polynomial-time scheme for single constraint