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

Kobe Grobben, Phablo F. S. Moura, Hande Yaman

TL;DR

The paper studies a class of covering mixed-integer programs with a fixed number of constraints, showing that the optimal structure permits a decomposition into generalized multidimensional knapsack cover problems with a single continuous variable per dimension. This enables a PTAS for the general p when m is fixed, and a complementary FPTAS and ε-approximate linear formulation for the one-constraint case, improving prior 2-approximation results. It also provides a perfect, compact formulation for the one-dimensional case with uniform bounds and extends the results to packing and assignment variants. Collectively, the work connects classic knapsack techniques to fixed-constraint mixed-integer programs, offering both approximation schemes and tight formulations for broader classes like facility location and supplier selection problems.

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.

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

TL;DR

The paper studies a class of covering mixed-integer programs with a fixed number of constraints, showing that the optimal structure permits a decomposition into generalized multidimensional knapsack cover problems with a single continuous variable per dimension. This enables a PTAS for the general p when m is fixed, and a complementary FPTAS and ε-approximate linear formulation for the one-constraint case, improving prior 2-approximation results. It also provides a perfect, compact formulation for the one-dimensional case with uniform bounds and extends the results to packing and assignment variants. Collectively, the work connects classic knapsack techniques to fixed-constraint mixed-integer programs, offering both approximation schemes and tight formulations for broader classes like facility location and supplier selection problems.

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.

Paper Structure

This paper contains 9 sections, 18 theorems, 46 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Decomposition into multidimensional knapsack problems
  3. A polynomial-time approximation scheme
  4. Extension to packing problems
  5. Extension to assignment problems
  6. FPTAS and approximate formulation for $m=1$
  7. Perfect compact formulation for $m = 1$ and uniform bounds
  8. Conclusion and further research
  9. Appendix

Key Result

Proposition 2.1

Let $I$ be an instance of p, and let $S=\{i \in [n] : f_i=0\}$. It holds that the optimal value $\mathop{\mathrm{OPT}}\nolimits(I)=0$ if and only if, for each $j \in [m]$, $\sum_{i \in Z_j} c_{ij} \ge d_j$, where $Z_j=\{i \in S : v_{ij}=0\}$.

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Theorem 3.1
  • ...and 23 more