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Strengthening convex relaxations of 0/1-sets using Boolean formulas.

Samuel Fiorini1, Tony Huynh2, Stefan Weltge3

  • 1Université libre de Bruxelles, Brussels, Belgium.

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This study introduces an efficient method to strengthen convex relaxations in integer programming by using Boolean formulas to define feasible integer points. This approach enhances optimization problem-solving by combining general and specific techniques.

Keywords:
52Bxx68Q0690Cxx

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Area of Science:

  • Optimization
  • Integer Programming
  • Computational Mathematics

Background:

  • Convex relaxations are crucial for solving integer programming problems.
  • Existing methods for strengthening relaxations are either general-purpose or highly specific to particular problem structures.
  • A gap exists in methods that bridge general and specific strengthening techniques.

Purpose of the Study:

  • To develop a novel, efficient method for strengthening convex relaxations in integer programming.
  • To integrate specific information about feasible integer points (defined by Boolean formulas) into general convex sets.
  • To analyze the theoretical strength and practical implications of the proposed procedure.

Main Methods:

  • Proposing a new procedure that "feeds" a convex set into a Boolean formula defining the target set of feasible integer points.
  • Analyzing the properties and strength of the resulting strengthened relaxation.
  • Applying the method to combinatorial optimization problems, specifically covering problems.

Main Results:

  • The proposed method effectively strengthens convex relaxations by incorporating Boolean formula information.
  • Iterated application of the procedure forms a hierarchy, simplifying and improving prior results.
  • The method offers a unified approach interpolating between general and specific relaxation strengthening techniques.

Conclusions:

  • The new method provides an efficient way to enhance convex relaxations in integer programming.
  • It offers a flexible framework that can be adapted to various combinatorial optimization problems.
  • The findings extend and improve upon existing work on covering problems and relaxation hierarchies.