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Advances on strictly -modular IPs.

Martin Nägele1, Christian Nöbel1, Richard Santiago1

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Summary
This summary is machine-generated.

This study advances integer programming (IP) by developing new techniques for solving strictly modular IPs beyond prime cases. It introduces a randomized algorithm for feasibility checks when k is even.

Keywords:
Bounded subdeterminantsCongruency constraintsGroup constraintsInteger programmingTotal unimodularity

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Area of Science:

  • Optimization
  • Computer Science
  • Discrete Mathematics

Background:

  • Integer programs (IPs) with bounded subdeterminants are a recent focus.
  • A conjecture posits efficient solvability for k-modular IPs, where constraint matrix A has bounded k-minors.
  • Progress often relies on solving strictly k-modular IPs, a restricted subclass.

Purpose of the Study:

  • To extend efficient solvability of strictly k-modular IPs beyond the prime k case.
  • To develop novel techniques not reliant on strong number-theoretic results specific to primes.
  • To address the open conjecture for strictly k-modular IPs.

Main Methods:

  • Development of new algorithmic techniques for integer programming.
  • Focus on methods that do not depend on number-theoretic properties of prime numbers.
  • Introduction of a randomized feasibility check for strictly k-modular IPs.

Main Results:

  • First progress on strictly k-modular IPs beyond the prime k case.
  • Demonstration of techniques applicable to non-prime k.
  • A randomized algorithm for checking feasibility of strictly k-modular IPs when k is even, running in strongly polynomial time.

Conclusions:

  • The study makes significant inroads into the challenging problem of k-modular integer programming.
  • The developed techniques offer a path forward for non-prime cases, broadening applicability.
  • The feasibility algorithm for even k represents a notable advancement in computational complexity for this class of problems.