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Probability of unique integer solution to a system of linear equations

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  • Mangasarian, O.L.
  • Recht, Benjamin

Abstract

We consider a system of m linear equations in n variables Ax = d and give necessary and sufficient conditions for the existence of a unique solution to the system that is integer: x [set membership, variant] {-1, 1}n. We achieve this by reformulating the problem as a linear program and deriving necessary and sufficient conditions for the integer solution to be the unique primal optimal solution. We show that as long as m is larger than n/2, then the linear programming reformulation succeeds for most instances, but if m is less than n/2, the reformulation fails on most instances. We also demonstrate that these predictions match the empirical performance of the linear programming formulation to very high accuracy.

Suggested Citation

  • Mangasarian, O.L. & Recht, Benjamin, 2011. "Probability of unique integer solution to a system of linear equations," European Journal of Operational Research, Elsevier, vol. 214(1), pages 27-30, October.
    • Handle: RePEc:eee:ejores:v:214:y:2011:i:1:p:27-30
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    References listed on IDEAS

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    1. G Appa, 2002. "On the uniqueness of solutions to linear programs," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 53(10), pages 1127-1132, October.
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    Cited by:

    1. Fukshansky, Lenny & Needell, Deanna & Sudakov, Benny, 2019. "An algebraic perspective on integer sparse recovery," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 31-42.

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