Probability of unique integer solution to a system of linear equations

Abstract

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 ∈ {−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.

The Convex algebraic geometry of linear inverse problems

“…3) For a unique sign-vector solution in R p to a linear system of equations, we require that the linear system contain at least p 2 random equations. This agrees with results obtained in [8], [14]. 4) In order to recover a p × p permutation matrix via a convex formulation involving the Birkhoff polytope, we appeal to Proposition 3.2 and the fact that there are p!…”

“…Suppose that there is such a sign-vector, and we wish to recover this vector given linear measurements. This corresponds to a version of the multiknapsack problem [14]. In this case the algebraic set A is the set of set of all sign-vectors, and the convex hull C(A ) is the hypercube.…”

“…• When does a system of linear equations have a unique integer solution? This question is related to the integrality gap between a class of integer linear programs and their linear-programming relaxations, and has previously been studied in [8], [14]. • When can we recover a measure given a few linear measurements of its moments?…”

Signal representations with minimum &#x2113;<inf>&#x221E;</inf>-norm

“…In such situations, one is typically interested in solving problems of the form (P ε ∞ ) rather than solving minimum-energy problems; corresponding applications have been described in, e.g., [6]- [8]. Note that the problem (P ε ∞ ) with ε = 0 can also be used to recover antipodal solutions, i.e., vectors with coefficients belonging to {−α, +α}, from an underdetermined system of linear equations y = Dx provided that certain conditions on the matrix D are met (see, e.g., [9], [10]). In this paper, however, we focus on the computation of signal representations having minimal ∞ -norm and small dynamic range, rather than on the recovery a given vector x from y = Dx.…”

BCS: Compressive sensing for binary sparse signals

“…The solution is not able to exactly reconstruct the original signal in X ∈ B N . In a very recent work [7], the authors employ integer programming model to solve the under-determined binary systems of linear equations. The work is basically the solution for the general binary systems of equations.…”