This paper by L. Babai addresses the application of Lovász' lattice reduction to find a point within a factor of \(c^d\) of a given point in \(\mathbf{R}^d\), where \(c\) is a constant. The author proves that two straightforward heuristic procedures achieve this goal when applied to a lattice given by a Lovász-reduced basis. One of these procedures requires proving a geometric property of Lovász-reduced bases: a lower bound on the angle between any basis vector and the hyperplane generated by the other vectors. As applications, the paper provides a solution to the nonhomogeneous simultaneous diophantine approximation problem and improves the Grötschel—Lovász—Schrijver version of H. W. Lenstra's integer linear programming algorithm. The algorithms run in polynomial time when applied to rational input vectors. The paper also discusses the importance of Lovász' lattice reduction in various applications, including integer programming, polynomial factorization, and number theory.This paper by L. Babai addresses the application of Lovász' lattice reduction to find a point within a factor of \(c^d\) of a given point in \(\mathbf{R}^d\), where \(c\) is a constant. The author proves that two straightforward heuristic procedures achieve this goal when applied to a lattice given by a Lovász-reduced basis. One of these procedures requires proving a geometric property of Lovász-reduced bases: a lower bound on the angle between any basis vector and the hyperplane generated by the other vectors. As applications, the paper provides a solution to the nonhomogeneous simultaneous diophantine approximation problem and improves the Grötschel—Lovász—Schrijver version of H. W. Lenstra's integer linear programming algorithm. The algorithms run in polynomial time when applied to rational input vectors. The paper also discusses the importance of Lovász' lattice reduction in various applications, including integer programming, polynomial factorization, and number theory.