Las Vegas algorithms for linear and integer programming when the dimension is small

Abstract

This paper gives an algorithm for solving linear programming problems. For a problem with n constraints and d variables, the algorithm requires an expected O(d 2 n) + (log n)O(d) d/2+O(1) + O(d 4 √nlog n) arithmetic operations, as n→∞ . The constant factors do not depend on d. Also, an algorithm is given for integer linear programming. Let φ bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector. Let n and d be as before. Then, the algorithm requires expected O(2 d dn + 8 d d√n ln ln n) + d O(d) φ ln n operations on numbers with d O(1) φ bits, as n→∞ , where the constant factors do not depend on d or φ to other convex programming problems. For example, an algorithm for finding the smallest sphere enclosing a set of n points in E d has the same time bound.