On the Complexity of Integer Programming in the Blum-Shub-Smale Computational Model

Abstract

In the framework of the Blum-Shub-Smale real number model [3], we study the algebraic complexity of the integer linear programming problem (ILPR): Given a matrix A ∈ Rm×n and vectors b ∈ R m, d ∈ R n, decide whether there is x ∈ Z n such that Ax ≤ b, where 0 ≤ x ≤ d. The main contributions of the paper are the following: An O (m log ∥d∥) algorithm for ILPR, when the value of n is fixed. As a corollary, we obtain under the same restriction a tight algebraic complexity bound Θ (log 1/amin, amin = min{a1,..., a n }, for the knapsack problem (KP R): Given a ∈ R + n, decide whether there is x ∈ Zn such that a T x = 1. We achieve these results in particular through a careful analysis of the algebraic complexity of the Lovász’ basis reduction algorithm and the Kannan-Bachem’s Hermite normal form algorithm, which may be of interest in its own. An O mn5 log n (n + log ∥d∥)) depth algebraic decision tree for ILPR, for every m and n. A new lower bound for 0/1 KP R. More precisely, no algorithm can solve 0/1 KP R in o (n log n) f (a1,..., a n ) time, even if f is an arbitrary continuous function of n variables. This result appears as an alternative to the well-known Ben-Or’s bound Ω(n2) [1] and is independent upon it. Keywords Algebraic complexity kw]Complexity bounds Integer programmingKnapsack problem