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

• Authors: Valentin E. Brimkov;Stefan S. Dantchev
• Venue: TCS '00 Proceedings of the International Conference IFIP on Theoretical Computer Science, Exploring New Frontiers of Theoretical Informatics
• Year: 2000

In the framework of the Blum-Shub-Smale real number model, we study the algebraic complexity of the integer linear programming problem (ILPR) : Given a matrix A ∈ Rm×n and vectors b ∈ Rm, d ∈ Rn, decide whether there is x ∈ Zn 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, ..., an}, for the knapsack problem (KPR) : Given a ∈ R+n, decide whether there is x ∈ Zn such that aT 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 KPR. More precisely, no algorithm can solve 0/1 KPR in o (n log n) f(a1, ..., an) 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) and is independent upon it.