Integrality Gaps of Integer Knapsack Problems

Abstract

Abstract. We obtain optimal lower and upper bounds for the (additive) integrality gaps of integer knapsack problems. In a randomised setting, we show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario.

Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma

“…Thus our bound, which is independent of n, is an improvement by a factor of n 2 for integer programs in standard form and fixed m. c) We use this to generalize a recent bound on the absolute integrality gap for the case m = 1 by Aliev et al [2] that states that c T (x…”

“…A classical bound of Cook et al [11] implies, in the standard-form setting, z * − x * ∞ n · ( m · ∆) m and thus z * − x * 1 n 2 · ( m · ∆) m . Thus our bound, which is independent of n, is an improvement by a factor of n 2 for integer programs in standard form and fixed m. c) We use this to generalize a recent bound on the absolute integrality gap for the case m = 1 by Aliev et al [2] that states that c T (x * − z * ) c ∞ · 2 · ∆. Our distance bound shows that the absolute integrality gap is bounded by c ∞ · O(m) m+1 · O(∆) m .…”

“…An integer program ( 1) is called an unbounded knapsack problem if m =1. In this case, Aliev et al [2] show that one has…”

“…The out-degree of each node is bounded by u * i +l * i 2·L 1 +1. Therefore, the number of arcs is bounded by n · |U | · (2 · L 1 + 1) = n · O(m) (m+1) 2 · O(∆) m·(m+2) (26) which would lead to a corresponding running time of n · O(m) (m+1) 2 · O(∆) m·(m+2) since longest path in an acyclic digraph can be computed in linear time in the number of nodes and arcs. However, a standard technique can be applied to significantly decrease the number of arcs.…”