Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma

Abstract

We consider integer programming problems in standard form max { c T x : Ax = b , x ⩾ 0, x ∈ Z n } where A ∈ Z m × n , b ∈ Z m , and c ∈ Z n . We show that such an integer program can be solved in time ( m ⋅ Δ) O ( m ) ⋅ \Vert b\Vert ∞ 2 , where Δ is an upper bound on each absolute value of an entry in A . This improves upon the longstanding best bound of Papadimitriou [27] of ( m ⋅ Δ) O ( m 2 ) , where in addition, the absolute values of the entries of b also need to be bounded by Δ. Our result relies on a lemma of Steinitz that states that a set of vectors in R m that is contained in the unit ball of a norm and that sum up to zero can be ordered such that all partial sums are of norm bounded by m . We also use the Steinitz lemma to show that the ℓ 1 -distance of an optimal integer and fractional solution, also under the presence of upper bounds on the variables, is bounded by m ⋅ (2, m ⋅ Δ +1) m . Here Δ is again an upper bound on the absolute values of the entries of A . The novel strength of our bound is that it is independent of n . We provide evidence for the significance of our bound by applying it to general knapsack problems where we obtain structural and algorithmic results that improve upon the recent literature.