Emergent Mind

Optimizing Low Dimensional Functions over the Integers

(2303.02474)
Published Mar 4, 2023 in cs.DS

Abstract

We consider box-constrained integer programs with objective $g(Wx) + cT x$, where $g$ is a "complicated" function with an $m$ dimensional domain. Here we assume we have $n \gg m$ variables and that $W \in \mathbb Z{m \times n}$ is an integer matrix with coefficients of absolute value at most $\Delta$. We design an algorithm for this problem using only the mild assumption that the objective can be optimized efficiently when all but $m$ variables are fixed, yielding a running time of $nm(m \Delta){O(m2)}$. Moreover, we can avoid the term $nm$ in several special cases, in particular when $c = 0$. Our approach can be applied in a variety of settings, generalizing several recent results. An important application are convex objectives of low domain dimension, where we imply a recent result by Hunkenschr\"oder et al. [SIOPT'22] for the 0-1-hypercube and sharp or separable convex $g$, assuming $W$ is given explicitly. By avoiding the direct use of proximity results, which only holds when $g$ is separable or sharp, we match their running time and generalize it for arbitrary convex functions. In the case where the objective is only accessible by an oracle and $W$ is unknown, we further show that their proximity framework can be implemented in $n (m \Delta){O(m2)}$-time instead of $n (m \Delta){O(m3)}$. Lastly, we extend the result by Eisenbrand and Weismantel [SODA'17, TALG'20] for integer programs with few constraints to a mixed-integer linear program setting where integer variables appear in only a small number of different constraints.

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