Integer Maximization over ${\ell }_p$ Balls: Hardness and Exact Algorithms

Abstract

We study the problem of maximizing a linear function over the integer points of an origin-centered ${\ell }_p$ ball, which we call \BallIPp{p}. For every fixed integer $p\ge 2$, we prove that the decision problem over an ${\ell }_p$-ball is NP-complete. We then focus on the Euclidean case and study how the difficulty of the problem depends on the numerical parameters of the instance. We give two complementary pseudo-polynomial exact algorithms. The first specializes a known radius-budget dynamic program; the same nonlinear-knapsack framework also applies to every fixed finite integer $p$. We then develop a complementary dynamic program over candidate objective values for the Euclidean case. The latter is polynomial in the encoding size of the radius and pseudo-polynomial in the magnitude of the cost coefficients. For a fixed objective value, feasibility can be formulated as a closest vector problem (CVP) instance. This connection gives an exact algorithm whose running time depends on the covering radius of that lattice. Conversely, we show that a rank-$k$ Euclidean closest-vector instance in ambient dimension $d$ reduces to the decision version of a Euclidean \BallIP{} instance in dimension $d+k\le 2d$, transferring known bounds under the Exponential Time Hypothesis (ETH). Finally, we study dimension-dependent approaches based on proximity to the continuous optimizer and describe the fixed-level constructions that extend to ellipsoids.