整数最大化在 $\ell_p$ 球上：困难性与精确算法

中文摘要 AI 辅助

我们研究了在原点为中心的 $\ell_p$ 球的整数点上最大化线性函数的问题，我们称之为 \BallIPp{p}。对于每个固定的整数 $p\ge2$，我们证明了在 $\ell_p$ 球上的决策问题是 NP 完全的。然后我们关注欧几里得情形，并研究问题的难度如何依赖于实例的数值参数。我们给出了两个互补的伪多项式精确算法。第一个算法专门化了一个已知的半径预算动态规划；相同的非线性背包框架也适用于每个固定的有限整数 $p$。然后我们为欧几里得情形开发了一个基于候选目标值的互补动态规划。后者在半径的编码大小上是多项式的，在成本系数的幅度上是伪多项式的。对于固定的目标值，可行性可以被表述为一个最近向量问题（CVP）实例。这种联系给出了一个精确算法，其运行时间取决于该格子的覆盖半径。相反，我们证明了在环境维度 $d$ 中的秩为 $k$ 的欧几里得最近向量实例可以归约到维度 $d+k\le2d$ 的欧几里得 \BallIP{} 实例的决策版本，从而在指数时间假说（ETH）下转移已知界限。最后，我们研究了基于与连续优化器接近度的维度相关方法，并描述了扩展到椭球的固定水平构造。

英文摘要

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 $p\ge2$, 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 $d+k\le2d$, 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.

发表机构

• Grado Department of Industrial and Systems Engineering, Virginia Tech（弗吉尼亚理工大学工业与系统工程格拉多系）

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