通过边界超平面覆盖实现反凸集整数编程的复杂性

arXiv - MATH - Optimization and Control Pub Date : 2024-09-09 DOI:arxiv-2409.05308

Robert Hildebrand, Adrian Göß

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引用次数: 0

摘要

我们研究了识别反凸集整数可行性的复杂性。我们介绍了在维数固定的情况下，复杂度可以是 NP-Hard，也可以是高效求解的各种情况。我们特别感兴趣的是多面体域的有界反向凸约束。我们引入了一种结构，即有界超平面覆盖（\emph{Boundary Hyperplane Cover}），只要非线性反向凸集的数量固定，就能在固定维度下以多项式时间求解这个问题。

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Complexity of Integer Programming in Reverse Convex Sets via Boundary Hyperplane Cover

We study the complexity of identifying the integer feasibility of reverse convex sets. We present various settings where the complexity can be either NP-Hard or efficiently solvable when the dimension is fixed. Of particular interest is the case of bounded reverse convex constraints with a polyhedral domain. We introduce a structure, \emph{Boundary Hyperplane Cover}, that permits this problem to be solved in polynomial time in fixed dimension provided the number of nonlinear reverse convex sets is fixed.

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arXiv - MATH - Optimization and Control

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