具有1阶Chvátal秩的多面体上的整数规划与格问题难度相当

中文摘要 AI 辅助

若有理多面体经一轮Chvátal-Gomory割即可得到其整数壳，则其Chvátal秩至多为1。根据Boyd和Pulleyblank在20世纪80年代初的研究，这类多面体的整数可行性问题属于NP∩coNP，因此它不太可能是NP难的。自那时起，该问题是否存在多项式时间算法一直是个悬而未决的问题。我们基于格密码学中的两个标准假设，对该问题给出了否定答案：其一，若存在求解该问题的多项式时间算法，则可在确定性多项式时间内以多项式因子求解有界距离解码问题，这与被广泛认可的猜想相矛盾；其二，假设学习误差问题（有界距离解码的平均情况类似问题）是困难的，则对于可高效采样的多面体分布，该问题在平均情况下也是困难的。上述两个结果均基于一个基础充分条件：若多面体沿某幺模矩阵的每一行的宽度均小于1，则其Chvátal秩至多为1；对于我们构造的多面体，虽存在这样的矩阵，但难以找到它。

英文摘要

A rational polyhedron has Chvátal rank at most one if a single round of Chvátal-Gomory cuts yields its integer hull. For such polyhedra, integer feasibility is in NP $\cap$ coNP by a result of Boyd and Pulleyblank from the early 1980s, so it is unlikely to be NP-hard. Whether it is polynomial has remained open since then. We answer this question negatively, under either of two standard assumptions from lattice-based cryptography. First, a polynomial-time algorithm for this problem would solve bounded distance decoding with polynomial factors in deterministic polynomial time, contradicting a widely believed conjecture. Second, assuming the hardness of learning with errors, an average-case analogue of bounded distance decoding, the problem is also hard on average, for an efficiently samplable distribution of polytopes. Both results rest on an elementary sufficient condition: a polyhedron has Chvátal rank at most one if its width is less than one along every row of some unimodular matrix. For our polytopes, such a matrix exists but is hard to find.

发表机构

• Wisconsin Institute for Discovery, University of Wisconsin–Madison（威斯康星大学麦迪逊分校威斯康星发现研究所）

机构由 AI 辅助整理，请以论文原文为准。