两权图平衡中3/2配置线性规划间隙的最小见证，其大小具有唯一性
A minimum witness for the 3/2 configuration-linear-program gap in two-weight graph balancing, unique at its size
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中文总结 AI 辅助
本文针对两权图平衡问题，构造出达到3/2配置线性规划间隙的最小6作业见证实例I*，证明其大小唯一，并揭示不同作业数下见证实例的数量规律及相关复杂度结论。
中文摘要 AI 辅助
在受限分配-最大完成时间最小化问题中，每个作业具有一个尺寸和一组允许的机器，配置线性规划（configuration LP）是研究中最紧的松弛形式，其整数间隙在一般情况下仍未解决。对于两权图平衡问题（每个作业最多允许在两台机器上，尺寸仅来自两个值），Jansen、Land和Maack（2016）给出了该间隙的已知上下界：他们的表1实例达到3/2的间隙，而其推论11给出的尺寸s < b时的界2 - s/b在尺寸为{1,2}时与该间隙匹配。本文研究这类间隙见证实例的最小规模：我们给出I*，一个含6个作业的见证实例，其结构为4台机器上的完全图，哈密顿环上为单位尺寸作业，互补完美匹配上为尺寸2的作业，其整数最优解为3，而松弛值为2。该实例比已发表的最小实例少一个作业，我们证明它是最小的，且在其大小下具有唯一性：最多5个作业的同类型实例无法达到3/2的间隙；对于6个作业，无论机器数量多少，I*是唯一的见证实例（机器重命名和添加无作业可使用的机器后仍保持唯一性）；7个作业时唯一性失效，经分类共有恰好13个见证实例（包括Jansen-Land-Maack的实例）；8个作业时则有恰好154个见证实例。此外，任意尺寸下3台机器都无法达到该间隙，4台机器是达到该间隙的必要条件。同类结果已在一维下料问题中发表，其中极值非取整实例已被枚举并分类小需求情况，讨论部分阐述了两者的关联。所有见证实例的松弛值均为2，而识别松弛值为2的见证实例是coNP完全问题，因此除非NP = coNP，否则不存在最小-最大刻画。所有穷举断言背后的可行性判定均通过浮点运算和精确有理运算两次完成，结果完全一致，且在否定结论被认可前，计算流程必须能复现I*实例。
英文摘要
In restricted assignment - makespan minimization where each job has one size and a set of allowed machines - the configuration LP is the tightest studied relaxation, and its integrality gap is open in general. On two-weight graph balancing - each job allowed on at most two machines, sizes from two values - the value is known, both bounds due to Jansen, Land, and Maack (2016): their Table 1 instance attains 3/2, and their Corollary 11 bound of 2 - s/b for sizes s < b meets it at {1,2}. We ask how small such an instance - a witness - can be. We give I*, a six-job witness: the complete graph on four machines, unit jobs on a Hamiltonian cycle, weight-2 jobs on the complementary perfect matching, with integral optimum 3 against relaxation value 2. That is one job fewer than the smallest previously in print, and we prove it minimum and unique at its size. No instance of the class with at most five jobs reaches gap 3/2, on any number of machines; at six jobs, again on any number of machines, I* is the only witness, up to relabeling machines and adding machines no job can use. At seven jobs uniqueness fails: exactly thirteen witnesses, classified - the Jansen-Land-Maack instance among them - and at eight jobs exactly 154. Three machines never suffice, at any size: four are necessary for the gap. Results of this shape are in print for the same relaxation in one-dimensional cutting stock, where the extremal non-round-up instances have been enumerated and classified for small demand; the Discussion sets out the relation. Recognizing witnesses at relaxation value 2 - where all of ours live - is coNP-complete, so no min-max characterization exists unless NP = coNP. Every feasibility decision behind the exhaustive claims was made twice, in floating point and in exact rational arithmetic, with full agreement, and the pipeline must rediscover I* before its negatives are believed.
发表机构
- Hunter College and the Graduate Center, CUNY（亨特学院和纽约市立大学研究生中心）
机构由 AI 辅助整理，请以论文原文为准。
