Given an n -dimensional integer vector $$\mathbf {v} = (v_1, v_2, \ldots , v_n)$$ v = ( v 1 , v 2 , … , v n ) with $$v_1 \le v_2 \le \cdots \le v_n$$ v 1 ≤ v 2 ≤ ⋯ ≤ v n , a pinwheel schedule for $$\mathbf {v}$$ v is an infinite symbol sequence $$S_1 S_2 S_3 \cdots$$ S 1 S 2 S 3 ⋯ , which satisfies that $$S_j \in \{ 1, 2, \ldots , n \}, \forall j \in \mathbb {Z}^{+}$$ S j ∈ { 1 , 2 , … , n } , ∀ j ∈ Z + and every $$i \in \{ 1, 2, \ldots , n \}$$ i ∈ { 1 , 2 , … , n } occurs at least once in every $$v_i$$ v i consecutive symbols $$S_{j + 1} S_{j + 2} \cdots S_{j + v_i}, \forall j \in \mathbb {Z}^{+} \cup \{ 0 \}$$ S j + 1 S j + 2 ⋯ S j + v i , ∀ j ∈ Z + ∪ { 0 } . If $$\mathbf {v}$$ v has a pinwheel schedule then $$\mathbf {v}$$ v is called (pinwheel) schedulable . The density of $$\mathbf {v}$$ v is defined as $$d(\mathbf {v}) = \sum _{i = 1}^{n} \frac{1}{v_i}$$ d ( v ) = ∑ i = 1 n 1 v i . Chan and Chin (Algorithmica 9(5):425–462, 1993) made a conjecture that every vector $$\mathbf {v}$$ v with $$d(\mathbf {v}) \le \frac{5}{6}$$ d ( v ) ≤ 5 6 is schedulable. In this paper, we check the conjecture from the perspective of low-dimensional vectors, including 3-, 4- and 5-dimensional ones. We first find some simple but important properties of schedulable vectors, and then propose two comparing rules according to these properties. Also, we define a vector-space tree to represent all k -dimensional vectors, for any given integer $$k \ge 2$$ k ≥ 2 . Under the framework of the vector-space tree, we use the comparing rules to develop a Branch-and-Cut Approach to examining the schedulability of all k -dimensional vectors. As a result, we prove that the maximum density guarantee for the pinwheel schedulability of low-dimensional vectors is $$\frac{5}{6}$$ 5 6 , which partially supports Chan and Chin’s conjecture.

中文翻译：

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一种检查低维向量风车可调度性的最大密度保证的分支和切割方法

给定一个 n 维整数向量 $$\mathbf {v} = (v_1, v_2, \ldots , v_n)$$ v = ( v 1 , v 2 , ... , vn ) 和 $$v_1 \le v_2 \le \ cdots \le v_n$$ v 1 ≤ v 2 ≤ ⋯ ≤ vn ，$$\mathbf {v}$$ v 的风车表是一个无限符号序列 $$S_1 S_2 S_3 \cdots$$ S 1 S 2 S 3 ⋯ ，满足 $$S_j \in \{ 1, 2, \ldots , n \}, \forall j \in \mathbb {Z}^{+}$$ S j ∈ { 1 , 2 , … , n } , ∀ j ∈ Z + 并且每个 $$i \in \{ 1, 2, \ldots , n \}$$ i ∈ { 1 , 2 , ... , n } 在每个 $$v_i$$ 中至少出现一次vi 连续符号 $$S_{j + 1} S_{j + 2} \cdots S_{j + v_i}, \forall j \in \mathbb {Z}^{+} \cup \{ 0 \}$$ S j + 1 S j + 2 ⋯ S j + vi , ∀ j ∈ Z + ∪ { 0 } 。如果 $$\mathbf {v}$$ v 有一个风车时间表，那么 $$\mathbf {v}$$ v 被称为 (pinwheel) schedulable 。$$\mathbf {v}$$ v 的密度定义为 $$d(\mathbf {v}) = \sum _{i = 1}^{n} \frac{1}{v_i}$$ d ( v ) = ∑ i = 1 n 1 vi 。Chan 和 Chin (Algorithmica 9(5):425–462, 1993) 做出了一个猜想，即每个向量 $$\mathbf {v}$$ v 和 $$d(\mathbf {v}) \le \frac{5} {6}$$ d ( v ) ≤ 5 6 是可调度的。在本文中，我们从低维向量的角度来检验这个猜想，包括 3 维、4 维和 5 维向量。我们首先找到可调度向量的一些简单但重要的特性，然后根据这些特性提出两个比较规则。此外，我们定义了一个向量空间树来表示所有 k 维向量，对于任何给定的整数 $$k \ge 2$$ k ≥ 2 。在向量空间树的框架下，我们使用比较规则开发了一种分支和切割方法来检查所有 k 维向量的可调度性。因此，我们证明了低维向量风车可调度性的最大密度保证为 $$\frac{5}{6}$$ 5 6 ，这部分支持了 Chan 和 Chin 的猜想。