A branch-and-cut approach to examining the maximum density guarantee for pinwheel schedulability of low-dimensional vectors

Abstract

Given an n-dimensional integer vector $\mathbf {v} = (v_1, v_2, \ldots , v_n)$ with $v_1 \le v_2 \le \cdots \le v_n$, a pinwheel schedule for $\mathbf {v}$ is an infinite symbol sequence $S_1 S_2 S_3 \cdots$, which satisfies that $S_j \in \{ 1, 2, \ldots , n \}, \forall j \in \mathbb {Z}^{+}$ and every $i \in \{ 1, 2, \ldots , n \}$ occurs at least once in every $v_i$ consecutive symbols $S_{j + 1} S_{j + 2} \cdots S_{j + v_i}, \forall j \in \mathbb {Z}^{+} \cup \{ 0 \}$. If $\mathbf {v}$ has a pinwheel schedule then $\mathbf {v}$ is called (pinwheel) schedulable. The density of $\mathbf {v}$ is defined as $d(\mathbf {v}) = \sum _{i = 1}^{n} \frac{1}{v_i}$. Chan and Chin (Algorithmica 9(5):425–462, 1993) made a conjecture that every vector $\mathbf {v}$ with $d(\mathbf {v}) \le \frac{5}{6}$ 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 $k \ge 2$. 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 $\frac{5}{6}$, which partially supports Chan and Chin’s conjecture.