Approximating Unrelated Machine Weighted Completion Time Using Iterative Rounding and Computer Assisted Proofs

TL;DR

It is proved that the algorithm achieves a $(1.36 + \epsilon)$-approximation, improving upon the previous best approximation ratio of $1.4$ due to Harris.

Abstract

We revisit the unrelated machine scheduling problem with the weighted completion time objective. It is known that independent rounding achieves a 1.5 approximation for the problem, and many prior algorithms improve upon this ratio by leveraging strong negative correlation schemes. On each machine $i$, these schemes introduce strong negative correlation between events that some pairs of jobs are assigned to $i$, while maintaining non-positive correlation for all pairs. Our algorithm deviates from this methodology by relaxing the pairwise non-positive correlation requirement. On each machine $i$, we identify many groups of jobs. For a job $j$ and a group $B$ not containing $j$, we only enforce non-positive correlation between $j$ and the group as a whole, allowing $j$ to be positively-correlated with individual jobs in $B$. This relaxation suffices to maintain the 1.5-approximation, while enabling us to obtain a much stronger negative correlation within groups using an iterative rounding procedure: at most one job from each group is scheduled on $i$. We prove that the algorithm achieves a $(1.36 + ε)$-approximation, improving upon the previous best approximation ratio of $1.4$ due to Harris. While the improvement may not be substantial, the significance of our contribution lies in the relaxed non-positive correlation condition and the iterative rounding framework. Due to the simplicity of our algorithm, we are able to derive a closed form for the weighted completion time our algorithm achieves with a clean analysis. Unfortunately, we could not provide a good analytical analysis for the quantity; instead, we rely on a computer assisted proof.

Figures (4)

• Figure 1: The edges in $G$ between $i$ and $J_k$ for the case $\mathrm{vol}(\delta^k_i) \geq \beta \rho^k$, where we have $\mathrm{vol}(\delta^{k, \mathrm{mk}}_i) = \beta \rho^k$.
• Figure 2: The three cases for the machine $i_0$ on a pseudo-marked path in $\bar{G} = (M, J^k, \bar{E})$. They also apply to the machine $i_t$.
• Figure 3: The 6 cases for Lemma \ref{['lemma:sub-program2-cases']}. Each row of line segments represents a case in Lemma \ref{['lemma:sub-program2-cases']}. Each line segment between two dots represent an element whose value is the length of the line segment. Each ray represents a flexible element (whose length is not determined) and the cross represents the rightmost endpoint it can reach. Dashed line segments represent type-s or type-b elements, black solid line segments represent type-1 elements, and red solid line segments represent type-2 elements.
• Figure 4: The 23 cases in Lemma \ref{['lemma:sub-program3-cases']}. The legends are the same as those in Figure \ref{['fig:configuration-1']}, except now we have type-0 elements, denoted by thin solid line segments.

Theorems & Definitions (42)

• Theorem 1.1
• Lemma 2.1
• proof
• Definition 2.2
• Claim 2.3
• Definition 2.4
• Claim 2.5
• proof
• Claim 2.6
• proof