Improved Online Load Balancing in the Two-Norm

TL;DR

The paper tackles online load balancing on unrelated machines with the objective to minimize the sum of squared loads. It introduces a primal–dual framework based on a natural semidefinite programming relaxation and an online correlated rounding scheme, achieving a new best competitive ratio of $4.9843$. It also provides simple, unified analyses for the classical greedy $(3+2\sqrt{2})$-competitive algorithm, the $5$-competitive independent-rounding algorithm, and a new $4$-competitive optimal fractional algorithm, plus several matching lower bounds. The approach leverages online water-filling, grouping of hard/easy jobs, and negatively correlated rounding to reduce variance terms, offering a powerful technique potentially applicable to other online quadratic objectives. Overall, the work significantly advances the understanding of randomness and correlation in online scheduling and demonstrates the value of semidefinite programming in online algorithm design.

Abstract

We study the online load balancing problem on unrelated machines, with the objective of minimizing the square of the $\ell_2$ norm of the loads on the machines. The greedy algorithm of Awerbuch et al. (STOC'95) is optimal for deterministic algorithms and achieves a competitive ratio of $3 + 2 \sqrt{2} \approx 5.828$, and an improved $5$-competitive randomized algorithm based on independent rounding has been shown by Caragiannis (SODA'08). In this work, we present the first algorithm breaking the barrier of $5$ on the competitive ratio, achieving a bound of $4.9843$. To obtain this result, we use a new primal-dual framework to analyze this problem based on a natural semidefinite programming relaxation, together with an online implementation of a correlated randomized rounding procedure of Im and Shadloo (SODA'20). This novel primal-dual framework also yields new, simple and unified proofs of the competitive ratio of the $(3 + 2 \sqrt{2})$-competitive greedy algorithm, the $5$-competitive randomized independent rounding algorithm, and that of a new $4$-competitive optimal fractional algorithm. We also provide lower bounds showing that the previous best randomized algorithm is optimal among independent rounding algorithms, that our new fractional algorithm is optimal, and that a simple greedy algorithm is optimal for the closely related online scheduling problem $R || \sum w_j C_j$.

Figures (6)

• Figure 1: An example of the grouping procedure for a fixed machine $i \in M$. The length of a job $j \in J$ reflects its fractional value $x_{ij} \in [0,1]$. Each easy job (in green) is contained in its own group, while the hard jobs (in red) are partitioned in the groups $G_{i,1},G_{i,2}$ and $G_{i,3}$. The large vertical lines indicate once a hard group becomes full. The first two hard groups $G_{i,1}$ and $G_{i,2}$ are full, while $G_{i,3}$ is not yet full: the next arriving hard job would thus be assigned to it.
• Figure 2: Illustration of the first case in the proof of Lemma \ref{['lem:lower_bound_v0_end_group']}. The length of each job represents its contribution $w_{ij}x_{ij}$ to the expected load of machine $i$. In this case, the contribution of the easy jobs $E \subseteq \{s+1, \dots, j\}$ is small, implying that $\hat{v}^{(j)}$ is only a small constant factor bigger than $v^{(s)}$, see \ref{['eq_vj_ve_relation_case1']}. This implies that the bonus added at $j \in H$ satisfies $B(H) = \Omega(L_h)$, see \ref{['eq_bonus_lower_bound']} and \ref{['eq_Lh_upperb']}, where $L_h$ denotes the total contribution from the hard jobs $H \subseteq \{s+1,\dots, j\}$.
• Figure 3: Illustration of the second case in the proof of Lemma \ref{['lem:lower_bound_v0_end_group']}. In this case, the contribution of the easy jobs $E \subseteq \{s+1, \dots, j\}$ is large, meaning that it satisfies $v_e \geq \tau \hat{v}^{(j)}$. This is enough to prove the lemma without using the bonus as every easy job $k \in E$ give an increase of $(\beta + \delta) w_{ik} x_{ik}$ to the dual vector, where $\delta > \tilde{\varepsilon}$.
• Figure 4: The slack of \ref{['eq_dual_feas_expanded']} for different values of $x_{ij}$ under the assumption that $\delta = \varepsilon = 0$ and $\gamma = 1/5$. Observe that the inequality is tight for $q_{ij}=2\sqrt{2/5}\approx 1.265$ and $x_{ij}=0$. The plot also shows the interval $[a,b]$ in red.
• Figure 5: The four different cases where dual feasibility needs to be checked. The two green regions correspond to easy jobs, whereas the two red regions correspond to hard jobs. The horizontal line at $\sqrt{2}$ differentiates between the cases where $q_{ij} < \sqrt{2}$, implying $\alpha_{ij} = q_{ij}$, and the case $q_{ij} \geq \sqrt{2}$, which implies $\alpha_{ij} = \sqrt{2}$.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• Remark