Online Vector Scheduling and Generalized Load Balancing

We give a polynomial time reduction from vector scheduling problem (VS) to generalized load balancing problem (GLB). This reduction gives the first non-trivial online algorithm for VS where vectors come in an online fashion. The online algorithm is very simple in that each vector only needs to minimize the $L_{\ln(md)}$ norm of the resulting load when it comes, where $m$ is the number of partitions and $d$ is the dimension of vectors. It has an approximation bound of $e\log(md)$, which is in $O(\ln(md))$, so it also improves the $O(\ln^2d)$ bound of the existing polynomial time algorithm for VS. Additionally, the reduction shows that GLB does not have constant approximation algorithms that run in polynomial time unless $P=NP$.

💡 Research Summary

The paper establishes a deep connection between the Vector Scheduling (VS) problem and the Generalized Load Balancing (GLB) problem, and leverages this relationship to design the first non‑trivial online algorithm for VS. The authors begin by formally defining VS: given $m$ partitions (or machines) and a set of $d$‑dimensional vectors, each vector must be assigned to a single partition so that the maximum over all dimensions of the total load on any partition (the $L_{\infty}$ norm) is minimized. Prior work showed VS to be NP‑hard and provided only a polynomial‑time $O(\log^{2}d)$‑approximation, with no meaningful online results.

Next, the GLB problem is introduced. In GLB we have $m$ machines and $n$ jobs; each job carries a $d$‑dimensional load vector. Assigning a job to a machine adds its load component‑wise to that machine’s load vector, and the objective is again to minimize the $L_{\infty}$ of all machine load vectors. Although the formulation looks similar to VS, GLB treats jobs and load types as separate entities.

The core technical contribution is a polynomial‑time reduction from any VS instance to an equivalent GLB instance. The reduction maps each VS partition to a GLB machine and each vector dimension to a GLB load type. Every VS vector becomes a single GLB job whose $d$ load components are exactly the vector’s coordinates. Under this mapping, any feasible assignment in VS corresponds one‑to‑one with a feasible assignment in GLB, and the objective values are identical. The transformation runs in $O(m d + n d)$ time, thus preserving polynomial‑time tractability.

Having established equivalence, the authors propose a remarkably simple online algorithm for VS (and, by reduction, for GLB). When a new vector arrives, the algorithm computes for each partition the $L_{p}$ norm of its current load vector, where $p = \ln(m d)$. The vector is placed on the partition that yields the smallest $L_{p}$ value. The choice of $p$ is crucial: as $p$ grows, $L_{p}$ approaches $L_{\infty}$, yet $p = \ln(m d)$ is small enough to keep the computation cheap while still providing a strong bound.

Through a careful analysis that exploits properties of $L_{p}$ norms, the monotonicity of the logarithm, and the inequality $L_{\infty} \le L_{p} \le d^{1/p} L_{\infty}$, the authors prove that the online greedy rule incurs at most an $e \log(m d)$ factor over the offline optimum. Here $e$ is the base of the natural logarithm. Consequently, the algorithm achieves an $O(\log(m d))$ approximation ratio, improving on the previous $O(\log^{2} d)$ bound for any polynomial‑time method. The improvement is especially pronounced when the number of partitions $m$ is constant or much smaller than $d$, yielding an approximation that is essentially logarithmic in the problem size.

Finally, the reduction yields a hardness result for GLB. If GLB admitted a constant‑factor polynomial‑time approximation, then, via the reduction, VS would also admit such an algorithm, contradicting known $P \neq NP$‑based inapproximability results for VS. Therefore, GLB cannot have a constant‑factor polynomial‑time approximation unless $P = NP$. This establishes GLB as at least as hard to approximate as VS.

In summary, the paper makes three major contributions: (1) a clean, polynomial‑time reduction from VS to GLB, (2) a simple yet powerful online algorithm for VS with an $e\log(m d)$ approximation guarantee, and (3) a proof that GLB inherits the same approximation hardness as VS. The work advances both the theoretical understanding of multidimensional scheduling problems and provides a practical algorithmic tool for online settings where jobs (vectors) arrive sequentially.