Simpler and Better Algorithms for Minimum-Norm Load Balancing

Recently, Chakrabarty and Swamy (STOC 2019) introduced the {\em minimum-norm load-balancing} problem on unrelated machines, wherein we are given a set $J$ of jobs that need to be scheduled on a set of $m$ unrelated machines, and a monotone, symmetric norm; We seek an assignment $\text{sg}:J\mapsto [m]$ that minimizes the norm of the resulting load vector ${\text{lvec}}_{\text{sg}}\in {\mathbb{R}}_{+}^m$, where ${\text{lvec}}_{\text{sg}}(i)$ is the load on machine $i$ under the assignment $\text{sg}$. Besides capturing all ${\mathrm{\ell }}_p$ norms, symmetric norms also capture other norms of interest including top-$\mathrm{\ell }$ norms, and ordered norms. Chakrabarty and Swamy (STOC 2019) give a $(38+\text{ve})$-approximation algorithm for this problem via a general framework they develop for minimum-norm optimization that proceeds by first carefully reducing this problem (in a series of steps) to a problem called \minmax ordered load balancing, and then devising a so-called deterministic oblivious LP-rounding algorithm for ordered load balancing. We give a direct, and simple $4$-approximation algorithm for the minimum-norm load balancing based on rounding a (near-optimal) solution to a novel convex-programming relaxation for the problem. Whereas the natural convex program encoding minimum-norm load balancing problem has a large non-constant integrality gap, we show that this issue can be remedied by including a key constraint that bounds the “norm of the job-cost vector.” Our techniques also yield a (essentially) $4$-approximation for: (a) {\em multi-norm load balancing}, wherein we are given multiple monotone symmetric norms, and we seek an assignment respecting a given budget for each norm; (b) the best {\em simultaneous approximation factor} achievable for all symmetric norms for a given instance.