Multidimensional resource packing

Let each job have a vector of requested resources and each node a capacity vector. An assignment is feasible only if the sum on every node fits every resource dimension and all hard constraints hold. Counting aggregate free capacity is a useful preliminary check, but it discards the distribution across nodes. An eight-GPU request that must stay within one node cannot use one free GPU on each of eight nodes unless the workload and allocation model explicitly support that decomposition.

Even classical one-dimensional bin packing is computationally difficult to optimize exactly in general; multidimensional and constrained versions retain that difficulty. A production scheduler also works online with incomplete future information and limited decision time. This motivates heuristics rather than a claim that one score finds a global optimum. Separate a proven feasibility filter from a preference score: a high score must never authorize placing a workload that violates a hard capacity or compatibility constraint.

For identical nodes, total demand divided by node capacity gives a lower bound in each dimension; round up and take the largest. That bound does not prove a placement exists. Individual job sizes, complementary shapes, and constraints can force more nodes. Verify a proposed assignment explicitly, and distinguish an optimality proof from an example that merely fits. If a heuristic uses fewer machines than a baseline on one trace, that result does not establish a general approximation guarantee.

Packing policies can favor fuller nodes to leave others available, spread work to reduce interference, or balance several resources. These objectives conflict. Aggressive packing may improve allocation utilization while increasing shared-bandwidth contention or failure concentration. A scheduler score also depends on resource weights and normalization: CPU and memory units must not be compared as raw magnitudes. Preserve enough scoring diagnostics to explain why a feasible alternative was chosen and whether that choice actually improved the intended workload objective.

Test adversarial arrival orders as well as common traces. A sequence of small jobs can fragment the fleet before a large gang arrives; a static optimum with all arrivals known in advance has information the online scheduler lacks. Compare policies under the same admission, preemption, and migration rules. Track unschedulable demand by shape, not only free resource percentages. A migration that repairs packing has a cost and can invalidate a simplistic claim that a better final arrangement was obtained for free.

The vector lab illustrates a componentwise packing check for two identical normalized requests. Reset, set angle to 45 degrees and magnitude to 1, predict the sum of two copies of each component, then Step. Repeat from Reset with both angle at 45 degrees and magnitude at 0.5. Assume unit capacity on both axes. The instrument draws only one request; add its components on paper. It cannot search assignments or certify an optimal packing, but it makes clear why each individual request fitting is insufficient.