Roundportions

Roundportions is a conceptual term used in mathematics and computer science to describe the process of converting a set of fractional portions into discrete quantities while preserving the overall total. The concept appears in resource allocation, scheduling, data quantization, and related optimization problems, but there is no single standardized definition.

Typically, the objective is to produce integer portions bi that are as close as possible to the original ai, with the sum of all bi matching a target total, usually the rounded sum of ai. A common construction starts with bi = floor(ai) for all i, and then increases certain bi by one to reach the desired total. The choice of which indices to increment can be based on the fractional parts fi = ai - floor(ai), taking the largest fi first in deterministic schemes or using probabilities proportional to fi in randomized schemes.

Properties of roundportions include preservation of nonnegativity, and bounded deviation from the original proportion. In many settings, additional constraints apply, such as keeping bi within allowed ranges or minimizing individual or total error.

Applications range from fair division of indivisible goods and resource scheduling to bin packing, data quantization, and load balancing in distributed systems. A simple example: for a1 = 2.4, a2 = 3.6, a3 = 1.2, the sum is 7.2 and the rounded total is 7. Floors sum to 6; increment the index with the largest fractional part (0.6 for a2) to obtain (2, 4, 1), which sums to 7.

See also: rounding, floor and ceiling functions, randomized rounding, dependent rounding, resource allocation.