Deterministic Optimal Rounding

• Deterministic optimal rounding is a process that converts real values to integers while minimizing Lp error and strictly satisfying global sum constraints.
• It employs a greedy selection approach, as in the ORIC algorithm, by rounding up the K largest fractional parts to achieve the optimal solution.
• This method is pivotal in applications such as integer programming, apportionment, and distributed algorithms, delivering efficient, non-randomized precision.

Deterministic optimal rounding refers to the process of converting a set of real values, typically subject to global constraints (such as integer-sum or feasibility in an integer program), into integers in a manner that minimizes a well-specified objective—most often an error norm—while abiding by the constraints. Unlike traditional threshold or stochastic approaches, deterministic optimal rounding guarantees both constraint satisfaction and minimization of overall rounding error via an efficiently computable, non-random procedure, and occupies a central role in numerical optimization, combinatorial optimization, and the theory of pseudodeterministic algorithms.

1. Problem Definition: Optimal Rounding under Integer Constraints

Let $y = (y_1, \dots, y_N) \in \mathbb{R}^N$ satisfy a global integer constraint $\sum_{i=1}^N y_i = S \in \mathbb{Z}$. For any $p > 1$, the deterministic optimal rounding problem seeks

$$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$

That is, find integer vectors $x$ with $\sum_i x_i = S$ minimizing the $L^p$-norm error to $y$. The constrained $L^p$-error is denoted

$$V_p(y) = \min_{x\in\mathbb{Z}^N,\, \sum x_i=S} \sum_{i=1}^N |x_i-y_i|^p$$

A key structural result is that each $\sum_{i=1}^N y_i = S \in \mathbb{Z}$0, reducing the solution space to subsets of size $\sum_{i=1}^N y_i = S \in \mathbb{Z}$1 (the integer shortfall) (Cont et al., 2014). The problem is thus combinatorial but amenable to efficient solution.

2. Structural Insights and the ORIC Algorithm

The main insight (Proposition 1 in (Cont et al., 2014)) is that in any minimizer, each coordinate is rounded either up or down. With $\sum_{i=1}^N y_i = S \in \mathbb{Z}$2, let $\sum_{i=1}^N y_i = S \in \mathbb{Z}$3, and set $\sum_{i=1}^N y_i = S \in \mathbb{Z}$4. The optimal integer vector rounds $\sum_{i=1}^N y_i = S \in \mathbb{Z}$5 coordinates up to $\sum_{i=1}^N y_i = S \in \mathbb{Z}$6 and the rest down. To minimize the $\sum_{i=1}^N y_i = S \in \mathbb{Z}$7 error, select the $\sum_{i=1}^N y_i = S \in \mathbb{Z}$8 largest $\sum_{i=1}^N y_i = S \in \mathbb{Z}$9 to round up.

This yields the Optimal Rounding under Integer Constraints (ORIC) algorithm:

1. Compute $p > 1$0, $p > 1$1
2. Compute $p > 1$2
3. Sort the $p > 1$3 in descending order (break ties by $p > 1$4 descending for a canonical minimizer)
4. Assign $p > 1$5 for the top $p > 1$6 indices and $p > 1$7 for the rest
5. Output $p > 1$8 with $p > 1$9, guaranteeing minimum $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$0 error

The entire procedure runs in $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$1 time due to the sort step. The algorithm is exact, deterministic, and its solution is unique minimizing the $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$2 error among all feasible integer vectors (Cont et al., 2014).

3. Comparison to Stochastic and Heuristic Rounding

• Classical threshold rounding might round $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$3 up if $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$4 for some fixed $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$5, but in general this fails to achieve the sum constraint for arbitrary $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$6.
• Randomized rounding independently rounds $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$7 up with probability $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$8, ensuring expectation but yielding only a probabilistic match to the sum and typically yielding biased estimators of the optimal $$x^* = \arg\min_{x \in \mathbb{Z}^N,\, \sum_i x_i = S} \sum_{i=1}^N |x_i - y_i|^p$$9.
• In contrast, deterministic optimal rounding produces the unique unbiased minimizer of total $x$0 error and always satisfies the sum constraint (Cont et al., 2014).

The following table captures key distinctions:

Scheme	Sum-Constraint Satisfied	Minimizes $x$1 error	Bias	Complexity
Threshold	No (in general)	No	Yes (to $x$2)	$x$3
Randomized	In expectation	No	Yes (in mean)	$x$4
Deterministic Optimal (ORIC)	Always	Yes	No (to $x$5), but unbiased among minimizers	$x$6

4. Relationship to High-Dimensional and Geometric Rounding

Deterministic rounding admits a geometric interpretation: For $x$7, rounding can be viewed as a partitioning of $x$8 (“secluded partitions”) such that no small $x$9-ball intersects too many cells. A $\sum_i x_i = S$0-secluded partition associates every $\sum_i x_i = S$1 to a cell (rounded value), so any ball of radius $\sum_i x_i = S$2 hits at most $\sum_i x_i = S$3 values. Explicit, efficiently computable $\sum_i x_i = S$4-secluded partitions exist and are optimal up to constants (Woude et al., 2023, Woude et al., 2022). The design of deterministic rounding schemes in high dimension is therefore closely tied to geometric bounds on partitions.

Lower bounds show any deterministic, bounded-diameter rounding scheme with $\sum_i x_i = S$5 outputs must have granularity $\sum_i x_i = S$6. Constructions using reclusive matrices yield matching upper bounds (Woude et al., 2022).

5. Algorithmic Variants and Extensions

• Generalization to arbitrary $\sum_i x_i = S$7: The ORIC approach relies on convexity and extends to any strictly convex norm; the greedy selection of largest $\sum_i x_i = S$8 remains optimal (Cont et al., 2014).
• Ties in $\sum_i x_i = S$9: Any subset among tied $L^p$0 yields the same $L^p$1 error; for a canonical (e.g., minimum relative-error) output, sort also by $L^p$2 decreasing.
• Edge cases: When $L^p$3 is already integer, $L^p$4 and the original vector is optimal.

Other extensions appear in high-dimensional rounding for pseudodeterministic or distributed settings (see (Faour et al., 2022, Bhattacharya et al., 2021)).

6. Applications and Context

Integer Programming: For solving relaxed LPs and rounding back to feasible integral points while controlling $L^p$5 error, deterministic optimal rounding provides a critical rounding step without sacrificing the global constraint.

Apportionment and Resource Allocation: Problems such as apportioning seats by proportional votes or distributing budget subject to integer constraints are solved optimally with ORIC (recovering the Hare–Niemeyer largest-remainders rule, but now with $L^p$6-optimality) (Cont et al., 2014).

Statistical Rounding: Used in financial settings for rounding floating-point numbers to fixed denominators (e.g., cents) while maintaining overall totals and accuracy.

Distributed and Dynamic Graph Algorithms: Deterministic rounding underpins distributed algorithms for matching, independent sets, and set cover, with polylogarithmic round-complexity matching randomized methods in several regimes (Fischer, 2017, Faour et al., 2022, Bhattacharya et al., 2021, Bhattacharya et al., 2023).

High-Dimensional Clustering and Pseudodeterminism: Geometric interpretations lead to derandomization tradeoffs and bounds for pseudodeterministic algorithms in estimation and learning (Woude et al., 2023, Woude et al., 2022).

7. Limitations and Theoretical Barriers

The tradeoff between granularity and number of outputs is inherent: No deterministic rounding scheme with output degree $L^p$7 and cell diameter $L^p$8 can ensure any $L^p$9-ball of radius $y$0 meets fewer than $y$1 outputs (Woude et al., 2023). For $y$2 polynomial in $y$3, $y$4. In the integer-sum constraint setting, deterministic optimal rounding is subject to these lower bounds when generalized to high dimension.

Randomized or pseudodeterministic schemes can achieve coarser granularity but only in expectation and with increased complexity or with non-deterministic output. In contrast, deterministic optimal rounding yields tight $y$5-error minimization and is computationally efficient, but faces hard lower bounds on output granularity and degree in high dimensions (Woude et al., 2023, Woude et al., 2022).