Split Rounding Strategy

• Split Rounding Strategy is a family of integer rounding algorithms that transform fractional allocations into integer solutions while preserving sum constraints.
• It employs efficient methods like the largest-remainder and dynamic programming-based rounding to minimize p-norm error under defined linear constraints.
• Its applications span apportionment, resource allocation, fair division with subsidies, and graph optimization, providing unbiased and optimal results.

The split rounding strategy encompasses a family of integer rounding algorithms designed to transform real-valued vectors or fractional allocations into integer-valued solutions while satisfying linear constraints and optimizing a prescribed objective. This approach is fundamental in contexts where sum preservation, fairness, or optimal distance minimization is required, including apportionment, resource allocation, stable set rounding from convex relaxations, and proportional allocation with subsidies. Notable variants include the optimal split (largest-remainder) rounding (Cont et al., 2014), tree-splitting based schemes for fair allocation with subsidy (Wu et al., 2024), and dynamic programming-based rounding from SDP relaxations in split and generalized-split graphs (Gong et al., 9 Apr 2025).

1. Mathematical Structure of Split Rounding

Consider the canonical problem of rounding a real vector $x = (x_1, \ldots, x_N) \in \mathbb{R}^N$, where $\sum_{n=1}^N x_n = M \in \mathbb{Z}$, to an integer vector $y = (y_1, \ldots, y_N) \in \mathbb{Z}^N$ with $\sum_{n=1}^N y_n = M$, minimizing $p$-norm error: $$\min_{y\in\mathbb{Z}^N} \|x - y\|_p = \left(\sum_{n=1}^N |x_n - y_n|^p\right)^{1/p}, \qquad \text{subject to} \;\; \sum_{n=1}^N y_n = M \tag{P}$$ A fundamental property is that any optimal solution $y^*$ must satisfy $$y^*_n \in \{\lfloor x_n \rfloor, \lceil x_n \rceil\}$$ for all $n$, reducing the optimization to a combinatorial selection of which coordinates to round up versus down (Cont et al., 2014).

2. Efficient Algorithms for Split Rounding

The largest-remainder method (split rounding) is implemented by first writing each $x_n$ as

$$x_n = m_n + r_n, \quad m_n := \lfloor x_n \rfloor, \quad r_n = x_n - m_n \in [0,1)$$

The deficit $D = M - \sum_{n=1}^N m_n$ gives the number of components that must be rounded up. Sorting indices in descending order of $(r_n, m_n)$, the top $D$ coordinates are rounded up; the remainder are left at their floor values. Time complexity is $O(N\log N)$ and space complexity is $O(N)$ (Cont et al., 2014). This split rounding procedure is optimal for any $\ell_p$ norm (strictly convex, $p \geq 1$).

Step	Operation	Complexity
Compute	Floors and remainders	$O(N)$
Sort	Descending $(r_n, m_n)$	$O(N\log N)$
Assign	Round up top $D$ and down rest	$O(N)$

This strategy generalizes the Hare–Niemeyer method for apportionment and removes the systematic errors inherent in midpoint and randomized rounding.

3. Split Rounding in Graph Optimization and SDP Rounding

In the context of rounding fractional solutions to graph optimization problems, variants of split rounding integrate with convex relaxations. For the Lovász theta SDP for the maximum weighted stable set problem on perfect graphs, dynamic programming (DP) rounding recovers an integer stable set exactly whenever the convex relaxation is tight. For split, generalized-split, chordal, and co-chordal graphs, the DP-based split rounding recovers an optimal stable set by exploiting value-function approximations from the dual SDP solution. A graph is unipolar (split) if the vertex set admits a clique-independent set partition, and generalized-split if it or its complement is unipolar (Gong et al., 9 Apr 2025).

The VFA-based DP rounding method involves:

• Phase I: Discarding non-optimal vertices based on primal/dual slackness.
• Phase II: Sequentially constructing a stable set through candidate selection and a one-step look-ahead, guided by the VFA function (based on principals of split rounding).

On these graph classes, the procedure provably recovers an optimal solution (Gong et al., 9 Apr 2025).

4. Split and Tree-Splitting Rounding in Fair Division and Allocation

In weighted proportional allocation with indivisible items and money, tree-splitting based split rounding is used to minimize the total subsidy required to achieve proportionality. Starting from a fractional bid-and-take allocation, items are organized into a forest structure—each tree corresponding to the fractional assignment graph. The tree is split into canonical components (size-2 paths and expanded atom-paths) by specific splitting routines:

• Simple Splitting: Partitions trees with no item shared by more than two agents.
• Atom-Path Splitting: Handles items assigned fractionally to multiple agents along a path.

Each component is rounded independently, and the incurred subsidy in each is bounded above by $z/3$ where $z$ is the component's size. Summing across all components, the total required subsidy to achieve weighted proportional allocation is bounded by $n/3 - 1/6$ (Wu et al., 2024).

5. Comparison to Other Rounding Strategies

Split rounding outperforms standard midpoint and randomized rounding in objectives that require sum preservation or bias minimization. Randomized rounding can result in violations of constraints or systematic bias due to finite sample effects. In apportionment settings and as the rounding step in relaxed integer programming (such as resource allocation, scheduling, portfolio rounding), the split rounding strategy is both exact and computationally efficient.

Rounding Method	Sum Constraint Enforced	Bias	Optimality (w.r.t $\ell_p$)
Midpoint Rounding	No	Systematic	No
Randomized Rounding	No/Approximate	Finite-sample bias	No
Split Rounding	Yes	Unbiased	Yes

6. Broad Applicability and Extensions

Split rounding serves as the optimal projection from continuous to integer solutions for linear constraints under $p$-norm error, with applications in:

• Parliamentary seat apportionment (largest-remainder method, Hare–Niemeyer)
• Portfolio rounding with conservation properties
• Resource and task allocation with fixed quotas
• Constrained rounding post-relaxation in integer programming

It also underlies efficient rounding schemes in more structured and stochastic allocation settings where the required fairness or constraint consistency is a hard requirement. Tree-splitting variants extend the paradigm to network and allocation problems with combinatorial structure on the set of fractional assignments (Wu et al., 2024).

7. Significance in Optimization and Computational Efficiency

The significance of split rounding strategies lies in their theoretical optimality—guaranteeing sum constraints, minimum norm error, and unbiasedness—alongside practical efficiency, with $O(N\log N)$ complexity for general vector rounding and $O(n^3)$ for structured SDPs in graph optimization. As such, split rounding and its tree-based and DP-value-function extensions constitute foundational building blocks for rounding algorithms in modern combinatorial, convex, and fair division optimization (Cont et al., 2014, Gong et al., 9 Apr 2025, Wu et al., 2024).