Ratio Spread (RS): Optimization & Coding

• Ratio Spread (RS) is the optimization of the ratio between two submodular functions, balancing gain and cost in both combinatorial optimization and coding theory.
• In combinatorial settings, RS uses a reduction to difference-of-submodular objectives and greedy algorithms with binary search to achieve improved approximation guarantees.
• In coding theory, RS forms the basis of Constrained Reed Solomon codes that reduce PAPR in FBMC systems, ensuring efficient power control and robust error correction.

Ratio Spread (RS) is a concept that arises in both combinatorial optimization and coding theory, with distinct technical realizations in submodular function maximization and in multi-carrier communications. In combinatorial optimization, RS refers to the maximization of the ratio $f(S)/g(S)$, where $f$ and $g$ are non-negative, monotone, and submodular set functions, and $g$ is strictly positive. In coding theory, RS coders, particularly Constrained Reed Solomon (CRS) codes, are leveraged to optimize power efficiency and robustness in filter bank multi-carrier (FBMC) communication systems. Across both domains, RS formalism enables rigorous trade-off analysis and algorithmic development for objectives involving gain-to-cost ratios or signal peak minimization.

1. RS Optimization of Submodular Functions

The RS optimization problem seeks to maximize $f(S)/g(S)$ over subsets $S$, where $f$ and $g$ are set functions exhibiting submodularity and monotonicity. The canonical RS objective is

$\max_{S \subseteq \mathcal{N}} \frac{f(S)}{g(S)}$

where $\mathcal{N}$ is the ground set, $f:\mathcal{N} \rightarrow \mathbb{R}_{\geq 0}$, and $g:\mathcal{N} \rightarrow \mathbb{R}_{>0}$ are monotone submodular.

A foundational algorithmic approach reduces the ratio problem to a parametric family of difference objectives $f(S) - \lambda g(S)$, where $\lambda$ is iteratively tuned, typically via binary search, to approximate the optimal RS value. If one can, for a particular $\lambda$, identify $S$ that (approximately) maximizes $f(S) - \lambda g(S)$, then setting $\lambda^* = f(S^*)/g(S^*)$ for the true optimum $S^*$ provides a solution to the original RS problem. This reduction establishes a formal approximation relationship between RS and difference-of-submodular (DS) problems (Perrault et al., 2021).

2. Greedy Algorithms for RS and DS Objectives

The paper introduces and analyzes a greedy framework for both RS and DS objectives, formalizing the GreedRatio algorithm for RS:

• At each iteration, select the element $i$ that maximizes the marginal ratio

$$\frac{f(S \cup \{i\}) - f(S)}{g(S \cup \{i\}) - g(S)}$$

This approach generalizes to objectives $\Psi(f(S), g(S))$, where $\Psi$ is quasiconvex and non-decreasing in $f$. Specifically, setting $\Psi(f, g) = f/g$ recovers RS; setting $\Psi(f, g) = f-g$ recovers DS. The unified greedy analysis corrects issues in previous works for non-normalized functions and extends algorithmic guarantees to cases with additional function parameters such as the curvature of $g$. For RS, the greedy approach aligns with state-of-the-art procedures, but the analysis introduces new bounds and applies for more general settings (Perrault et al., 2021).

3. Approximation Guarantees and Theoretical Bounds

Approximation factors for RS optimization can be achieved using a fully polynomial-time approximation scheme (FPTAS) that combines interval-doubling or binary search over $\lambda$ with greedy submodular maximization. For RS, the best-known approximation ratios are $O(n^{-1/2} \log^{-1}(n))$, where $n$ is the ground set size. DS optimization, by contrast, is inapproximable in the classical sense; however, by adopting a weaker "relative approximation" guarantee (e.g., ensuring $f(S) - g(S) \geq \alpha(f(S^*) - g(S^*))$ for some $\alpha$), one can leverage RS approximation algorithms to construct comparable guarantees for DS problems (Perrault et al., 2021).

This provides a tight link between the complexity of RS and DS optimization, and demonstrates that improvements in ratio spread algorithms directly transfer to difference-of-submodular settings under relaxed approximation notions.

4. Applications and Impact Across Domains

Combinatorial Optimization

The RS formulation enables practical algorithmic solutions for problems balancing trade-offs between desirables (f) and costs (g), such as maximizing utility per cost or selecting subsets with optimal gain-cost ratios. The equivalence between RS and DS optimization under relaxed guarantees expands the toolkit for problems where difference objectives are classically inapproximable, facilitating solution approaches in domains ranging from data summarization to resource allocation.

Multi-Carrier Communication Systems

In coding theory, RS refers to Reed Solomon block codes, and specifically in FBMC-OQAM systems, to Constrained Reed Solomon (CRS) coding schemes designed to minimize Peak to Average Power Ratio (PAPR):

• CRS coding strategically distributes bits between message and parity symbols, expanding the overall symbol space and reducing per-symbol information density. The key constraint is $p < q$ (message bits per symbol $<$ parity bits per symbol), which ensures lower PAPR.
• The constraint formalized in

$k' p = r q$

with $k', r, p, q$ chosen to ensure robust error correction and reduced PAPR.

• Simulation in (Chunkath et al., 2023) demonstrates that CRS(31,19) coding, possibly augmented with u-law companding, confines PAPR variations to as little as 0.55 dB—critical for avoiding nonlinear distortion in RF amplifiers.
• CRS coding preserves maximum distance separability (MDS), maintaining strong BER performance across channel models, even when further PAPR minimization is achieved.

5. Hybrid Techniques: Coding and Companding

Hybridization of CRS coding with nonlinear companding transforms (e.g., u-law) offers additional PAPR reduction. The u-law companding function:

$$F(y) = \text{sgn}(y) \cdot \frac{\ln(1 + u|y|)}{\ln(1 + u)}, \quad -1 \leq y \leq 1$$

with $u=25$ provides amplitude compression, lowering signal peaks. The receiver applies the inverse transform for accurate amplitude recovery. When paired with CRS coding, such hybrid approaches yield PAPR reductions from approximately 5.15 dB (full load) to 4.6 dB (random load), while sustaining BER performance benefits in both Pedestrian B and Vehicular A channel models (Chunkath et al., 2023). The joint application guarantees both efficient peak control and strong error correction, outperforming alternatives like BCH or conventional RS codes at low SNR.

6. Interconnections, Generalizations, and Future Directions

The formal reduction between RS and DS optimization facilitates bidirectional transfer of algorithmic advances. The unified greedy framework for general $\Psi(f, g)$ objectives hints at broader applicability across other quasiconvex or trade-off-driven formulations.

In communications, the RS formalism in coding theory—especially when enhanced by parameter constraints and hybridization—improves multi-carrier systems’ spectral efficiency and robustness to amplifier imperfections.

A plausible implication is that continued advances in algorithmic theory for RS optimization may yield new classes of near-optimal coding schemes and power control algorithms, especially in systems where gain-cost trade-offs underpin aggregate system performance.

7. Summary Table: Comparative Aspects of RS Formalism

Domain	RS Definition	Algorithmic Approach
Submodular Optimization	$\max_S f(S)/g(S)$	GreedRatio, FPTAS
Error-Control Coding (CRS)	Constrained RS parameters	Symbol distribution, hybrid coding & companding

The table above summarizes RS in the principal domains addressed in the referenced works. In both cases, control over ratio spread leads to optimized system performance—either through bounded approximability in combinatorial settings or minimized signal distortion and error rates in coded communications.

RS serves as a unifying measure for quantifying and algorithmically balancing trade-offs in gain versus cost, with broad theoretical and practical significance.