A Stable Polygamy Approach to Spectrum Access with Channel Reuse

TL;DR

This work introduces the Stable Polygamy Problem (SPP), a broad generalization of stable matching for spectrum access with channel reuse, allowing many users from a large set $\mathcal{L}$ to share a channel in $\mathcal{S}$ while respecting social constraints $\\mathcal{C}_{\\ell}$. It develops a graph-theoretic vertex-coloring view of SPP, proving NP-hardness for common-utility SPP and providing distributed and centralized algorithms—DSSAR for common utility and RP&R for ranking—that solve SPP in practical, spectrum-access scenarios. The paper also establishes solvability in several structured graph regimes and demonstrates near-optimal performance via extensive simulations, highlighting the approach’s viability for 5G-and-beyond spectrum management with channel reuse. Overall, the results offer a principled, scalable framework for stable spectrum allocation under interference constraints, with significant implications for efficient channel reuse and network throughput in wireless deployments.

Abstract

We introduce a new and broader formulation of the stable marriage problem (SMP), called the stable polygamy problem (SPP), where multiple individuals from a larger group $L$ of $|L|$ individuals can be matched with a single individual from a smaller group $S$ of $|S|$ individuals. Each individual $\ell \in L$ possesses a social constraint set $C_{\ell}$ that contains a subset of $L$ with whom they cannot coexist harmoniously. We define a generalized concept of stability based on the preference and constraints of the individuals. We explore two common settings: common utility, where the utility of a match is the same for individuals from both sets, and preference ranking, where each individual has a preference ranking for every other individual in the opposite set. Our analysis is conducted using a novel graph-theoretical framework. The classic SMP has been investigated in recent years for spectrum access to match cells or users to channels, where only one-to-one matching is allowed. By contrast, the new SPP formulation allows us to solve more general models with channel reuse, where multiple users may access the same channel simultaneously. Interestingly, we show that classic algorithms, such as propose and reject (P&R), and Hungarian method are no longer efficient in the polygamy setting. We develop efficient algorithms to solve the SPP in polynomial time, tailored for implementations in spectrum access with channel reuse. We analytically show that our algorithm always solves the SPP with common utility. While the SPP with preference ranking cannot be solved by any algorithm in all cases, we prove that our algorithm effectively solves it in specific graph structures representing strong and weak interference regimes. Simulation results demonstrate the efficiency of our algorithms across various spectrum access scenarios.

Figures (9)

• Figure 1: A sample visualization of an empty social constraints graph with 6 nodes ($L=6$).
• Figure 2: A sample visualization of a complete social constraints graph with 6 nodes ($L=6$).
• Figure 3: The results for common utility for a small network, measured by data rate as a function of the number of cells ($L$), averaged across 10,000 random experiments for various algorithms.
• Figure 4: The results for total social welfare for a small network as a function of the number of cells ($L$), averaged across 10,000 random experiments for various algorithms.
• Figure 5: The results for common utility for a larger network, measured by data rate as a function of the number of cells ($L$), averaged across 10,000 random experiments for various algorithms.

Theorems & Definitions (23)

• Definition 1: Social compatibility
• Definition 2: Social availability
• Definition 3: Admissible polygamy
• Definition 4: Harmonious polygamy
• Definition 5: Stable polygamy
• Definition 6: Stable polygamy problem
• Corollary 1: SMP as a special case of SPP
• proof
• Theorem 1
• proof