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Selection Improvements on the Parallel Iterative Algorithm for Stable Matching

Scott Wynn, Alec Kyritsis, Stephora Alberi, Enyue Lu

TL;DR

This paper tackles scalable stable matching in large-scale settings by leveraging parallelism through an $n^2$-processor architecture. It augments the Parallel Iterative Improvement (PII) algorithm with two targeted selection strategies, Right-Minimum Selection and Dynamic Selection, plus a fast Quick Initialization preprocessing step, to achieve robust convergence and reduced runtime. Empirically, the proposed PII-RMD method attains $100\%$ convergence across $3.6\times 10^6$ trials and scales favorably with larger $n$, consistently achieving $O(n \log n)$ average runtime. The work has practical implications for high-throughput systems such as switch scheduling in data centers, where fast, reliable stable matchings are essential.

Abstract

Sequential algorithms for the Stable Matching Problem are often too slow in the context of some large scale applications like switch scheduling. Parallel architectures can offer a notable decrease in runtime complexity. We propose a stable matching algorithm using $n^2$ processors that converges in $O(n log(n))$ average runtime. The algorithm is structurally based on the Parallel Iterative Improvement (PII) algorithm, where we improve the convergence rate from $90\%$ to $100\%$ over a large number of trials. We suggest alternative selection methods for pairs in the PII algorithm, called Right-Minimum and Dynamic Selection, as well as a faster preprocessing step, called Quick Initialization, resulting in full convergence over $3.6$ million trials and significantly improved runtime.

Selection Improvements on the Parallel Iterative Algorithm for Stable Matching

TL;DR

This paper tackles scalable stable matching in large-scale settings by leveraging parallelism through an -processor architecture. It augments the Parallel Iterative Improvement (PII) algorithm with two targeted selection strategies, Right-Minimum Selection and Dynamic Selection, plus a fast Quick Initialization preprocessing step, to achieve robust convergence and reduced runtime. Empirically, the proposed PII-RMD method attains convergence across trials and scales favorably with larger , consistently achieving average runtime. The work has practical implications for high-throughput systems such as switch scheduling in data centers, where fast, reliable stable matchings are essential.

Abstract

Sequential algorithms for the Stable Matching Problem are often too slow in the context of some large scale applications like switch scheduling. Parallel architectures can offer a notable decrease in runtime complexity. We propose a stable matching algorithm using processors that converges in average runtime. The algorithm is structurally based on the Parallel Iterative Improvement (PII) algorithm, where we improve the convergence rate from to over a large number of trials. We suggest alternative selection methods for pairs in the PII algorithm, called Right-Minimum and Dynamic Selection, as well as a faster preprocessing step, called Quick Initialization, resulting in full convergence over million trials and significantly improved runtime.
Paper Structure (16 sections, 3 theorems, 3 figures)

This paper contains 16 sections, 3 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parallel Architecture
  4. Statement of Results
  5. Overview
  6. Previous Work
  7. Iterative Stable Matching Algorithms
  8. The PII Algorithm
  9. The PII-SC Algorithm
  10. Selection Methods
  11. Right-Minimum-Selection
  12. Dynamic Selection
  13. Preprocessing
  14. The PII-RMD Algorithm
  15. Results
  16. ...and 1 more sections

Key Result

Lemma 3.1.1

Let $A$ be an instance of the $PII$ algorithm. If $A$ uses Right-Minimum-Selection over its entire duration, then $A$ is $NM1$-cycle free.

Figures (3)

  • Figure 1: Comparison of a single iteration of Right-Minimum Selection (from 1 to 2) and a single iteration of the original PII algorithm (from 3 to 4). Each sub-figure (1,2,3,4) contains the entries of the preference matrix, with shapes indicating the matching at a given iteration. The restriction imposed by Right-Minimum selection allows immediate convergence, while PII continues to allow multiple blocking pairs.
  • Figure 2: Example of Dynamic Selection. Each sub-figure (1,2,3,4) contains the entries of the preference matrix, with shapes indicating the matching at a given iteration. Iterations of the original PII algorithm are done for the first 2 iterations (from 1-2 and 2-3). Dynamic Selection is done for the 3rd iteration (from 3-4) leading to convergence. Note that another iteration of the original PII algorithm after the matching at 3 would return to the initial matching (1) causing a cycle.
  • Figure 3: Probability of successfully finding a stable matching within $5n$ iterations with $100,000$ trials for various $n$ ranging from $10-100$. All methods were tested using random initialization, except PII-RMD (Quick) and PII-RMD (Smart) which used Quick and Smart Initialization respectively.

Theorems & Definitions (9)

  • Definition 3.1.1: Right-Minimum-Selection
  • Lemma 3.1.1: $NM1$-Cycle Freeness
  • proof
  • Lemma 3.1.2
  • proof
  • Theorem 3.1.1: Cycle Freeness
  • proof
  • Definition 3.2.1: Dynamic Selection
  • proof