insight - Algorithms and Data Structures - # Popular Matching Algorithms

Finding the Cheapest Popular Perfect Matching in Many-to-Many Matching Scenarios

Q: How does the complexity of the algorithm change if we introduce ties in the preference lists of agents and jobs?

Introducing ties in the preference lists of agents and jobs significantly impacts the complexity of the algorithm for finding a min-cost popular perfect matching. Here's why: Breaking Symmetry: The original algorithm relies on the strict preference order of agents and jobs to construct the colorful multigraph and establish the mapping between stable matchings in this auxiliary instance and popular perfect matchings in the original instance. Ties introduce ambiguity in this mapping, as multiple stable matchings in the colorful instance might correspond to the same popular perfect matching in the original instance, or vice versa. NP-Hardness: Finding a popular matching (even without the min-cost constraint) becomes NP-hard when ties are allowed, even in the one-to-one setting [4,11]. This complexity carries over to the many-to-many setting. Need for Alternative Approaches: Due to the NP-hardness, polynomial-time algorithms are unlikely to exist for the problem with ties. We would need to explore alternative approaches like: Preference List Completion: Preprocessing the instance to resolve ties in a consistent manner, potentially using additional criteria or randomization. However, this might not always be feasible or desirable. Approximation Algorithms: Designing algorithms that find "near-popular" matchings, i.e., matchings that minimize the degree of unpopularity, measured by the number of blocking edges or other metrics. Fixed-Parameter Tractability: Exploring if the problem becomes tractable when certain parameters, such as the maximum length of a tie or the number of ties in the instance, are bounded.

Q: Could there be alternative approaches, such as approximation algorithms, that might be more efficient for large-scale instances where finding the absolute min-cost matching is computationally expensive?

Yes, approximation algorithms present a promising avenue for large-scale instances where finding the absolute min-cost popular perfect matching is computationally expensive. Here are some potential directions: Relaxing Popularity: Instead of seeking a strictly popular perfect matching, we could aim for an almost popular matching. This could involve: Bounded Unpopularity: Finding a perfect matching that minimizes the maximum vote difference against any other matching, i.e., minimizing the value of max_N Δ(M, N). Probabilistic Guarantees: Designing algorithms that return a matching that is popular with high probability, potentially using randomized rounding techniques on a fractional solution. Exploiting Structure: If the underlying graph or preference profiles exhibit specific structures (e.g., bounded treewidth, sparsity, correlated preferences), specialized algorithms or data reduction techniques might be applicable. Heuristics and Metaheuristics: Employing techniques like local search, simulated annealing, or genetic algorithms to explore the solution space efficiently and converge to near-optimal solutions. These methods often provide good trade-offs between solution quality and runtime but might not offer theoretical guarantees on the approximation factor.

Core Concepts

A min-cost popular perfect matching in a many-to-many matching scenario with preferences and edge costs can be found efficiently by transforming the problem into a min-cost stable matching problem in a specially constructed auxiliary graph.

Abstract

Bibliographic Information: Kavitha, T., & Makino, K. (2024). Perfect Matchings and Popularity in the Many-to-Many Setting. arXiv preprint arXiv:2411.00384v1.
Research Objective: To develop an efficient algorithm for finding a minimum-cost popular perfect matching in a many-to-many matching instance with preferences and edge costs.
Methodology: The paper presents a reduction of the min-cost popular perfect matching problem in a many-to-many setting to the min-cost stable matching problem in an auxiliary many-to-many instance. The authors leverage the existing polynomial-time algorithm by Fleiner [15] for computing min-cost stable matchings in multigraphs to solve the transformed problem.
Key Findings: The paper establishes a novel characterization of popular perfect matchings in many-to-many settings based on stable matchings in a colorful auxiliary instance. This characterization enables the reduction to the min-cost stable matching problem, leading to a polynomial-time algorithm for the original problem.
Main Conclusions: The research demonstrates that finding a min-cost popular perfect matching in a many-to-many setting with preferences and edge costs is tractable. This result has implications for various applications, including conference management systems and project allocation, where finding optimal and fair matchings is crucial.
Significance: This work extends the understanding of popular matchings in more general settings and provides a practical solution for a problem relevant to real-world scenarios.
Limitations and Future Research: The paper focuses on instances where a perfect matching exists. Exploring the problem's complexity when perfect matchings are not guaranteed or considering other optimality criteria beyond cost minimization could be interesting avenues for future research.

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Perfect Matchings and Popularity in the Many-to-Many Setting

by Telikepalli ... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00384.pdf

Perfect Matchings and Popularity in the Many-to-Many Setting

Deeper Inquiries

How does the complexity of the algorithm change if we introduce ties in the preference lists of agents and jobs?

Introducing ties in the preference lists of agents and jobs significantly impacts the complexity of the algorithm for finding a min-cost popular perfect matching. Here's why:

Breaking Symmetry: The original algorithm relies on the strict preference order of agents and jobs to construct the colorful multigraph and establish the mapping between stable matchings in this auxiliary instance and popular perfect matchings in the original instance. Ties introduce ambiguity in this mapping, as multiple stable matchings in the colorful instance might correspond to the same popular perfect matching in the original instance, or vice versa.
NP-Hardness:  Finding a popular matching (even without the min-cost constraint) becomes NP-hard when ties are allowed, even in the one-to-one setting [4,11]. This complexity carries over to the many-to-many setting.
Need for Alternative Approaches:  Due to the NP-hardness, polynomial-time algorithms are unlikely to exist for the problem with ties. We would need to explore alternative approaches like:

Preference List Completion:  Preprocessing the instance to resolve ties in a consistent manner, potentially using additional criteria or randomization. However, this might not always be feasible or desirable.
Approximation Algorithms: Designing algorithms that find "near-popular" matchings, i.e., matchings that minimize the degree of unpopularity, measured by the number of blocking edges or other metrics.
Fixed-Parameter Tractability:  Exploring if the problem becomes tractable when certain parameters, such as the maximum length of a tie or the number of ties in the instance, are bounded.

Could there be alternative approaches, such as approximation algorithms, that might be more efficient for large-scale instances where finding the absolute min-cost matching is computationally expensive?

Yes, approximation algorithms present a promising avenue for large-scale instances where finding the absolute min-cost popular perfect matching is computationally expensive. Here are some potential directions:

Relaxing Popularity: Instead of seeking a strictly popular perfect matching, we could aim for an almost popular matching. This could involve:

Bounded Unpopularity: Finding a perfect matching that minimizes the maximum vote difference against any other matching, i.e., minimizing the value of  max_N Δ(M, N).
Probabilistic Guarantees: Designing algorithms that return a matching that is popular with high probability, potentially using randomized rounding techniques on a fractional solution.


Exploiting Structure: If the underlying graph or preference profiles exhibit specific structures (e.g., bounded treewidth, sparsity, correlated preferences), specialized algorithms or data reduction techniques might be applicable.
Heuristics and Metaheuristics:  Employing techniques like local search, simulated annealing, or genetic algorithms to explore the solution space efficiently and converge to near-optimal solutions. These methods often provide good trade-offs between solution quality and runtime but might not offer theoretical guarantees on the approximation factor.

What are the ethical implications of using popularity as a measure of fairness in matching problems, particularly in contexts where access to resources or opportunities is at stake?

Using popularity as a measure of fairness in matching problems, especially when access to resources or opportunities is involved, raises several ethical considerations:

Bias Amplification: Popularity is inherently susceptible to biases present in the preference lists. If agents or jobs exhibit discriminatory preferences, a popular matching might perpetuate or even exacerbate existing inequalities. For instance, in a job matching scenario, if employers have unconscious biases against certain demographic groups, a popular matching could disadvantage those groups.
Tyranny of the Majority: Popularity favors matchings that satisfy the preferences of the majority, potentially at the expense of minorities or individuals with less common preferences. This could lead to unfair outcomes for those who are already marginalized or underrepresented.
Lack of Transparency: The concept of popularity, while mathematically well-defined, might not be easily interpretable or justifiable to individuals who are not familiar with the underlying algorithm or voting mechanism. This lack of transparency could erode trust in the matching process and raise concerns about potential manipulation or unfairness.
Alternatives to Consider:  Given these ethical concerns, it's crucial to consider alternative fairness notions alongside popularity, such as:

Stability: Ensuring no agent-job pair has a strong incentive to deviate from the matching, promoting a sense of fairness and preventing unraveling of the matching.
Equity and Diversity:  Incorporating constraints or objectives that promote a more equitable distribution of opportunities, ensuring representation of different groups, and mitigating the impact of biases.
Individual Fairness:  Considering the specific needs and circumstances of individuals, potentially through personalized rankings or weighting factors, to ensure that the matching process is fair at an individual level.
In conclusion, while popularity offers a tractable approach to finding desirable matchings, it's essential to be aware of its limitations and potential ethical implications. A holistic approach to fairness in matching should consider multiple criteria, including ethical considerations, transparency, and the potential impact on different stakeholders.