1. Introduction

Coordinating a meeting location among multiple participants is a recurring problem in social, professional, and urban contexts. While digital maps provide routing capabilities, they rarely offer group-aware meeting point optimization.

Most existing solutions implicitly assume:

• Euclidean distance as a proxy for effort
• Static participants
• Single-objective optimization

These assumptions fail in dense urban environments, where travel time, fairness, and accessibility dominate user experience.

We propose Where2Meet, a system that treats meeting location selection as a multi-objective optimization problem under real-world constraints.

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2. Problem Formulation

Let the set of participants be:

$$U = \{u_1, u_2, \dots, u_n\}$$

Each participant $u_i$ is associated with a geographic coordinate:

$$L_i = (lat_i, lng_i)$$

Our goal is to compute an optimal meeting location $M$ that minimizes collective inconvenience while maintaining fairness and practical feasibility.

We define the optimization objective as:

$$\min_{M} \quad \alpha \cdot f_{geo}(M) + \beta \cdot f_{travel}(M) + \gamma \cdot f_{poi}(M)$$

where:

• $f_{geo}$ measures geometric fairness
• $f_{travel}$ captures transportation cost
• $f_{poi}$ evaluates location suitability
• $\alpha, \beta, \gamma$ are tunable weights

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3. Baseline Method: Geometric Centroid

The simplest baseline computes the arithmetic centroid of all participant locations:

$$M_{centroid} = \left( \frac{1}{n}\sum_{i=1}^{n} lat_i, \frac{1}{n}\sum_{i=1}^{n} lng_i \right)$$

Limitations

Despite its simplicity, this approach suffers from:

• Sensitivity to outliers
• Ignorance of transportation networks
• Potential placement in non-accessible regions

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4. Geometric Optimization via Minimum Enclosing Circle

To improve fairness, we adopt the Minimum Enclosing Circle (MEC) formulation.

The MEC center minimizes the maximum distance any participant must travel:

$$M_{mec} = \arg\min_{M} \max_{i} d(M, L_i)$$

where $d(\cdot)$ denotes geographic distance.

We employ Welzl’s randomized algorithm, which computes the MEC in expected linear time:

$$\mathcal{O}(n)$$

This optimization ensures worst-case fairness and robustness against skewed distributions.

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5. Travel-Time Aware Optimization

Geometric distance alone is insufficient in urban settings. We replace $d(\cdot)$ with a network-based travel time function.

For a candidate location $c$, the travel cost is defined as:

$$f_{travel}(c) = \sum_{i=1}^{n} w_i \cdot T(L_i \rightarrow c)$$

where:

• $T(\cdot)$ denotes estimated travel time
• $w_i$ represents participant-specific weights

To improve performance, we use:

• Isochrone-based candidate pruning
• Batched routing queries
• Redis-based memoization of travel costs

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6. Candidate Location Filtering with POI Constraints

Instead of arbitrary coordinates, Where2Meet restricts recommendations to valid Points of Interest (POIs):

$$C = \{c_1, c_2, \dots, c_k\}$$

Each candidate is scored as:

$$Score(c) = -\alpha \sum_{i} T(L_i \rightarrow c) -\beta \cdot \mathrm{Var}(T_i) +\gamma \cdot Q(c)$$

where:

• $\mathrm{Var}(T_i)$ penalizes unfair travel distributions
• $Q(c)$ represents POI quality signals (rating, category, transit access)

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7. Real-Time Incremental Optimization

Where2Meet supports dynamic participant updates:

• New users trigger incremental recomputation
• MEC boundaries are updated without full recomputation
• Candidate scores are partially recomputed
• Results are streamed to clients via Server-Sent Events (SSE)

This design enables low-latency collaboration and scalability across sessions.

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8. System Architecture

The system follows a modular pipeline:

1. User location ingestion
2. Geometric pre-optimization
3. Candidate POI generation
4. Travel-time estimation
5. Multi-objective scoring
6. Real-time visualization

Each component can be independently optimized or replaced.

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9. Experimental Evaluation

We evaluate our approach using the following metrics:

• Average travel time
• Maximum travel time
• Travel-time variance
• POI suitability score

Method	Avg Time ↓	Max Time ↓	Variance ↓
Centroid	✗	✗	✗
MEC	✓	✓✓	✓
MEC + Travel-Time	✓✓	✓✓	✓✓
Full Model	✓✓✓	✓✓	✓✓✓

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10. Discussion

Our results highlight three key insights:

1. Fairness cannot be captured by averages alone
2. Transportation networks dominate real-world usability
3. Candidate restriction significantly improves user experience

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11. Conclusion and Future Work

Where2Meet demonstrates that meeting location recommendation benefits from the integration of computational geometry, transportation modeling, and real-time system design.

Future work includes:

• Personalized utility functions
• Multi-modal routing fusion
• Learning-based weight adaptation
• Preference negotiation among participants

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References

1. Welzl, E. Smallest Enclosing Disks (Balls and Ellipsoids), 1991.
2. Okabe et al. Spatial Tessellations, Wiley, 2000.
3. Zheng, Y. Urban Computing, MIT Press, 2019.