[Paper Review] Online Matching in a Ride-Sharing Platform

Chinmoy Dutta, Chris Sholley|arXiv (Cornell University)|Jun 27, 2018

Optimization and Search Problems8 references3 citations

TL;DR

This paper introduces Online Windowed Non-Bipartite Matching (OWNBM), a novel graph-theoretic model for online ride-matching in ride-sharing platforms. It presents a randomized 1/4-competitive algorithm for the edge-weighted version using combinatorial auction theory, and a 1/2(1−1/e)-competitive algorithm for the vertex-weighted version via randomized primal-dual analysis, both under temporal and spatial constraints.

ABSTRACT

We propose a formal graph-theoretic model for studying the problem of matching rides online in a ride-sharing platform. Unlike most of the literature on online matching, our model, that we call {\em Online Windowed Non-Bipartite Matching} ($\mbox{OWNBM}$), pertains to online matching in {\em non-bipartite} graphs. We show that the edge-weighted and vertex-weighted versions of our model arise naturally in ride-sharing platforms. We provide a randomized $\frac{1}{4}$-competitive algorithm for the edge-weighted case using a beautiful result of Lehmann, Lehmann and Nisan (EC 2001) for combinatorial auctions. We also provide an $\frac{1}{2} (1 - \frac{1}{e})$-competitive algorithm for the vertex-weighted case (with some constraint relaxation) using insights from an elegant randomized primal-dual analysis technique of Devanur, Jain and Kleinberg (SODA 2013).

Motivation & Objective

• To formalize the online ride-matching problem in ride-sharing platforms using a graph-theoretic framework.
• To model the trade-offs between matching efficiency and matching rate in real-time, spatially and temporally constrained environments.
• To develop competitive online algorithms for non-bipartite matching under adversarial arrival sequences.
• To address the lack of formal mathematical modeling for ride-sharing matching in existing literature.
• To establish theoretical guarantees for online matching in non-bipartite graphs, a gap in prior work.

Proposed method

• Proposes the Online Windowed Non-Bipartite Matching (OWNBM) model to capture spatial, temporal, and online constraints in ride-sharing matching.
• Uses a randomized algorithm for the edge-weighted OWNBM case, leveraging a result from Lehmann, Lehmann, and Nisan (EC 2001) on combinatorial auctions.
• Applies a randomized primal-dual analysis technique from Devanur, Jain, and Kleinberg (SODA 2013) for the vertex-weighted case with relaxed constraints.
• Introduces the concept of semi-matchings and half-weight to analyze expected performance in the vertex-weighted setting.
• Employs a three-stage process: constructing a semi-matching, transforming it into a 3-matching, and ensuring edge selection within the matching window.
• Defines origin-type and destination-type semi-matchings to compute half-weights, enabling lower bounding of expected performance.

Experimental results

Research questions

• RQ1 How can online ride-matching in ride-sharing platforms be formally modeled as an online non-bipartite matching problem?
• RQ2 What competitive ratio can be achieved for edge-weighted online matching in non-bipartite graphs under temporal and spatial constraints?
• RQ3 Can a randomized primal-dual approach yield improved competitive ratios for vertex-weighted online non-bipartite matching with window constraints?
• RQ4 What is the theoretical performance limit of online matching in non-bipartite graphs when decisions must be made in real-time with incomplete future information?
• RQ5 How do spatial overlap and time window constraints affect the design and competitiveness of online matching algorithms in ride-sharing systems?

Key findings

• The paper presents a randomized 1/4-competitive algorithm for the edge-weighted OWNBM problem, leveraging results from combinatorial auction theory.
• For the vertex-weighted OWNBM variant, it achieves a competitive ratio of 1/2(1−1/e) using a randomized primal-dual technique with constraint relaxation.
• The expected half-weight of the semi-matching constructed by Algorithm 3 is proven to be at least 1/2(1−1/e) times the optimal matching weight.
• A transformation from semi-matching to 3-matching ensures that the final matching weight is at least the half-weight of the semi-matching.
• The algorithm guarantees that any matched edge is selected no later than d time steps after the arrival of the terminal vertex, satisfying the matching window constraint.
• The proposed OWNBM model is the first formal framework for online non-bipartite matching in adversarial settings with temporal and spatial constraints.