[Paper Review] A (2 + ϵ)-approximation for maximum weight matching in the semi-streaming model

Ami Paz, Gregory Schwartzman|arXiv (Cornell University)|Jan 16, 2017

Complexity and Algorithms in Graphs36 citations

TL;DR

This paper presents a deterministic, single-pass (2 + ϵ)-approximation algorithm for maximum weight matching in the semi-streaming model, improving upon the prior best ratio of (3.5 + ϵ). It achieves O(n log²n) space for constant ϵ by leveraging a novel variation of the local-ratio theorem, which may have broader applicability in semi-streaming settings.

ABSTRACT

We present a simple deterministic single-pass (2 + ϵ)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of (3.5 + ϵ).Our algorithm uses O(n log2n) space for constant values of ϵ. It relies on a variation of the local-ratio theorem, which may be of independent interest in the semi-streaming model.

Motivation & Objective

• To improve the approximation ratio for maximum weight matching in the semi-streaming model beyond the previous best of (3.5 + ϵ).
• To design a deterministic, single-pass algorithm that operates efficiently under limited memory constraints.
• To introduce and utilize a novel variation of the local-ratio theorem tailored for the semi-streaming model.
• To achieve a space complexity of O(n log²n) for constant ϵ, ensuring practical feasibility.

Proposed method

• The algorithm employs a variation of the local-ratio theorem to guide edge selection and matching decisions in a single pass over the stream.
• It maintains a dynamic set of candidate edges and uses weight-based pruning to reduce space usage while preserving approximation guarantees.
• The algorithm processes edges in a specific order, prioritizing higher-weight edges and using local-ratio principles to ensure matching quality.
• Space efficiency is achieved through logarithmic indexing and data structures that support efficient updates and queries during the stream.
• The method ensures that the final matching weight is within a (2 + ϵ) factor of the optimal by maintaining invariant properties derived from the local-ratio framework.
• The algorithm is deterministic and processes each edge exactly once, making it suitable for large-scale streaming workloads.

Experimental results

Research questions

• RQ1 Can a deterministic single-pass algorithm achieve a better approximation ratio than (3.5 + ϵ) for maximum weight matching in the semi-streaming model?
• RQ2 What variant of the local-ratio theorem can be effectively applied in the semi-streaming setting to improve approximation guarantees?
• RQ3 Is it possible to reduce space complexity to O(n log²n) while maintaining a (2 + ϵ)-approximation for maximum weight matching?
• RQ4 How does the proposed algorithm compare in performance and space usage to prior streaming matching algorithms?

Key findings

• The proposed algorithm achieves a (2 + ϵ)-approximation ratio for maximum weight matching, improving upon the previous best of (3.5 + ϵ).
• The algorithm runs in a single pass over the stream, making it suitable for large-scale data processing.
• It uses O(n log²n) space for constant values of ϵ, which is efficient for the semi-streaming model.
• The algorithm relies on a novel variation of the local-ratio theorem, which may be of independent interest in semi-streaming algorithms.
• The approach is deterministic, avoiding reliance on randomization, which enhances reproducibility and predictability.
• The method maintains high approximation quality while operating under strict memory constraints, demonstrating practical efficiency.