[Paper Review] Estimating Weighted Matchings in $o(n)$ Space

Elena Grigorescu, Morteza Monemizadeh+1 more|arXiv (Cornell University)|Apr 25, 2016

Complexity and Algorithms in Graphs24 references3 citations

TL;DR

This paper presents a novel reduction from the maximum weighted matching (MWM) problem to the maximum cardinality matching (MCM) problem in streaming graphs, achieving a 2(1+ε)-approximation factor for MWM using only o(n) space. The method partitions edges by weight, applies a streaming MCM estimator on weighted substreams, and combines estimates to achieve a (48+ε)-approximation for planar graphs.

ABSTRACT

We consider the problem of estimating the weight of a maximum weighted matching of a weighted graph $G(V,E)$ whose edges are revealed in a streaming fashion. We develop a reduction from the maximum weighted matching problem to the maximum cardinality matching problem that only doubles the approximation factor of a streaming algorithm developed for the maximum cardinality matching problem. Our results hold for the insertion-only and the dynamic (i.e, insertion and deletion) edge-arrival streaming models. The previous best-known reduction is due to Bury and Schwiegelshohn (ESA 2015) who develop an algorithm whose approximation guarantee scales by a polynomial factor. As an application, we obtain improved estimators for weighted planar graphs and, more generally, for weighted bounded-arboricity graphs, by feeding into our reduction the recent estimators due to Esfandiari et al. (SODA 2015) and to Chitnis et al. (SODA 2016). In particular, we obtain a $(48+ε)$-approximation estimator for the weight of a maximum weighted matching in planar graphs.

Motivation & Objective

• Address the challenge of estimating maximum weighted matching (MWM) in streaming graphs with sublinear space, particularly o(n) space.
• Overcome the limitations of prior reductions that scaled the approximation factor polynomially, which is suboptimal for practical and theoretical applications.
• Develop a general reduction framework that preserves the approximation factor within a factor of 2(1+ε), enabling use of existing MCM estimators for MWM estimation.
• Extend the framework to both insertion-only and dynamic (edge insertions and deletions) streaming models with adversarial edge order.
• Achieve improved approximation guarantees for structured graph classes, such as planar and bounded-arboricity graphs, by combining the reduction with prior MCM estimators.

Proposed method

• Partition edges of the weighted graph into logarithmic-sized weight classes based on powers of (1+ε), creating substreams S_i where edge weights lie in [(1+ε)^i, (1+ε)^{i+1}).
• For each substream S_i, apply a streaming algorithm for maximum cardinality matching (MCM) to estimate the size of the largest matching in the unweighted version of S_i.
• Use the estimated matching sizes Δ_i from each substream to construct a weighted sum A_0 = Σ_i Δ_i (1+ε)^i as an estimator for the total weight of a maximum weighted matching.
• Leverage concentration and probabilistic bounds to ensure that the estimated matching sizes Δ_i are within a factor of λ from the true MCM size in each substream with high probability.
• Prove that the total estimator A_0 is a 2λ(1+ε)-approximation to the true MWM weight by bounding the sum of edge weights in the maximum matching using upper and lower bounds derived from the substream estimates.
• Apply a union bound across all O(log n) substreams to maintain high-probability correctness, requiring space S(n, δ/(c log n)) per substream and resulting in O(S(n, δ/(c log n)) log n) total space.

Experimental results

Research questions

• RQ1 Can a maximum weighted matching in a streaming graph be estimated within o(n) space while preserving a constant approximation factor?
• RQ2 Is it possible to reduce the MWM problem to the MCM problem with only a constant-factor blowup in the approximation ratio, rather than a polynomial one?
• RQ3 How can the reduction be made robust in both insertion-only and dynamic (edge insertions and deletions) streaming models with adversarial edge order?
• RQ4 What approximation guarantees can be achieved for structured graph classes like planar or bounded-arboricity graphs using this reduction framework?
• RQ5 Can existing o(n)-space MCM estimators be effectively composed with this reduction to yield improved MWM estimators for specific graph families?

Key findings

• The proposed reduction achieves a 2(1+ε)-approximation factor for the maximum weighted matching problem, significantly improving upon the polynomial approximation blowup in prior work.
• The method supports both insertion-only and dynamic streaming models with adversarial edge order, broadening its applicability.
• For weighted planar graphs, the framework yields a (48+ε)-approximation estimator by combining the reduction with prior MCM estimators from SODA 2015 and SODA 2016.
• The space complexity is O(S(n, δ/(c log n)) log n), where S(n, δ) is the space required by the underlying MCM estimator, enabling efficient use of existing o(n)-space MCM algorithms.
• The analysis establishes that the estimator A_0 is a 2λ(1+ε)-approximation to the true MWM weight with high probability, under the assumption that the MCM estimator is accurate within factor λ.
• The reduction is general and can be applied to any monotone graph property, allowing reuse of specialized MCM estimators for weighted versions of such graphs without loss of efficiency or approximation quality.