Structural Results on Matching Estimation with Applications to Streaming

We study the problem of estimating the size of a matching when the graph is revealed
in a streaming fashion. Our results are multifold:
1. We give a tight structural result relating the size of a maximum matching to the
arboricity \(\alpha\) of a graph, which has been one of the most studied graph parameters
for matching algorithms in data streams. One of the implications is an algorithm
that estimates the matching size up to a factor of \((\alpha+2)\cdot(1+\varepsilon)\) using \(\tilde{O}(n^{2/3})\)
space in insertion-only graph streams and \(\tilde{O}(n^{4/5})\) space in dynamic streams,
where n is the number of nodes in the graph. We also show that in the vertex
arrival insertion-only model, an \((\alpha+2)\) approximation can be achieved using only
\(O(\log n)\) space.
2. We further show that the weight of a maximum weighted matching can be efficiently estimated by augmenting any routine for estimating the size of an unweighted
matching. Namely, given an algorithm for computing a \(\lambda\)-approximation
in the unweighted case, we obtain a \(2(1+\varepsilon)\lambda\) approximation for the weighted case,
while only incurring a multiplicative logarithmic factor in the space bounds. The
algorithm is implementable in any streaming model, including dynamic streams.
3. We also investigate algebraic aspects of computing matchings in data streams, by
proposing new algorithms and lower bounds based on analyzing the rank of the
Tutte-matrix of the graph. In particular, we present an algorithm determining
whether there exists a matching of size k using \(O(k^2\log n)\) space.
4. We also show a lower bound of \(\Omega(n^{1-\varepsilon})\) space for small approximation factors to
the maximum matching size in insertion-only streams. This lower bound also holds
for approximating the rank of a matrix.