On Estimating Maximum Matching Size in Graph Streams

Abstract

We study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams, which only contain edge insertions, and dynamic streams that allow both insertions and deletions of the edges, and present new upper and lower bound results for both cases.On the upper bound front, we show that an α-approximate estimate of the matching size can be computed in dynamic streams using O(n 2 /α 4 ) space, and in insertiononly streams using O(n/α 2 )-space. These bounds respectively shave off a factor of α from the space necessary to compute an α-approximate matching (as opposed to only size), thus proving a non-trivial separation between approximate estimation and approximate computation of matchings in data streams.On the lower bound front, we prove that any α-approximation algorithm for estimating matching size in dynamic graph streams requires Ω( √ n/α 2.5 ) bits of space, even if the underlying graph is both sparse and has arboricity bounded by O(α). We further improve our lower bound to Ω(n/α 2 ) in the case of dense graphs. These results establish the first non-trivial streaming lower bounds for superconstant approximation of matching size.Furthermore, we present the first super-linear space lower bound for computing a (1 + ε)-approximation of matching size even in insertion-only streams. In particular, we prove that a (1 + ε)-approximation to matching size requires RS(n) · n 1−O(ε) space; here, RS(n) denotes the maximum number of edge-disjoint induced matchings of size Θ(n) in an n-vertex graph. It is a major open problem with far-reaching implications to determine the value of RS(n), and current results leave open the possibility that RS(n) may be as large as n/ log n. Moreover, using the best known lower bounds for RS(n), our result already rules out any O(n · poly(log n/ε))-space algorithm for (1 + ε)-approximation of matchings. We also show how to avoid the dependency on the parameter RS(n) in proving lower bound for dynamic streams and present a near-optimal lower bound of n 2−O(ε) for (1 + ε)-approximation in this model. Using a well-known connection between matching size and matrix rank, all our lower bounds also hold for the problem of estimating matrix rank. In particular our results imply a near-optimal n 2−O(ε) bit lower bound for (1 + ε)-approximation of matrix ranks for dense matrices in dynamic streams, answering an open question of Li and Woodruff (STOC 2016).

Testing Matrix Rank, Optimally

“…Our work has two parts: (1) we resolve the query complexity of non-adaptive matrix rank testing, a wellstudied problem in this model, and (2) we develop a new framework for testing numerical properties of real matrices, including the stable rank, the Schatten-p norms and the SVD entropy. Our results are summarized in Table 1.…”

“…• After processing the i-th column of A, the restricted column A R (i) ,i is in the column space of S R (i) ,: . 2 In the number of parts d + d log( 1 η ), the first term follows from the operation of augmenting 1 × 1 submatrix to d × d. The second term follows from moving the submatrix towards the upper left corner (from the lower-right corner in the worst case).…”

“…Previous papers [2,11,17] used the BHH 0 n,p problem to prove lower bounds for estimating matching size in the data stream: given an instance (x, M) in BHH 0 n,p (Denote by D BHH the hard distribution of BHH 0 n,p ), we create a graph G(V ∪ W, E) with |V | = |W | = n via the following algorithm.…”

Polynomial pass lower bounds for graph streaming algorithms

“…where H (A | B = B) is defined in a standard way by using the distribution of A conditioned on the event B = B in Eq (13). The mutual information of two random variables A and B is denoted by I (A ; B) and is defined as:…”

“…Using a reduction from hidden-pointer chasing, we prove that any algorithm for submodular function minimization needs to make n 2−o(1) value queries to the function unless it has a polynomial degree of adaptivity.A vast body of work in graph streaming lower bounds concerns algorithms that make only one or a few passes over the stream. Examples of single-pass lower bounds include the ones for diameter [60], approximate matchings [13,14,63,84], exact minimum/maximum cuts [119], and maximal independent sets [10,46]. Examples of multi-pass lower bounds include the ones for BFS trees [60], perfect matchings [67], shortest path [67], and minimum vertex cover and dominating set [71].…”