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We consider the maximum matching problem in the semi-streaming model formalized by Feigenbaum, Kannan, McGregor, Suri, and Zhang that is inspired by giant graphs of today. As our main result, we give a two-pass (1/2 + 1/16)-approximation algorithm for triangle-free graphs and a two-pass (1/2 + 1/32)-approximation algorithm for general graphs; these improve the approximation ratios of 1/2 + 1/52 for bipartite graphs and 1/2 + 1/140 for general graphs by Konrad, Magniez, and Mathieu. In three passes, we are able to achieve approximation ratios of 1/2 + 1/10 for triangle-free graphs and 1/2 + 1/19.753 for general graphs. We also give a multi-pass algorithm where we bound the number of passes precisely—we give a (2/3 − ε)- approximation algorithm that uses 2/(3ε) passes for triangle-free graphs and 4/(3ε) passes for general graphs. Our algorithms are simple and combinatorial, use O(n log n) space, and (can be implemented to) have O(1) update time per edge. For general graphs, our multi-pass algorithm improves the best known deterministic algorithms in terms of the number of passes: * Ahn and Guha give a (2/3−ε)-approximation algorithm that uses O(log(1/ε)/ε2) passes, whereas our (2/3 − ε)-approximation algorithm uses 4/(3ε) passes; * they also give a (1 − ε)-approximation algorithm that uses O(log n · poly(1/ε)) passes, where n is the number of vertices of the input graph; although our algorithm is (2/3−ε)-approximation, our number of passes do not depend on n. Earlier multi-pass algorithms either have a large constant inside big-O notation for the number of passes or the constant cannot be determined due to the involved analysis, so our multi-pass algorithm should use much fewer passes for approximation ratios bounded slightly below 2/3. 

We revisit the much-studied problem of space-efficiently estimating the number of triangles in a graph stream, and extensions of this problem to counting fixed-sized cliques and cycles, obtaining a number of new upper and lower bounds. For the important special case of counting triangles, we give a $4$-pass, $(1\pm\varepsilon)$-approximate, randomized algorithm that needs at most $\widetilde{O}(\varepsilon^{-2}\cdot m^{3/2}/T)$ space, where $m$ is the number of edges and $T$ is a promised lower bound on the number of triangles. This matches the space bound of a very recent algorithm (McGregor et al., PODS 2016), with an arguably simpler and more general technique. We give an improved multi-pass lower bound of $\Omega(\min\{m^{3/2}/T, m/\sqrt{T}\})$, applicable at essentially all densities $\Omega(n) \le m \le O(n^2)$. We also prove other multi-pass lower bounds in terms of various structural parameters of the input graph. Together, our results resolve a couple of open questions raised in recent work (Braverman et al., ICALP 2013). Our presentation emphasizes more general frameworks, for both upper and lower bounds. We give a sampling algorithm for counting arbitrary subgraphs and then improve it via combinatorial means in the special cases of counting odd cliques and odd cycles. Our results show that these problems are considerably easier in the cash-register streaming model than in the turnstile model, where previous work had focused (Manjunath et al., ESA 2011; Kane et al., ICALP 2012). We use Tur{\'a}n graphs and related gadgets to derive lower bounds for counting cliques and cycles, with triangle-counting lower bounds following as a corollary.