In this paper, we study Ranking, a well-known randomized greedy matching algorithm, for general graphs. The algorithm was originally introduced by Karp, Vazirani, and Vazirani [STOC 1990] for the online bipartite matching problem with one-sided vertex arrivals, where it achieves a tight approximation ratio of \(1 -1/e\). It was later extended to bipartite graphs with random vertex arrivals by Mahdian and Yan [STOC 2011] and to general graphs by Goel and Tripathi [FOCS 2012]. The Ranking algorithm for general graphs is as follows: a permutation \(\sigma\) over the vertices is chosen uniformly at random. The vertices are then processed sequentially according to this order, with each vertex being matched to the first available neighbor (if any) according to the same permutation \(\sigma\).

We study streaming algorithms for the maximum directed cut problem. The edges of an n-vertex directed graph arrive one by one in an arbitrary order, and the goal is to estimate the value of the maximum directed cut using a single pass and small space. With O(n) space, a (1−ε)-approximation can be trivially obtained for any fixed ε > 0 using additive cut sparsifiers. The question that has attracted significant attention in the literature is the best approximation achievable by algorithms that use truly sublinear (i.e., n1−Ω(1)) space. A lower bound of Kapralov and Krachun (STOC’19) implies .5-approximation is the best one can hope for. The current best algorithm for general graphs obtains a .485-approximation due to the work of Saxena, Singer, Sudan, and Velusamy (FOCS’23). The same authors later obtained a (1/2−ε)-approximation, assuming that the graph is constant-degree (SODA’25). In this paper, we show that for any ε > 0, a (1/2−ε)-approximation of maximum dicut value can be obtained with n1−Ωε(1) space in *general graphs*. This shows that the lower bound of Kapralov and Krachun is generally tight, settling the approximation complexity of this fundamental problem. The key to our result is a careful analysis of how correlation propagates among high- and low-degree vertices, when simulating a suitable local algorithm.

In this paper, we study the weighted stochastic matching problem. Let $G=(V, E)$ be a given edge-weighted graph and let its realization $\mathcal{G}$ be a random subgraph of $G$ that includes each edge $e\in E$ independently with a known probability $p_e$. The goal in this problem is to pick a sparse subgraph $Q$ of $G$ without prior knowledge of $G$'s realization, such that the maximum weight matching among the realized edges of $Q$ (i.e. the subgraph $Q\cap \mathcal{G}$) in expectation approximates the maximum weight matching of the entire realization $\mathcal{G}$. Attaining any constant approximation ratio for this problem requires selecting a subgraph of max-degree $Ω(1/p)$ where $p=\min_{e\in E} p_e$. On the positive side, there exists a $(1-ε)$-approximation algorithm by Behnezhad and Derakhshan, albeit at the cost of max-degree having exponential dependence on $1/p$. Within the $\text{poly}(1/p)$ regime, however, the best-known algorithm achieves a $0.536$ approximation ratio due to Dughmi, Kalayci, and Patel improving over the $0.501$ approximation algorithm by Behnezhad, Farhadi, Hajiaghayi, and Reyhani. In this work, we present a 0.68 approximation algorithm with $O(1/p)$ queries per vertex, which is asymptotically tight. This is even an improvement over the best-known approximation ratio of $2/3$ for unweighted graphs within the $\text{poly}(1/p)$ regime due to Assadi and Bernstein. The $2/3$ approximation ratio is proven tight in the presence of a few correlated edges in $\mathcal{G}$, indicating that surpassing the $2/3$ barrier should rely on the independent realization of edges. Our analysis involves reducing the problem to designing a randomized matching algorithm on a given stochastic graph with some variance-bounding properties.

Given a graph $G$, an edge-coloring is an assignment of colors to edges of $G$ such that any two edges sharing an endpoint receive different colors. By Vizing's celebrated theorem, any graph of maximum degree $Δ$ needs at least $Δ$ and at most $(Δ+ 1)$ colors to be properly edge colored. In this paper, we study edge colorings in the streaming setting. The edges arrive one by one in an arbitrary order. The algorithm takes a single pass over the input and must output a solution using a much smaller space than the input size. Since the output of edge coloring is as large as its input, the assigned colors should also be reported in a streaming fashion. The streaming edge coloring problem has been studied in a series of works over the past few years. The main challenge is that the algorithm cannot "remember" all the color assignments that it returns. To ensure the validity of the solution, existing algorithms use many more colors than Vizing's bound. Namely, in $n$-vertex graphs, the state-of-the-art algorithm with $\widetilde{O}(n s)$ space requires $O(Δ^2/s + Δ)$ colors. Note, in particular, that for an asymptotically optimal $O(Δ)$ coloring, this algorithm requires $Ω(nΔ)$ space which is as large as the input. Whether such a coloring can be achieved with sublinear space has been left open. In this paper, we answer this question in the affirmative. We present a randomized algorithm that returns an asymptotically optimal $O(Δ)$ edge coloring using $\widetilde{O}(n \sqrtΔ)$ space. More generally, our algorithm returns a proper $O(Δ^{1.5}/s + Δ)$ edge coloring with $\widetilde{O}(n s)$ space, improving prior algorithms for the whole range of $s$.