Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.02105

Faster Semi-Streaming and Dynamic Approximate Maximum Weight Matchings in General Graphs: A Unified Framework

Jiale Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the $(1-\varepsilon)$-approximate maximum weight matching problem in general graphs with $n$ vertices and integer edge weights in $[1,W]$. For $W=n^{O(1)}$, we obtain faster algorithms in the semi-streaming and dynamic settings: (1) Semi-streaming: Deterministic $O(\varepsilon^{-2})$-pass and $O(\varepsilon^{-2}\log(1/\varepsilon))$-pass algorithms for unweighted and weighted matching. (2) Incremental: A deterministic algorithm for weighted matching processing $m$ edge insertions in total time $O((m\varepsilon^{-3}+n\varepsilon^{-6})\log^3 n)$. (3) Decremental: A randomized algorithm for weighted matching processing all edge deletions from an $m$-edge graph in total time $O((m\varepsilon^{-8}+n\varepsilon^{-47})\log^{O(1)}n)$ with high probability. (4) Fully dynamic: A deterministic reduction from weighted matching in general graphs to unweighted matching in bipartite graphs incurring $O(\varepsilon^{-4}\log^{O(1)}(1/\varepsilon))$ multiplicative update-time overhead, independent of $n$. Underlying these results is a unified framework that reduces the problem to repeatedly solving two simpler matching subproblems: a $(1-\varepsilon)$-approximate maximum weight matching on a sparse subgraph with $O(n/\varepsilon)$ edges, and a constant approximation on the remaining edges. The framework builds on dual entropy regularization, a smoothing technique underlying efficient solvers for packing linear programs. To handle the exponentially many odd-set constraints, we adapt the smoothing technique to regularize only the vertex duals and develop a new cut partition that preserves odd-set feasibility during optimization.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jiale Chen. 2026-11-01. Faster Semi-Streaming and Dynamic Approximate Maximum Weight Matchings in General Graphs: A Unified Framework. https://arxiv.org/abs/2611.02105

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Loop unrolling of UCA models: distance labeling

A proper circular-arc (PCA) model is a pair $M = (C, A)$ where $C$ is a circle and $A$ is a family of inclusion-free arcs on $C$ whose extremes are pairwise different. The model $M$ represents a digraph $D$ that has one vertex $v(a)$ for each $a \in A$ and one edge $v(a) \to v(b)$ for each pair of arcs $a,b \in A(M)$ such that the beginning point of $b$ belongs to $a$. For $k \geq 0$, the $k$-th power $D^k$ of $D$ has the same vertices as $D$ and $v(a) \to v(b)$ is an edge of $D^k$ when $a\neq b$ and the distance from $v(a)$ to $v(b)$ in $D$ is at most $k$. A unit circular-arc (UCA) model is a PCA model $U = (C,A)$ in which all the arcs have the same length $\ell+1$. If $\ell$, the length $c$ of $C$, and the extremes of the arcs of $A$ are integer, then $U$ is a $(c,\ell)$-CA model. For $i \geq 0$, the model $i \times U$ of $U$ is obtained by replacing each arc $(s,s+\ell+1)$ with the arc $(s,s+i\ell+1)$. If $U$ represents a digraph $D$, then $U$ is $k$-multiplicative when $i \times U$ represents $D^i$ for every $0 \leq i \leq k$. In this article we design a linear time algorithm to decide if a PCA model $M$ is equivalent to a $k$-multiplicative UCA model when $k$ is given as input. The algorithm either outputs a $k$-multiplicative UCA model $U$ equivalent to $M$ or a negative certificate that can be authenticated in linear time. Our main technical tool is a new characterization of those PCA models that are equivalent to $k$-multiplicative UCA models. For $k=1$, this characterization yields a new algorithm for the classical representation problem that is simpler than the previously known algorithms.

cs.DS↗
arXiv2026

Improved Algorithms for Unrelated Crowd Worker Scheduling in Mobile Social Networks

This paper studies task scheduling with unrelated crowd workers in mobile social networks, where the service time of a task depends on the crowd worker that performs it, and the objective is to minimize the total weighted completion time. For identical crowd workers, we tighten the approximation ratio of the Largest-Ratio-First (LRF) algorithm and update the competitive ratio of its online version. For unrelated crowd workers, we show that the time-indexed rounding framework of Im and Li remains valid when worker-dependent contact delays are added, which yields a randomized algorithm with an expected approximation ratio of 1.45; this improves upon the expected ratio of $1.5+ε$ obtained in our previous work for identical crowd workers. We also derandomize this algorithm. To improve computational efficiency, we further propose an algorithm that exploits the optimality of the Smith-ratio order on a single crowd worker, and we prove that it is a deterministic 1.5-approximation algorithm and that its polynomial-time version achieves an approximation ratio arbitrarily close to 1.5. Experiments on synthetic instances and on instances derived from real contact traces show that this algorithm outperforms three variants of LRF in all tested settings, with the largest advantage when the crowd workers are highly heterogeneous, performs on par with a local-search heuristic for unrelated machines, and is within 0.4\% of the optimum on average on small instances that can be solved exactly.

cs.DS↗
arXiv2026

Fully Scalable MPC Algorithms for Euclidean Clustering: Constant Approximation in Constant Rounds

We study Euclidean $k$-Means and $k$-Median under the Massively Parallel Computation (MPC) model, focusing on the \emph{fully scalable} setting. Previously, fully scalable algorithms for $k$-Means and/or $k$-Median either required $O(\log\log n\cdot\log\log\log n)$ rounds [Cohen-Addad, Kuhn, and Parsaeian; SODA'26] or guaranteed only an $O(\log n)$-approximation [Goranci, Jiang, Kiss, Kong, Qian, and Szilagyi; SODA'26]. In this paper, we close this gap by giving the first fully scalable algorithms for both problems, which work even when $k$ is polynomially larger than the local memory, and \emph{simultaneously} achieve an $O(1)$-approximation and $O(1)$ round complexity. Technically, we devise a new variant of the MP algorithm [Mettu and Plaxton, SICOMP'03] that computes a \emph{fractional solution satisfying the Lagrangian Multiplier Preserving (LMP) property}. Importantly, this LMP guarantee holds even under arbitrary distance distortions, a feature that is crucial for efficient MPC implementations. We then give a novel MPC rounding algorithm that first rounds to a bicriteria solution with $k+O(n^{1-ε})$ centers and then uses an \emph{iterative kernelization} idea to convert it into one with exactly $k$ centers.

cs.DS↗