Fully Dynamic Maximal Matching in $O(\log n)$ Update Time
TLDR
An algorithm for maintaining maximal matching in a graph under addition and deletion of edges that can maintain a factor 2 approximate maximum matching in expected amortized $O(\log n )$ time per update as a direct corollary of the maximal matching scheme.Abstract:
We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our algorithm is randomized and it takes expected amortized $O(\log n)$ time for each edge update, where $n$ is the number of vertices in the graph. While there exists a trivial $O(n)$ time algorithm for each edge update, the previous best known result for this problem is due to Ivkovicź and Lloyd [Lecture Notes in Comput. Sci. 790, Springer-Verlag, London, 1994, pp. 99--111]. For a graph with $n$ vertices and $m$ edges, they gave an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sublinear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfeld [Proceedings of STOC'10, Cambridge, MA, 2010, pp. 457--464] designed a randomized algorithm that achieves expected amortized $O(\log^2 n)$ time for each update for maintaining a $c$-approximate maximum matching for some unspecified large constant $c$. In contrast, we can maintain a factor 2 approximate maximum matching in expected amortized $O(\log n )$ time per update as a direct corollary of the maximal matching scheme. This in turn also implies a 2-approximate vertex cover maintenance scheme that takes expected amortized $O(\log n )$ time per update.read more
Citations
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Proceedings ArticleDOI
On Fully Dynamic Graph Sparsifiers
TL;DR: In this paper, a fully dynamic algorithm for graph sparsification with amortized update time poly(log n, e−1) was proposed. But the algorithm requires poly(n, e)-logarithmic time after each update in the graph.
Proceedings ArticleDOI
Randomized Composable Coresets for Matching and Vertex Cover
Sepehr Assadi,Sanjeev Khanna +1 more
TL;DR: In this article, it was shown that the intractability of matching and vertex cover in the simultaneous communication model is inherently connected to an adversarial partitioning of the underlying graph across machines.
Proceedings ArticleDOI
Dynamic set cover: improved algorithms and lower bounds
TL;DR: These are the first algorithms that obtain an approximation factor linear in f for dynamic set cover, thereby almost matching the best bounds known in the offline setting and improving upon the previous best approximation of O(f2) in the dynamic setting.
Proceedings ArticleDOI
Fully dynamic spectral vertex sparsifiers and applications
TL;DR: In this article, a data structure that supports insertions and deletions of edges, and terminal additions, all in sublinear time, is presented. But the complexity of the data structure is O(min(m3/4,n5/6 ǫ−2) ǔ−4 ) on an unweighted graph, and O(n 5/6 Ô−6 Ò−6 ) on weighted graphs.
Posted Content
Simple dynamic algorithms for Maximal Independent Set and other problems
Manoj Gupta,Shahbaz Khan +1 more
TL;DR: A surprisingly simple deterministic centralized algorithm which improves the amortized update time to $O(\min\{\Delta,m^{2/3}\})$ and some other minor results related to dynamic MIS, Maximum Flow, and Maximum Matching are presented.
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