Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBeating the Folklore Algorithm for Dynamic Matching
130
0
0.0
(
0
)
Added by
David Wajc
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence $2$-approximate) matching in $O(n)$ worst-case update time in $n$-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of {em both} approximation ratio and worst-case update time. Specifically, we give a $(2-Omega(1))$-approximate algorithm with $O(sqrt{n}sqrt[8]{m})=O(n^{3/4})$ worst-case update time in $n$-node, $m$-edge graphs. For sufficiently small constant $epsilon>0$, no deterministic $(2+epsilon)$-approximate algorithm with worst-case update time $O(n^{0.99})$ was known. Our second result is the first deterministic $(2+epsilon)$-approximate (weighted) matching algorithm with $O_epsilon(1)cdot O(sqrt[4]{m}) = O_epsilon(1)cdot O(sqrt{n})$ worst-case update time.
rate research
Read More
In fully dynamic graphs, we know how to maintain a 2-approximation of maximum matching extremely fast, that is, in polylogarithmic update time or better. In a sharp contrast and despite extensive studies, all known algorithms that maintain a $2-Omega(1)$ approximate matching are much slower. Understanding this gap and, in particular, determining the best possible update time for algorithms providing a better-than-2 approximate matching is a major open question. In this paper, we show that for any constant $epsilon > 0$, there is a randomized algorithm that with high probability maintains a $2-Omega(1)$ approximate maximum matching of a fully-dynamic general graph in worst-case update time $O(Delta^{epsilon}+text{polylog } n)$, where $Delta$ is the maximum degree. Previously, the fastest fully dynamic matching algorithm providing a better-than-2 approximation had $O(m^{1/4})$ update-time [Bernstein and Stein, SODA 2016]. A faster algorithm with update-time $O(n^epsilon)$ was known, but worked only for maintaining the size (and not the edges) of the matching in bipartite graphs [Bhattacharya, Henzinger, and Nanongkai, STOC 2016].
We study the maximum matching problem in the random-order semi-streaming setting. In this problem, the edges of an arbitrary $n$-vertex graph $G=(V, E)$ arrive in a stream one by one and in a random order. The goal is to have a single pass over the stream, use $n cdot poly(log n)$ space, and output a large matching of $G$. We prove that for an absolute constant $epsilon_0 > 0$, one can find a $(2/3 + epsilon_0)$-approximate maximum matching of $G$ using $O(n log n)$ space with high probability. This breaks the natural boundary of $2/3$ for this problem prevalent in the prior work and resolves an open problem of Bernstein [ICALP20] on whether a $(2/3 + Omega(1))$-approximation is achievable.
The Micali-Vazirani (MV) algorithm for maximum cardinality matching in general graphs, which was published in 1980 cite{MV}, remains to this day the most efficient known algorithm for the problem. This paper gives the first complete and correct proof of this algorithm. Central to our proof are some purely graph-theoretic facts, capturing properties of minimum length alternating paths; these may be of independent interest. An attempt is made to render the algorithm easier to comprehend.
We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.
Dynamic Connectivity is a fundamental algorithmic graph problem, motivated by a wide range of applications to social and communication networks and used as a building block in various other algorithms, such as the bi-connectivity and the dynamic minimal spanning tree problems. In brief, we wish to maintain the connected components of the graph under dynamic edge insertions and deletions. In the sequential case, the problem has been well-studied from both theoretical and practical perspectives. However, much less is known about efficient concurrent solutions to this problem. This is the gap we address in this paper. We start from one of the classic data structures used to solve this problem, the Euler Tour Tree. Our first contribution is a non-blocking single-writer implementation of it. We leverage this data structure to obtain the first truly concurrent generalization of dynamic connectivity, which preserves the time complexity of its sequential counterpart, but is also scalable in practice. To achieve this, we rely on three main techniques. The first is to ensure that connectivity queries, which usually dominate real-world workloads, are non-blocking. The second non-trivial technique expands the above idea by making all queries that do not change the connectivity structure non-blocking. The third ingredient is applying fine-grained locking for updating the connected components, which allows operations on disjoint components to occur in parallel. We evaluate the resulting algorithm on various workloads, executing on both real and synthetic graphs. The results show the efficiency of each of the proposed optimizations; the most efficient variant improves the performance of a coarse-grained based implementation on realistic scenarios up to 6x on average and up to 30x when connectivity queries dominate.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
