Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching
نویسندگان
چکیده
We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph G = (V,E), with |V | = n and |E| = m, in o( √ m ) time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a (2+ ) approximation inO(log n/ ) amortized time per update. For maximum matching, we show how to maintain a (3 + ) approximation in O(min( √ n/ ,m/ )) amortized time per update, and a (4 + ) approximation in O(m/ ) worst-case time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld [13]. ∗An extended abstract of this paper, not containing the algorithm in Section 4 and not containing the proof of Theorem 2.11, has been accepted for publication in ACM-SIAM Symposium on Discrete Algorithms (SODA)’ 2015. †Email: [email protected]. The Institute of Mathematical Sciences, Chennai, India. This work was done while the author was in Faculty of Computer Science, University of Vienna, Austria. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programe (FP7/2007-2013) / ERC Grant Agreement number 340506. ‡Email: [email protected]. Faculty of Computer Science, University of Vienna, Austria. The research leading to these results has received funding from the European Unions Seventh Framework Programme (FP7/2007-2013) under grant agreement 317532 and from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement number 340506. §Email: [email protected]. Università di Roma ”Tor Vergata”, Rome, Italy. Partially supported by MIUR, the Italian Ministry of Education, University and Research, under Project AMANDA (Algorithmics for MAssive and Networked DAta). i ar X iv :1 41 2. 13 18 v1 [ cs .D S] 3 D ec 2 01 4
منابع مشابه
Deterministic Fully Dynamic Approximate Vertex Cover and Fractional Matching in O(1) Amortized Update Time
We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solom...
متن کاملFully Dynamic Almost-Maximal Matching: Breaking the Polynomial Barrier for Worst-Case Time Bounds
Despite significant research effort, the state-of-the-art algorithm for maintaining an approximate matching in fully dynamic graphs has a polynomial worst-case update time, even for very poor approximation guarantees. In a recent breakthrough, Bhattacharya, Henzinger and Nanongkai showed how to maintain a constant approximation to the minimum vertex cover, and thus also a constant-factor estima...
متن کاملMaximum Cardinality Matchings in Fully Dynamic Graphs
In this paper, we examine recent developments in the Maximum Cardinality Matching (MCM) problem for fully dynamic graphs. Matchings represent a classical combinatorial structure. They have been thoroughly studied and are well understood in the static case. Recently, there’s been an interest for understanding other basic structures such as vertex covers, shortest paths and spanning trees in dyna...
متن کاملBounding cochordal cover number of graphs via vertex stretching
It is shown that when a special vertex stretching is applied to a graph, the cochordal cover number of the graph increases exactly by one, as it happens to its induced matching number and (Castelnuovo-Mumford) regularity. As a consequence, it is shown that the induced matching number and cochordal cover number of a special vertex stretching of a graph G are equal provided G is well-covered bipa...
متن کاملFully Dynamic Approximate Maximum Matching and Minimum Vertex Cover in O(log3 n) Worst Case Update Time
We consider the problem of maintaining an approximately maximum (fractional) matching and an approximately minimum vertex cover in a dynamic graph. Starting with the seminal paper by Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. There remains, however, a polynomial gap between the best known worst case update time and the best known amortised u...
متن کامل