Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

Abstract

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 in O(log n/ 2 ) amortized time per update. For maximum matching, we show how to maintain a (3 + ) approximation in O(min( √ n/ , m 1/3 / 2 )) amortized time per update, and a (4 + ) approximation in O(m 1/3 / 2 ) 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.

Round compression for parallel matching algorithms

“…This was achieved by randomly matching high-degree vertices to their neighbors in consecutive phases while reducing the maximum degree in the remaining graph. This approach was further developed in the dynamic graph setting by a number of papers [12][13][14][15]. Ideas similar to those in the paper of Parnas and Ron [36] were also used to compute polylogarithmic approximation in the streaming model by Kapralov, Khanna, and Sudan [28].…”

Dynamic Parameterized Problems and Algorithms

“…The most studied dynamic problems are dynamic graph problems such as connectivity (e.g., [45,47,48,67]), reachability [41], shortest paths (e.g., [7,26,42]), and maximum matching [8,39,66]. For a dynamic graph algorithm, the updates are usually edge or vertex insertions and deletions.…”

“…By the discussion above, it seems clear that (unless P = NP), superpolynomial query/update times are necessary, and surely this is not as interesting as achieving near-constant time updates. If the problem is relaxed, and instead of exact solutions, approximation algorithms are sufficient, then efficient dynamic algorithms have been obtained for some polynomial time approximable problems such as dynamic approximate vertex cover [5,8,61]. What if we insist on exact solutions?…”

A Fully Dynamic Algorithm for k-Regret Minimizing Sets

“…Recently, there are some theoretical advances on covering and relevant problems (e.g., vertex cover, maximum matching, set cover, and maximal independent set) in dynamic settings [17]- [20]. Although these theoretical results have opened up new ways to design dynamic set cover algorithms, they cannot be directly applied to the update procedure of FD-RMS because of two limitations.…”