As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams. In this paper, we introduce a new approach to handling graph streams, by instead seeking solutions for the parameterized versions of these problems where we are given a parameter @math and the objective is to decide whether there is a solution bounded by @math . By combining kernelization techniques with randomized sketch structures, we obtain the first streaming algorithms for the parameterized versions of the Vertex Cover problem. We consider the following three models for a graph stream on @math nodes: 1. The insertion-only model where the edges can only be added. 2. The dynamic model where edges can be both inserted and deleted. 3. The dynamic model where we are guaranteed that at each timestamp there is a solution of size at most @math . In each of these three models we are able to design parameterized streaming algorithms for the Vertex Cover problem. We are also able to show matching lower bound for the space complexity of our algorithms. (Due to the arXiv limit of 1920 characters for abstract field, please see the abstract in the paper for detailed description of our results)
The question of finding maximal and maximum cardinality matchings has been heavily studied in the model of (insert-only) graph streams. A greedy algorithm trivially obtains a maximal matching (simply store every edge that links two currently unmatched nodes); this can also be shown to be a 0.5-approximation to the maximum cardinality matching @cite_3 . By taking multiple passes over the input streams, this can be improved to a @math approximation, by finding augmenting paths with successive passes @cite_16 @cite_13 .
@cite_16 @cite_13 @cite_3
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We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph problems. For general m-edge and n-vertex graphs, it is well-known to be solvable in @math time. We develop a linear-time algorithm to find maximum-cardinality matchings on cocomparability graphs, a prominent subclass of perfect graphs that contains interval graphs as well as permutation graphs. Our algorithm is based on the recently discovered Lexicographic Depth First Search (LDFS). We present algorithms for finding large graph matchings in the streaming model. In this model, applicable when dealing with massive graphs, edges are streamed-in in some arbitrary order rather than residing in randomly accessible memory. For e>0, we achieve a @math approximation for maximum cardinality matching and a @math approximation to maximum weighted matching. Both algorithms use a constant number of passes and @math space. We give simple, deterministic, distributed algorithms for computing maximal matchings, maximal independent sets and colourings. We show that edge colourings with at most 2Δ-1 colours, and maximal matchings can be computed within O(log* n + Δ) deterministic rounds, where Δ is the maximum degree of the network. We also show how to find maximal independent sets and (Δ + 1)-vertex colourings within O(log* n + Δ2) deterministic rounds. All hidden constants are very small and the algorithms are very simple.