In this paper we present a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams (where the input graph is defined by a sequence of edge/hyperedge insertions and deletions) and distributed systems such as MapReduce. In the case of dynamic graph streams, we use this primitive to prove the following results: • Matching: Our main result for matchings is that there exists añ O(k 2 ) space algorithm that returns the edges of a maximum matching on the assumption the cardinality is at most k. The best previous algorithm usedÕ(kn) space where n is the number of vertices in the graph and we prove our result is optimal up to logarithmic factors. Our algorithm hasÕ(1) update time. We also show that there exists anÕ(n 2 /α 3 ) space algorithm that returns an α-approximation for matchings of arbitrary size. In independent work, Assadi et al. (SODA 2016) proved this approximation algorithm is optimal and provided an alternative algorithm. We generalize our exact and approximate algorithms to weighted matching. For graphs with low arboricity such as planar graphs, the space required for constant approximation can be further reduced. While there has been a substantial amount of work on approximate matching in insert-only graph streams, these are the first nontrivial results in the dynamic setting. • Vertex Cover and Hitting Set: There exists anÕ(k d ) space algorithm that solves the minimum hitting set problem where d is the cardinality of the input sets and k is an upper bound on the size of the minimum hitting set. We prove this is optimal up to logarithmic factors. Our algorithm hasÕ(1) update time. The case d = 2 corresponds to minimum vertex cover. Finally, we consider a larger family of parameterized problems (including b-matching, disjoint paths, vertex coloring among others) for which our subgraph sampling primitive yields fast, small-space dynamic graph stream algorithms. We then show lower bounds for natural problems outside this family.
Algorithms for constructing a maximum matching of at most k edges in an unweighted or a weighted graph have also been studied under the dynamic streaming model REF .
Kernelization via Sampling with Applications to Finding Matchings and Related Problems in Dynamic Graph Streams
SODA
Computer Science Mathematics