Kernelization via sampling with applications to finding matchings and related problems in dynamic graph streams

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Authors

  • Graham Cormode
  • Hossein Esfandiari
  • MohammadTaghi Hajiaghayi
  • Andrew McGregor
  • Morteza Monemizadeh
  • Sofya Vorotnikova

Colleges, School and Institutes

External organisations

  • Weizmann Institute of Science
  • University of Warwick
  • University of Maryland
  • University of Massachusetts
  • Charles University in Prague

Abstract

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 an Õ(k2) space algorithm that returns the edges of a maximum matching on the assumption the cardinality is at most k. The best previous algorithm used Õ(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 has Õ(1) update time. We also show that there exists an Õ(n23) 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 an Õ(kd) 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 has Õ(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.

Details

Original languageEnglish
Title of host publicationProceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms
EditorsRobert Krauthgamer
Publication statusPublished - 10 Jan 2016
EventTwenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016) - Arlington, VA, United States
Duration: 10 Jan 201612 Jan 2016

Publication series

NameThe Annual ACM - SIAM Symposium on Discrete Algorithms. Proceedings
PublisherSociety for Industrial and Applied Mathematics (SIAM)
ISSN (Print)1071-9040
ISSN (Electronic)1557-9468

Conference

ConferenceTwenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016)
CountryUnited States
CityArlington, VA
Period10/01/1612/01/16