Fixed-Parameter Tractability of Graph Deletion Problems over Data Streams

The study of parameterized streaming complexity on graph problems was initiated by Fafianie et al. (MFCS’14) and Chitnis et al. (SODA’15 and SODA’16). In this work, we initiate a systematic study of parameterized streaming complexity of graph deletion problems – \(\mathcal {F}\) -Subgraph deletion, \(\mathcal {F}\) -Minor deletion in the four most well-studied streaming models: the \(\textsc {Ea}\) (edge arrival), \(\textsc {Dea}\) (dynamic edge arrival), \(\textsc {Va}\) (vertex arrival) and Al (adjacency list) models. Our main conceptual contribution is to overcome the obstacles to efficient parameterized streaming algorithms by utilizing the power of parameterization. We focus on the vertex cover size K as the parameter for the parameterized graph deletion problems we consider. At the same time, most of the previous work in parameterized streaming complexity was restricted to the Ea (edge arrival) or Dea (dynamic edge arrival) models. In this work, we consider the four most well-studied streaming models: the Ea, Dea, Va (vertex arrival) and Al (adjacency list) models.

[1]  Michal Pilipczuk,et al.  Preprocessing subgraph and minor problems: When does a small vertex cover help? , 2012, J. Comput. Syst. Sci..

[2]  Graham Cormode,et al.  The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs , 2016, ESA.

[3]  Michal Pilipczuk,et al.  Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help? , 2012, IPEC.

[4]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[5]  Mohammad Taghi Hajiaghayi,et al.  New Streaming Algorithms for Parameterized Maximal Matching & Beyond , 2015, SPAA.

[6]  Stefan Fafianie,et al.  Streaming Kernelization , 2014, MFCS.

[7]  Ioannis Z. Emiris,et al.  Randomized Embeddings with Slack and High-Dimensional Approximate Nearest Neighbor , 2018, ACM Trans. Algorithms.

[8]  Sofya Vorotnikova,et al.  Structural Results on Matching Estimation with Applications to Streaming , 2018, Algorithmica.

[9]  Zhengyuan Zhu,et al.  Spatial scan statistics: approximations and performance study , 2006, KDD '06.

[10]  Graham Cormode,et al.  Parameterized streaming: maximal matching and vertex cover , 2015, SODA 2015.

[11]  Arijit Ghosh,et al.  On the streaming complexity of fundamental geometric problems , 2018, ArXiv.

[12]  Sanjeev Khanna,et al.  (1 + Ω(1))-Αpproximation to MAX-CUT Requires Linear Space , 2017, SODA.

[13]  Graham Cormode,et al.  Towards a Theory of Parameterized Streaming Algorithms , 2019, IPEC.

[14]  E. Kushilevitz,et al.  Communication Complexity: Basics , 1996 .

[15]  Sofya Vorotnikova,et al.  Planar Matching in Streams Revisited , 2016, APPROX-RANDOM.

[16]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[17]  Graham Cormode,et al.  A unifying framework for ℓ0-sampling algorithms , 2013, Distributed and Parallel Databases.

[18]  Sofya Vorotnikova,et al.  Kernelization via Sampling with Applications to Finding Matchings and Related Problems in Dynamic Graph Streams , 2016, SODA.

[19]  Andrew McGregor,et al.  Graph stream algorithms: a survey , 2014, SGMD.

[20]  Graham Cormode,et al.  Independent Sets in Vertex-Arrival Streams , 2018, ICALP.

[21]  Sofya Vorotnikova,et al.  A Simple, Space-Efficient, Streaming Algorithm for Matchings in Low Arboricity Graphs , 2018, SOSA@SODA.

[22]  Venkatesan Guruswami,et al.  Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph , 2017, APPROX-RANDOM.

[23]  Sofya Vorotnikova,et al.  Better Algorithms for Counting Triangles in Data Streams , 2016, PODS.

[24]  Sepehr Assadi,et al.  Tight Bounds for Single-Pass Streaming Complexity of the Set Cover Problem , 2021, SIAM J. Comput..