Small Vertex Cover Helps in Fixed-Parameter Tractability of Graph Deletion Problems over Data Streams

Abstract

In the study of parameterized streaming complexity on graph problems, the main goal is to design streaming algorithms for parameterized problems such that O(f(k)logO(1)n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(f(k) \\log ^{\\mathcal {O}(1)} n)$$\\end{document} space is enough, where f is an arbitrary computable function depending only on the parameter k. However, in the past few years very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Vertex Cover or Maximum Matching parameterized by the size k of the solution we are seeking. Chitnis et al. (SODA’16) have shown that many important parameterized problems that form the backbone of traditional parameterized complexity are known to require Ω(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (n)$$\\end{document} bits of storage for any streaming algorithm; e.g. Feedback Vertex Set, Even Cycle Transversal, Odd Cycle Transversal, Triangle Deletion or the more general F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Subgraph Deletion when parameterized by solution size k. Our contribution lies in overcoming the obstacles to efficient parameterized streaming algorithms in graph deletion problems 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. In this work, we consider the four most well-studied streaming models: the Ea, Dea, Va (vertex arrival) and Al (adjacency list) models. Surprisingly, the consideration of vertex cover size K in the different models leads to a classification of positive and negative results for problems like F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Subgraph Deletion and F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Minor Deletion.