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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for d–Hitting Set that runs in time n^{O(d² + (d / ε))}, uses O(d² + (d / ε) log n) bits of space, and achieves an approximation ratio of O((d / ε) n^ε) for any positive ε ≤ 1 and any constant d ∈ ℕ. In particular, this yields a factor-O(d log n) approximation algorithm which uses O(log² n) bits of space. As a corollary, we obtain similar bounds on space and approximation ratio for Vertex Cover and several graph deletion problems. For graphs with maximum degree Δ, one can do better. We give a factor-2 approximation algorithm for Vertex Cover which runs in time n^{O(Δ)} and uses O(Δ log n) bits of space.
For Independent Set on graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses O(log n) bits of space. We also devise a factor-O(d²) approximation algorithm for Dominating Set on d-degenerate graphs which runs in time n^{O(log n)} and uses O(log² n) bits of space. For d-regular graphs, we observe that a known randomized algorithm which achieves an approximation ratio of O(log d) can be derandomized to run in polynomial time and use O(log n) bits of space.
Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.

Arindam Biswas, Venkatesh Raman, and Saket Saurabh. Approximation in (Poly-) Logarithmic Space. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{biswas_et_al:LIPIcs.MFCS.2020.16, author = {Biswas, Arindam and Raman, Venkatesh and Saurabh, Saket}, title = {{Approximation in (Poly-) Logarithmic Space}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {16:1--16:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.16}, URN = {urn:nbn:de:0030-drops-126852}, doi = {10.4230/LIPIcs.MFCS.2020.16}, annote = {Keywords: approximation, logspace, logarithmic, log, space, small, limited, memory, ROM, read-only} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

In this work, we initiate the study of the Min-Ones d-SAT problem in the parameterized streaming model. An instance of the problem consists of a d-CNF formula F and an integer k, and the objective is to determine if F has a satisfying assignment which sets at most k variables to 1. In the parameterized streaming model, input is provided as a stream, just as in the usual streaming model. A key difference is that the bound on the read-write memory available to the algorithm is O(f(k) log n) (f: N -> N, a computable function) as opposed to the O(log n) bound of the usual streaming model. The other important difference is that the number of passes the algorithm makes over its input must be a (preferably small) function of k.
We design a (k + 1)-pass parameterized streaming algorithm that solves Min-Ones d-SAT (d >= 2) using space O((kd^(ck) + k^d)log n) (c > 0, a constant) and a (d + 1)^k-pass algorithm that uses space O(k log n). We also design a streaming kernelization for Min-Ones 2-SAT that makes (k + 2) passes and uses space O(k^6 log n) to produce a kernel with O(k^6) clauses.
To complement these positive results, we show that any k-pass algorithm for or Min-Ones d-SAT (d >= 2) requires space Omega(max{n^(1/k) / 2^k, log(n / k)}) on instances (F, k). This is achieved via a reduction from the streaming problem POT Pointer Chasing (Guha and McGregor [ICALP 2008]), which might be of independent interest. Given this, our (k + 1)-pass parameterized streaming algorithm is the best possible, inasmuch as the number of passes is concerned.
In contrast to the results of Fafianie and Kratsch [MFCS 2014] and Chitnis et al. [SODA 2015], who independently showed that there are 1-pass parameterized streaming algorithms for Vertex Cover (a restriction of Min-Ones 2-SAT), we show using lower bounds from Communication Complexity that for any d >= 1, a 1-pass streaming algorithm for Min-Ones d-SAT requires space Omega(n). This excludes the possibility of a 1-pass parameterized streaming algorithm for the problem. Additionally, we show that any p-pass algorithm for the problem requires space Omega(n/p).

Akanksha Agrawal, Arindam Biswas, Édouard Bonnet, Nick Brettell, Radu Curticapean, Dániel Marx, Tillmann Miltzow, Venkatesh Raman, and Saket Saurabh. Parameterized Streaming Algorithms for Min-Ones d-SAT. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.FSTTCS.2019.8, author = {Agrawal, Akanksha and Biswas, Arindam and Bonnet, \'{E}douard and Brettell, Nick and Curticapean, Radu and Marx, D\'{a}niel and Miltzow, Tillmann and Raman, Venkatesh and Saurabh, Saket}, title = {{Parameterized Streaming Algorithms for Min-Ones d-SAT}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {8:1--8:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.8}, URN = {urn:nbn:de:0030-drops-115708}, doi = {10.4230/LIPIcs.FSTTCS.2019.8}, annote = {Keywords: min, ones, sat, d-sat, parameterized, kernelization, streaming, space, efficient, algorithm, parameter} }

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