Parameterized Streaming Algorithms for Min-Ones d-SAT
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).
min
ones
sat
d-sat
parameterized
kernelization
streaming
space
efficient
algorithm
parameter
Theory of computation~Streaming models
Theory of computation~Fixed parameter tractability
Theory of computation~Streaming, sublinear and near linear time algorithms
Mathematics of computing~Combinatorial algorithms
8:1-8:20
Regular Paper
Akanksha
Agrawal
Akanksha Agrawal
Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Arindam
Biswas
Arindam Biswas
The Institute of Mathematical Sciences, HBNI, Chennai, India
Édouard
Bonnet
Édouard Bonnet
CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Nick
Brettell
Nick Brettell
Department of Computer Science, Durham University, Durham, UK
Radu
Curticapean
Radu Curticapean
BARC, University of Copenhagen, Copenhagen, Denmark
ITU Copenhagen, Copenhagen, Denmark
Dániel
Marx
Dániel Marx
Institute for Computer Science and Control, MTA SZTAKI, Budapest, Hungary
Tillmann
Miltzow
Tillmann Miltzow
Department of Computer Science, Utrecht University, Utrecht, Netherlands
Venkatesh
Raman
Venkatesh Raman
The Institute of Mathematical Sciences, HBNI, Chennai, India
Saket
Saurabh
Saket Saurabh
The Institute of Mathematical Sciences, HBNI, Chennai, India
Department of Computer Science, University of Bergen, Bergen, Norway
10.4230/LIPIcs.FSTTCS.2019.8
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Akanksha Agrawal, Arindam Biswas, Édouard Bonnet, Nick Brettell, Radu Curticapean, Dániel Marx, Tillmann Miltzow, Venkatesh Raman, and Saket Saurabh
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