Analysis of Two-variable Recurrence Relations with Application to Parameterized Approximations

In this paper we introduce randomized branching as a tool for parameterized approximation and develop the mathematical machinery for its analysis. Our algorithms substantially improve the best known running times of parameterized approximation algorithms for Vertex Cover and 3-Hitting Set for a wide range of approximation ratios. The running times of our algorithms are derived from an asymptotic analysis of a broad class of two-variable recurrence relations. Our main theorem gives a simple formula for this asymptotics. The formula can be efficiently calculated by solving a simple numerical optimization problem, and provides the mathematical insight required for the algorithm design. To this end, we show an equivalence between the recurrence and a stochastic process. We analyze this process using the method of types, by introducing an adaptation of Sanov's theorem to our setting. We believe our novel analysis of recurrence relations which is of independent interest is a main contribution of this paper.

[1]  David Eppstein,et al.  Optimal point placement for mesh smoothing , 1997, SODA '97.

[2]  EppsteinDavid,et al.  3-coloring in time O(1.3289n) , 2005 .

[3]  David Eppstein,et al.  3-Coloring in Time O(1.3289^n) , 2000, J. Algorithms.

[4]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[5]  Reuven Bar-Yehuda,et al.  Random Algorithms for the Loop Cutset Problem , 1999, UAI.

[6]  David Eppstein,et al.  Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms , 2006, TALG.

[7]  Luca Trevisan,et al.  Mildly Exponential Time Approximation Algorithms for Vertex Cover, Uniform Sparsest Cut and Related Problems , 2018, ArXiv.

[8]  Weijia Jia,et al.  Vertex Cover: Further Observations and Further Improvements , 2001, J. Algorithms.

[9]  Ge Xia,et al.  Improved upper bounds for vertex cover , 2010, Theor. Comput. Sci..

[10]  I. N. Sanov.  On the probability of large deviations of random variables , 1958 .

[11]  Rolf Niedermeier,et al.  Upper Bounds for Vertex Cover Further Improved , 1999, STACS.

[12]  Fabrizio Grandoni,et al.  A measure & conquer approach for the analysis of exact algorithms , 2009, JACM.

[13]  Nikhil Bansal,et al.  New Tools and Connections for Exponential-Time Approximation , 2017, Algorithmica.

[14]  Saket Saurabh,et al.  Linear Time Parameterized Algorithms for Subset Feedback Vertex Set , 2018, ACM Trans. Algorithms.

[15]  Weijia Jia,et al.  Vertex Cover: Further Observations and Further Improvements , 1999, J. Algorithms.

[16]  W. Marsden.  I and J , 2012 .

[17]  R. Ellis,et al.  LARGE DEVIATIONS FOR A GENERAL-CLASS OF RANDOM VECTORS , 1984 .

[18]  Stephen P. Boyd,et al.  Disciplined quasiconvex programming , 2019, Optimization Letters.

[19]  P. Gács,et al.  Algorithms , 1992 .

[20]  Henning Fernau.  Parameterized algorithmics for d-Hitting Set , 2010, Int. J. Comput. Math..

[21]  Rahul Santhanam,et al.  Exponential Time Paradigms Through the Polynomial Time Lens , 2016, ESA.

[22]  W. Hoeffding.  Asymptotically Optimal Tests for Multinomial Distributions , 1965 .

[23]  Thomas M. Cover,et al.  Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing) , 2006 .

[24]  Henning Fernau,et al.  Parameterized Approximation Algorithms for Hitting Set , 2011, WAOA.

[25]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[26]  Vangelis Th. Paschos,et al.  On Subexponential and FPT-Time Inapproximability , 2013, Algorithmica.

[27]  Henning Fernau,et al.  A novel parameterised approximation algorithm for minimum vertex cover , 2013, Theor. Comput. Sci..

[28]  J. Gärtner.  On Large Deviations from the Invariant Measure , 1977 .

[29]  Magnus Wahlström,et al.  Algorithms, measures and upper bounds for satisfiability and related problems , 2007 .

[30]  Vangelis Th. Paschos,et al.  Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms , 2011, Discret. Appl. Math..

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

[32]  Luca Trevisan,et al.  Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut , 2018, APPROX-RANDOM.

[33]  Michael R. Fellows,et al.  Parameterized approximation via fidelity preserving transformations , 2012, J. Comput. Syst. Sci..

[34]  Dimitrios M. Thilikos,et al.  Invitation to fixed-parameter algorithms , 2007, Comput. Sci. Rev..

[35]  Fahad Panolan,et al.  Lossy kernelization , 2016, STOC.

[36]  O. F. Cook.  The Method of Types , 1898 .

[37]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .