Analysis of Two-variable Recurrence Relations with Application to Parameterized Approximations
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In this paper we introduce randomized branching as a tool for parameterized approximation and develop the mathematical machinery for its analysis. Our algorithms improve the best known running times of parameterized approximation algorithms for Vertex Cover and 3-Hitting Set for a wide range of approximation ratios. One notable example is a simple parameterized random 1.5-approximation algorithm for Vertex Cover, whose running time of O^*(1.01657^k) substantially improves the best known runnning time of O^*(1.0883^k) [Brankovic and Fernau, 2013]. For 3-Hitting Set we present a parameterized random 2-approximation algorithm with running time of O^*(1.125^k), improving the best known O^*(1.29^k) algorithm of [Brankovic and Fernau, 2012]. To evaluate the running times of our algorithms we develop mathematical tools for analyzing the asymptotic behavior of a family of two variables recurrence relations. To this end, we show an equivalence between the recurrence and a random process. We analyze this process via an adaptation of Sanov's theorem to our setting, which facilitates the use of method of types and information theory for the first time in the analysis of branching algorithms. We believe our novel analysis of recurrence relations may be of independent interest and is one of the main contributions of this paper.
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