Parameterized algorithms and computational lower bounds: a structural approach

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Title: Parameterized algorithms and computational lower bounds: a structural approach
Author: Xia, Ge
Abstract: Many problems of practical significance are known to be NP -hard , and hence , are unlikely to be solved by polynomial -time algorithms . There are several ways to cope with the NP -hardness of a certain problem . The most popular approaches include heuristic algorithms , approximation algorithms , and randomized algorithms . Recently , parameterized computation and complexity have been receiving a lot of attention . By taking advantage of small or moderate parameter values , parameterized algorithms provide new venues for practically solving problems that are theoretically intractable . In this dissertation , we design efficient parameterized algorithms for several wellknown NP -hard problems and prove strong lower bounds for some others . In doing so , we place emphasis on the development of new techniques that take advantage of the structural properties of the problems . We present a simple parameterized algorithm for Vertex Cover that uses polynomial space and runs in time O (1 .2738k + kn ) . It improves both the previous O (1 .286k + kn ) -time polynomial -space algorithm by Chen , Kanj , and Jia , and the very recent O (1 .2745kk4 + kn ) -time exponential -space algorithm , by Chandran and Grandoni . This algorithm stands out for both its performance and its simplicity . Essential to the design of this algorithm are several new techniques that use structural information of the underlying graph to bound the search space . For Vertex Cover on graphs with degree bounded by three , we present a still better algorithm that runs in time O (1 .194k + kn ) , based on an ? ? ? ? ? ?almost -global ? ? ? ? ? ? analysis of the search tree . We also show that an important structural property of the underlying graphs ? ? ? ? ? ? the graph genus ? ? ? ? ? ? largely dictates the computational complexity of some important graph problems including Vertex Cover , Independent Set and Dominating Set . We present a set of new techniques that allows us to prove almost tight computational lower bounds for some NP -hard problems , such as Clique , Dominating Set , Hitting Set , Set Cover , and Independent Set . The techniques are further extended to derive computational lower bounds on polynomial time approximation schemes for certain NP -hard problems . Our results illustrate a new approach to proving strong computational lower bounds for some NP -hard problems under reasonable conditions .
URI: http : / /hdl .handle .net /1969 .1 /4322
Date: 2006-10-30

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Parameterized algorithms and computational lower bounds: a structural approach. Available electronically from http : / /hdl .handle .net /1969 .1 /4322 .

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