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ЬЬЬ#ЬЬЬЬTTT$xTTTxTTT@@@@@@ Multivariate Investigation of NP-Hard Problems: Boundaries Between Parameterized Tractability and Intractability
The main goal when using computing to solve a problem is to develop a mechanism to solve it efficiently. In general, this efficiency is associated with solvability in polynomial time. The theory of NP-completeness was developed to show which problems probably do not have polynomial time algorithms. However, many NP-hard and NP-complete problems must still be solved in practice; therefore it is natural to ask if each of these problems admits an algorithm whose non-polynomial time complexity is purely a function of some subset of its aspects. Questions about the existence of such algorithms are addressed within the theory of parameterized computational complexity developed by Downey and Fellows.
In this thesis we present a multivariate investigation of the complexity of some NP-hard problems, i.e., we first develop a systematic complexity analysis of these problems, defining its subproblems and mapping which one belongs to each side of an Уimaginary boundaryФ between polynomial time solvability and intractability. After that, we analyze which sets of aspects of these problems are sources of their intractability, that is, subsets of aspects for which there exists an algorithm to solve the associated problem, whose non-polynomial time complexity is purely a function of those sets. Thus, we use classical and parameterized computational complexity in an alternate and complementary approach, to show which subproblems of the given problems are NP-hard and latter to diagnose for which sets of parameters the problems are fixed-parameter tractable, or in FPT.
This thesis exhibits a classical and parameterized complexity analysis of different groups of NP-hard problems. The addressed problems are divided into four groups of distinct nature, in the context of data structures, combinatorial games, and graph theory: (i) and/or graph solution and its variants; (ii) flooding-filling games; (iii) problems on P3-convexity; (iv) problems on induced matchings.
Keywords
Computational Complexity
Fixed-Parameter Tractability
Kernelization
Parameterized Intractability
And/Or Graph Solution
Flood-filling Games on Graphs
P3-Convexity
Induced Matching
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