Parameterized Compilability
Noel Arteche Echeverría
Abstract:
Compilability (also known as knowledge compilation) concerns the computational complexity of preprocessing intractable problems. For some hard computational problems, under the assumption that some part of the inputs will stay fixed over time, we allow this fixed part of the instances to be preprocessed in advance through some expensive precomputation whose output must be succinct (at most polynomially bigger than the input), hoping that this precomputation will help solve the complete instances efficiently (in polynomial time). The systematic study of compilability from a complexity-theoretic perspective was initiated by Cadoli, Donini, Liberatore and Schaerf (2002) and continued in an alternative style with Chen (2015), who modelled efficient compilation as a particular case of fixed-parameter tractability in his parameter compilation framework. Under such a complexity-theoretic framework, one can prove conditional lower bounds on the hardness of compilation.
In an attempt to go beyond polynomial-size compilation, this thesis studies parameterized compilability, where the compilation is allowed to be of fixed-parameter (fpt) size in the sense of parameterized complexity. This work introduces an extension of Chen’s parameter compilation framework to account for doubly parameterized problems. These are problems having two parameters, where the first one indicates what part of the input is available for precomputation and the second one imposes a fixed-parameter bound on the size of that precomputation. The framework introduces new complexity classes to model fpt-size compilation and gives a new notion of reduction to study the relations between problems regarding their parameterized compilability.
Furthermore, we apply the new framework to computational problems around the parameterized complexity classes W[1] and W[2], showing hardness results in their compilability. Amongst other problems, we study the parameterized compilability of completing a satisfying assignment for a Constraint Satisfaction Problem (CSP) as well as the complexity of different inference problems for CSP.