Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity

Lower bounds for kernelizations

"Vegeu el resum a l'inici del document del fitxer adjunt"

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

Many-one reductions and the category of multivalued functions

Multi-valued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees). A more general understanding is possible, if the category-theoretic properties of multi-valued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multi-valued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch). Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multi-valued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multi-valued functions.Comment: an earlier version was titled "Many-one reductions between search problems". in Mathematical Structures in Computer Science, 201

Parameterized algorithms and computational lower bounds: a structural approach

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 âÂÂalmost-globalâ analysis of the search tree. We also show that an important structural property of the underlying graphs â the graph genus â 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