Fast Parallel Fixed-Parameter Algorithms via Color Coding

Fixed-parameter algorithms have been successfully applied to solve numerous difficult problems within acceptable time bounds on large inputs. However, most fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no use of the parallel hardware present in modern computers. We show that parallel fixed-parameter algorithms do not only exist for numerous parameterized problems from the literature -- including vertex cover, packing problems, cluster editing, cutting vertices, finding embeddings, or finding matchings -- but that there are parallel algorithms working in \emph{constant} time or at least in time \emph{depending only on the parameter} (and not on the size of the input) for these problems. Phrased in terms of complexity classes, we place numerous natural parameterized problems in parameterized versions of AC$^0$. On a more technical level, we show how the \emph{color coding} method can be implemented in constant time and apply it to embedding problems for graphs of bounded tree-width or tree-depth and to model checking first-order formulas in graphs of bounded degree

A parameterized halting problem, the linear time hierarchy, and the MRDP theorem

The complexity of the parameterized halting problem for nondeterministic Turing machines p-Halt is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-Halt is in para-AC0, the parameterized version of the circuit complexity class AC0, then AC0, or equivalently, (+, x)-invariant FO, has a logic. Although it is widely believed that p-Halt ∉. para-AC0, we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊈ LINH. Here, LINH denotes the linear time hierarchy. On the other hand, we suggest an approach toward proving NE ⊈ LINH using bounded arithmetic. More specifically, we demonstrate that if the much celebrated MRDP (for Matiyasevich-Robinson-Davis-Putnam) theorem can be proved in a certain fragment of arithmetic, then NE ⊈ LINH. Interestingly, central to this result is a para-AC0 lower bound for the parameterized model-checking problem for FO on arithmetical structures.Peer ReviewedPostprint (author's final draft

Administration of galacto-oligosaccharide prebiotics in the Flinders Sensitive Line animal model of depression

INTRODUCTION: Major depressive disorder is the leading source of disability globally and current pharmacological treatments are less than adequate. Animal models such as the Flinders Sensitive Line (FSL) rats are used to mimic aspects of the phenotype in the human disorder and to characterise candidate antidepressant agents. Communication between the gut microbiome and the brain may play an important role in psychiatric disorders such as depression. Interventions targeting the gut microbiota may serve as potential treatments for depression, and this drives increasing research into the effect of probiotics and prebiotics in neuropsychiatric disorders. Prebiotics, galacto-oligosaccharides and fructooligosaccharides that stimulate the activity of gut bacteria have been reported to have a positive impact, reducing anxiety and depressive-like phenotypes and stress-related physiology in mice and rats, as well as in humans. Bimuno, the commercially available beta-galacto-oligosaccharide, has been shown to increase gut microbiota diversity. AIM: Here, we aim to investigate the effect of Bimuno on rat anxiety-like and depressive-like behaviour and gut microbiota composition in the FSL model, a genetic model of depression, in comparison to their control, the Flinders Resistant Line (FRL) rats. METHODS: Sixty-four male rats aged 5–7 weeks, 32 FSL and 32 FRL rats, will be randomised to receive Bimuno or control (4 g/kg) daily for 4 weeks. Animals will be tested by an experimenter unaware of group allocation on the forced swim test to assessed depressive-like behaviour, the elevated plus maze to assess anxiety-like behaviour and the open field test to assess locomotion. Animals will be weighed and food and water intake, per kilogram of bodyweight, will be recorded. Faeces will be collected from each animal prior to the start of the experiment and on the final day to assess the bacterial diversity and relative abundance of bacterial genera in the gut. All outcomes and statistical analysis will be carried out blinded to group allocation, group assignments will be revealed after raw data have been uploaded to Open Science Framework. Two-way analysis of variance will be carried out to investigate the effect of treatment (control or prebiotic) and strain (FSL or FRL) on depressive-like and anxiety-like behaviours

Computing Hitting Set Kernels By AC^0-Circuits

Given a hypergraph H = (V,E), what is the smallest subset X of V such that e and X are not disjoint for all e in E? This problem, known as the hitting set problem, is a basic problem in parameterized complexity theory. There are well-known kernelization algorithms for it, which get a hypergraph H and a number k as input and output a hypergraph H\u27 such that (1) H has a hitting set of size k if, and only if, H\u27 has such a hitting set and (2) the size of H\u27 depends only on k and on the maximum cardinality d of edges in H. The algorithms run in polynomial time, but are highly sequential. Recently, it has been shown that one of them can be parallelized to a certain degree: one can compute hitting set kernels in parallel time O(d) - but it was conjectured that this is the best parallel algorithm possible. We refute this conjecture and show how hitting set kernels can be computed in constant parallel time. For our proof, we introduce a new, generalized notion of hypergraph sunflowers and show how iterated applications of the color coding technique can sometimes be collapsed into a single application

Parallel Multivariate Meta-Theorems

Fixed-parameter tractability is based on the observation that many hard problems become tractable even on large inputs as long as certain input parameters are small. Originally, "tractable" just meant "solvable in polynomial time," but especially modern hardware raises the question of whether we can also achieve "solvable in polylogarithmic parallel time." A framework for this study of parallel fixed-parameter tractability is available and a number of isolated algorithmic results have been obtained in recent years, but one of the unifying core tools of classical FPT theory has been missing: algorithmic meta-theorems. We establish two such theorems by giving new upper bounds on the circuit depth necessary to solve the model checking problem for monadic second-order logic, once parameterized by the tree width and the formula (this is a parallel version of Courcelle\u27s Theorem) and once by the tree depth and the formula. For our proofs we refine the analysis of earlier algorithms, especially of Bodlaender\u27s, but also need to add new ideas, especially in the context where the parallel runtime is bounded by a function of the parameter and does not depend on the length of the input

Practical Access to Dynamic Programming on Tree Decompositions

Parameterized complexity theory has lead to a wide range of algorithmic breakthroughs within the last decades, but the practicability of these methods for real-world problems is still not well understood. We investigate the practicability of one of the fundamental approaches of this field: dynamic programming on tree decompositions. Indisputably, this is a key technique in parameterized algorithms and modern algorithm design. Despite the enormous impact of this approach in theory, it still has very little influence on practical implementations. The reasons for this phenomenon are manifold. One of them is the simple fact that such an implementation requires a long chain of non-trivial tasks (as computing the decomposition, preparing it,...). We provide an easy way to implement such dynamic programs that only requires the definition of the update rules. With this interface, dynamic programs for various problems, such as 3-coloring, can be implemented easily in about 100 lines of structured Java code. The theoretical foundation of the success of dynamic programming on tree decompositions is well understood due to Courcelle\u27s celebrated theorem, which states that every MSO-definable problem can be efficiently solved if a tree decomposition of small width is given. We seek to provide practical access to this theorem as well, by presenting a lightweight model-checker for a small fragment of MSO. This fragment is powerful enough to describe many natural problems, and our model-checker turns out to be very competitive against similar state-of-the-art tools

On the Descriptive Complexity of Color Coding

Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing - purely in terms of the syntactic structure of describing formulas - when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in fpt just because of the way they are commonly described using logical formulas

Jdrasil: A Modular Library for Computing Tree Decompositions

While the theoretical aspects concerning the computation of tree width - one of the most important graph parameters - are well understood, it is not clear how it can be computed practically. We present the open source Java library Jdrasil that implements several different state of the art algorithms for this task. By experimentally comparing these algorithms, we show that the default choices made in Jdrasil lead to an competitive implementation (it took the third place in the first PACE challenge) while also being easy to use and easy to extend