Parameterized Complexity Analysis of Randomized Search Heuristics

This chapter compiles a number of results that apply the theory of parameterized algorithmics to the running-time analysis of randomized search heuristics such as evolutionary algorithms. The parameterized approach articulates the running time of algorithms solving combinatorial problems in finer detail than traditional approaches from classical complexity theory. We outline the main results and proof techniques for a collection of randomized search heuristics tasked to solve NP-hard combinatorial optimization problems such as finding a minimum vertex cover in a graph, finding a maximum leaf spanning tree in a graph, and the traveling salesperson problem.Comment: This is a preliminary version of a chapter in the book "Theory of Evolutionary Computation: Recent Developments in Discrete Optimization", edited by Benjamin Doerr and Frank Neumann, published by Springe

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

Visualising and modelling flow processes in fractured carbonate rocks with X-ray computed tomography

Naturally Fractured Reservoirs (NFR) have typically very complex geometries from the pore scale to the field scale – discontinuities can be found at each scale. This makes NFRs hard to accurately be modelled for flow simulations. Fractures are especially difficult to incorporate in the simulations. The topology of a single fracture is usually simplified to a plane or disk, and apertures are usually averaged to be implemented in the simulation models. The fracture aperture distribution of a single fracture is already very heterogeneous though. Contact areas in fractures can detain flow, whereas connected fracture regions with larger apertures can result in preferred flow paths and lead to early breakthrough. To help understanding how well current Discrete Fracture and Matrix (DFM) models are suitable to retain fracture influences on flow in carbonates, this research project combines the simulation of miscible single-phase flow through fractures in carbonates with precise fracture measurements (comprising fracture aperture distributions and 3D topologies) and the visualization of real single and two-phase flow experiments in fractured carbonate cores. The simulation approach employs a DFM model with a hybrid finite element/ finite volume (FEFV) method. The fractured core samples and the flow experiments are imaged with high-resolution X-ray computer tomography (CT), or X-ray radiography respectively. The main goals are to develop and optimize an image processing workflow from the X-ray CT fracture measurement to an according mesh generation as input for simulations, and to be able to compare simulations and flow experiment studies qualitatively to analyse how well the DFM approach is able to capture the true nature of fluid flow in fractures with real aperture distributions. To obtain most relevant comparisons, we conduct numerical simulations and flow experiments on the same fracture geometries, which have been measured before non-destructivel

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

A Strongly-Uniform Slicewise Polynomial-Time Algorithm for the Embedded Planar Diameter Improvement Problem

In the embedded planar diameter improvement problem (EPDI) we are given a graph G embedded in the plane and a positive integer d. The goal is to determine whether one can add edges to the planar embedding of G in such a way that planarity is preserved and in such a way that the resulting graph has diameter at most d. Using non-constructive techniques derived from Robertson and Seymour\u27s graph minor theory, together with the effectivization by self-reduction technique introduced by Fellows and Langston, one can show that EPDI can be solved in time f(d)* |V(G)|^{O(1)} for some function f(d). The caveat is that this algorithm is not strongly uniform in the sense that the function f(d) is not known to be computable. On the other hand, even the problem of determining whether EPDI can be solved in time f_1(d)* |V(G)|^{f_2(d)} for computable functions f_1 and f_2 has been open for more than two decades [Cohen at. al. Journal of Computer and System Sciences, 2017]. In this work we settle this later problem by showing that EPDI can be solved in time f(d)* |V(G)|^{O(d)} for some computable function f. Our techniques can also be used to show that the embedded k-outerplanar diameter improvement problem (k-EOPDI), a variant of EPDI where the resulting graph is required to be k-outerplanar instead of planar, can be solved in time f(d)* |V(G)|^{O(k)} for some computable function f. This shows that for each fixed k, the problem k-EOPDI is strongly uniformly fixed parameter tractable with respect to the diameter parameter d

Accumulating delivered dose to the rectum to improve toxicity prediction in prostate radiotherapy

Accumulating delivered dose to the rectum to improve toxicity prediction in prostate radiotherapy Leila E. A. Shelley Gastrointestinal (GI) toxicity is a clinical issue suffered by up to 22% of prostate cancer radiotherapy patients. However, the relationship between radiation dose and toxicity is generally poorly understood. In prostate radiotherapy, the rectum is a dose-limiting structure to which treatment planning dose constraints are applied to minimise the risk of toxicity. Current normal tissue complication probability (NTCP) models are based on planned dose data and do not consider the effects of organ motion on true delivered dose. The VoxTox research programme has developed automated solutions for segmentation and dose calculation of the rectum for prostate cancer patients being treated with helical TomoTherapy. Daily image guidance scans, acquired primarily for the purposes of positional verification, are exploited by extracting quantitative information to facilitate the calculation of daily delivered and total accumulated dose to the rectum. Prospectively collected toxicity data at 2 years post-treatment were available for 295 patients across two separate cohorts. In this thesis, the hypothesis being tested is that delivered dose is a better predictor of rectal toxicity than planned dose in prostate radiotherapy. The research has successfully demonstrated, for the first time, that delivered dose produces stronger associations with rectal bleeding and proctitis than planned dose. Analysis was performed using dose surface maps (DSMs) of the rectal wall, allowing spatial aspects of dose to be retained during accumulation. A subsequent analysis on a separate cohort also found stronger links between delivered dose and GI toxicity, stool frequency, and bowel bother, in addition to rectal bleeding and proctitis. Biomechanical finite element (FE) modelling was introduced to provide a more anatomically plausible tool for dose accumulation and allowed more accurate tracking of dose at the voxel level. A sensitivity analysis was conducted which explored the effect of simulated rectal motion on dose, and corresponding change in NTCP. For VoxTox patient dose-toxicity analysis, further dose parameterisation approaches were explored in order to consider the increased resolution of information available. Voxel-based rectal subregions at risk (SRRs) were identified using geometric and statistical approaches. In general, discriminative power improved with FE modelling for both planned and accumulated delivered dose, and associations between accumulated dose and toxicity were strengthened by voxel-based subregion analysis. Multivariate NTCP models were constructed for 12 different toxicity endpoints based on planned and accumulated dose parameters. Model performance was compared between analysis approaches, and models were tested on a validation dataset. In general, FE-based dose models performed best, although the optimal dose parameter selected within the model varied with toxicity endpoint. Overall, results suggest that there is an advantage to incorporating delivered dose into NTCP modelling. However, the differences between planned and accumulated dose can be subtle. Meaningful parameterisation of accumulated dose needs careful consideration, as traditional methods for quantifying planned dose may not be directly transferable. Voxel-based analysis techniques are recommended in order to accurately preserve and register spatial dose information, and have been shown to improve the strength of dose-toxicity associations. Further research into quantifying voxel level dose distributions is encouraged. It is anticipated that the novel scientific contributions presented within this thesis will prove valuable for future development of clinical decision-making tools for adaptive radiotherapy, with the ultimate aim of reducing the incidence of radiation-induced toxicity for prostate cancer radiotherapy patients.Personal funding was provided by the WD Armstrong Trust. The VoxTox Research Programme was funded by Cancer Research UK