Graph Transversals for Hereditary Graph Classes: a Complexity Perspective

Within the broad field of Discrete Mathematics and Theoretical Computer Science, the theory of graphs has been of fundamental importance in solving a large number of optimization problems and in modelling real-world situations. In this thesis, we study a topic that covers many aspects of Graph Theory: transversal sets. A transversal set in a graph G is a vertex set that intersects every subgraph of G that belongs to a certain class of graphs. The focus is on vertex cover, feedback vertex set and odd cycle transversal. The decision problems Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal ask, for a given graph G and an integer k, whether there is a corresponding transversal of G of size at most k. These problems are NP-complete in general and our focus is to determine the complexity of the problems when various restrictions are placed on the input, both for the purpose of finding tractable cases and to increase our understanding of the point at which a problem becomes NP-complete. We consider graph classes that are closed under vertex deletion and in particular H-free graphs, i.e. graphs that do not contain a graph H as an induced subgraph. The first chapter is an introduction to the thesis. There we illustrate the motivation of our work and introduce most of the terminology we have used for our research. In the second chapter, we develop a number of structural results for some classes of H-free graphs. The third chapter looks at the Subset Transversal problems: there we prove that Feedback Vertex Set and Odd Cycle Transversal and their subset variants can be solved in polynomial time for both P_4-free and (sP_1+P_3)-free graphs, while for Subset Vertex Cover we show that it can be solved in polynomial time for (sP_1+P_4)-free graphs. The fourth chapter is entirely dedicated to the Connected Vertex Cover problem. The connectivity constraint requires additional proof techniques. We prove this problem can be solved in polynomial time for (sP_1+P_5)-free graphs, even when weights are given to the vertices of the graph. We continue the research on connected transversals in the fifth chapter: we show that Connected Feedback Vertex Set, Connected Odd Cycle Transversal and their extension variants can be solved in polynomial time for both P_4-free and (sP_1+P_3)-free graphs. In the sixth chapter we study the price of independence: can the size of a smallest independent transversal be bounded in terms of the minimum size of a transversal? We establish complete and almost-complete dichotomies which determine for which graph classes such a bound exists and for which cases such a bound is the identity

Snoring and arousals in full-night polysomnographic studies from sleep apnea-hypopnea syndrome patients

SAHS (Sleep Apnea-Hypopnea Syndrome) is recognized to be a serious disorder with high prevalence in the population. The main clinical triad for SAHS is made up of 3 symptoms: apneas and hypopneas, chronic snoring and excessive daytime sleepiness (EDS). The gold standard for diagnosing SAHS is an overnight polysomnographic study performed at the hospital, a laborious, expensive and time-consuming procedure in which multiple biosignals are recorded. In this thesis we offer improvements to the current approaches to diagnosis and assessment of patients with SAHS. We demonstrate that snoring and arousals, while recognized key markers of SAHS, should be fully appreciated as essential tools for SAHS diagnosis. With respect to snoring analysis (applied to a 34 subjects¿ database with a total of 74439 snores), as an alternative to acoustic analysis, we have used less complex approaches mostly based on time domain parameters. We concluded that key information on SAHS severity can be extracted from the analysis of the time interval between successive snores. For that, we built a new methodology which consists on applying an adaptive threshold to the whole night sequence of time intervals between successive snores. This threshold enables to identify regular and non-regular snores. Finally, we were able to correlate the variability of time interval between successive snores in short 15 minute segments and throughout the whole night with the subject¿s SAHS severity. Severe SAHS subjects show a shorter time interval between regular snores (p=0.0036, AHI cp(cut-point): 30h-1) and less dispersion on the time interval features during all sleep. Conversely, lower intra-segment variability (p=0.006, AHI cp: 30h-1) is seen for less severe SAHS subjects. Also, we have shown successful in classifying the subjects according to their SAHS severity using the features derived from the time interval between regular snores. Classification accuracy values of 88.2% (with 90% sensitivity, 75% specificity) and 94.1% (with 94.4% sensitivity, 93.8% specificity) for AHI cut-points of severity of 5 and 30h-1, respectively. In what concerns the arousal study, our work is focused on respiratory and spontaneous arousals (45 subjects with a total of 2018 respiratory and 2001 spontaneous arousals). Current beliefs suggest that the former are the main cause for sleep fragmentation. Accordingly, sleep clinicians assign an important role to respiratory arousals when providing a final diagnosis on SAHS. Provided that the two types of arousals are triggered by different mechanisms we hypothesized that there might exist differences between their EEG content. After characterizing our arousal database through spectral analysis, results showed that the content of respiratory arousals on a mild SAHS subject is similar to that of a severe one (p>>0.05). Similar results were obtained for spontaneous arousals. Our findings also revealed that no differences are observed between the features of these two kinds of arousals on a same subject (r=0.8, p<0.01 and concordance with Bland-Altman analysis). As a result, we verified that each subject has almost like a fingerprint or signature for his arousals¿ content and is similar for both types of arousals. In addition, this signature has no correlation with SAHS severity and this is confirmed for the three EEG tracings (C3A2, C4A1 and O1A2). Although the trigger mechanisms of the two arousals are known to be different, our results showed that the brain response is fairly the same for both of them. The impact that respiratory arousals have in the sleep of SAHS patients is unquestionable but our findings suggest that the impact of spontaneous arousals should not be underestimated

High Level Synthesis of Neural Network Chips

This thesis investigates the development of a silicon compiler dedicated to generate Application-Specific Neural Network Chips (ASNNCs) from a high level C-based behavioural specification language. The aim is to fully integrate the silicon compiler with the ESPRIT II Pygmalion neural programming environment. The integration of these two tools permits the translation of a neural network application specified in nC, the Pygmalion's C-based neural programming language, into either binary (for simulation) or silicon (for execution in hardware). Several applications benefit from this approach, in particular the ones that require real-time execution, for which a true neural computer is required. This research comprises two major parts: extension of the Pygmalion neural programming environment, to support automatic generation of neural network chips from the nC specification language; and implementation of the high level synthesis part of the neural silicon compiler. The extension of the neural programming environment has been developed to adapt the nC language to hardware constraints, and to provide the environment with a simulation tool to test in advance the performance of the neural chips. Firstly, new hardware-specific requisites have been incorporated to nC. However, special attention has been taken to avoid transforming nC into a hardware-oriented language, since the system assumes minimum (or even no) knowledge of VLSI design from the application developer. Secondly, a simulator for neural network hardware has been developed, which assesses how well the generated circuit will perform the neural computation. Lastly, a hardware library of neural network models associated with a target VLSI architecture has been built. The development of the neural silicon compiler focuses on the high level synthesis part of the process. The goal of the silicon compiler is to take nC as the input language and automatically translate it into one or more identical integrated circuits, which are specified in VHDL (the IEEE standard hardware description language) at the register transfer level. The development of the high level synthesis comprises four major parts: firstly, compilation and software-like optimisations of nC; secondly, transformation of the compiled code into a graph-based internal representation, which has been designed to be the basis for the hardware synthesis; thirdly, further transformations and hardware-like optimisations on the internal representation; and finally, creation of the neural chip's data path and control unit that implement the behaviour specified in nC. Special attention has been devoted to the creation of optimised hardware structures for the ASNNCs employing both phases of neural computing on-chip: recall and learning. This is achieved through the data path and control synthesis algorithms, which adopt a heuristic approach that targets the generated hardware structure of the neural chip in a specific VLSI architecture, namely the Generic Neuron. The viability, concerning the effective use of silicon area versus speed, has been evaluated through the automatic generation of a VHDL description for the neural chip employing the Back Propagation neural network model. This description is compared with the one created manually by a hardware designer

Sparse induced subgraphs in P_6-free graphs

We prove that a number of computational problems that ask for the largest sparse induced subgraph satisfying some property definable in CMSO2 logic, most notably Feedback Vertex Set, are polynomial-time solvable in the class of $P_6$-free graphs. This generalizes the work of Grzesik, Klimo\v{s}ov\'{a}, Pilipczuk, and Pilipczuk on the Maximum Weight Independent Set problem in $P_6$-free graphs~[SODA 2019, TALG 2022], and of Abrishami, Chudnovsky, Pilipczuk, Rz\k{a}\.zewski, and Seymour on problems in $P_5$-free graphs~[SODA~2021]. The key step is a new generalization of the framework of potential maximal cliques. We show that instead of listing a large family of potential maximal cliques, it is sufficient to only list their carvers: vertex sets that contain the same vertices from the sought solution and have similar separation properties