Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth

We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time $n^{\operatorname{polylog}(k)}$ where $n$ is the number of vertices and $k$ is the minimum treewidth of the given graphs and $\operatorname{polylog}(k)$ is some polynomial in $\operatorname{log}(k)$. Our result generalizes Babai's quasipolynomial-time graph isomorphism test.Comment: 52 pages, 1 figur

Interoceptive Awareness and the Insula – Application of Neuroimaging Techniques in Psychotherapy

Interoceptive awareness is defined as the awareness of stimuli originating inside one’s own body such as the heartbeat. The emergence of new brain imaging techniques like functional magnetic resonance imaging (fMRI) or magnetic resonance spectroscopy (MRS) has increased our knowledge of neural substrates underlying interoceptive awareness. In particular, the bilateral brain structure of the insula has been identified as a key region involved in interoceptive processes in healthy populations. In line with prominent theories of human emotion, the insula has an important function in connecting interoceptive awareness with affective experience. This connection hinging on the insula between interoception and emotional processing is suggestive of an involvement of the insula in mood disorders such as major depressive disorder (MDD). Multilayered deficits in the insula cortex of depressed individuals such as abnormal function, biochemistry, and anatomy support this hypothesis. The aim of the present article is a) to describe the importance of the insula for the interplay between interoception and emotional processing and b) how this might be figured into psychotherapeutic treatment of depressed patients using new imaging techniques like real-time fMRI. The article begins with a brief introduction about neuroanatomical settings of the insula (I. Introduction–Neuroanatomical background of the insula). Afterwards, early behavioral studies to investigate interoceptive awareness are described (II. A step Back–First attempts to investigate interoceptive awareness), followed by a description of more recent imaging studies outlining neural mechanisms underlying interoceptive awareness and emotional processing in the insula (III. The insula as key region involved in functional interoception and emotion.) Throughout, the article addresses the question of why the investigation of individuals suffering from depression might provide novel insights into the neural underpinnings of interoceptive awareness and its link to abnormal behavior (IV. Why study interoceptive awareness in depressed participants?). Following the description of a selected study that combines for the first time functional results of interoception (using fMRI) with biochemical results of the insula (using MRS) (V. Neuroimaging in interoceptive awareness combining fMRI and MRS – A specific study), the article concludes with a perspective outlining the potential for using imaging techniques like real-time fMRI to enhance neural activity in the insula during interoceptive awareness. This approach potentially leads to faster recovery in depressed patients and might be the first therapeutic application of functional imaging in psychiatry (VI. Perspectives: Neurofeedback in major depression using real-time fMRI)

Isomorphism Testing for Graphs Excluding Small Minors

We prove that there is a graph isomorphism test running in time $n^{\operatorname{polylog}(h)}$ on $n$-vertex graphs excluding some $h$-vertex graph as a minor. Previously known bounds were $n^{\operatorname{poly}(h)}$ (Ponomarenko, 1988) and $n^{\operatorname{polylog}(n)}$ (Babai, STOC 2016). For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments

Limb Lengthening Using the PRECICE^TM Nail System: Complications and Results

Background: Three types of telescopic nails are mainly used for intramedullary limb lengthening nowadays. Despite some important advantages of this new technology (e.g. controlled distraction rate, not restricted availability, possibility to perform accordion maneuvers), few articles exist on clinical results and complications after lengthening with the PRECICETM nail (Ellipse, USA). Objectives: The aim of the current study was to describe and analyze the complications associated with lengthening with the PRECICETM nail. Are the problems preventable when using the PRECICE, related to the distraction rate control, the lengthening goals and technique and handling? Methods: We retrospectively reviewed the charts of 9 patients operated between 2012 and 2013 with a PRECICETM nail for a leg length discrepancy (LLD). The mean age of the patients was 32 years (range, 17 - 48 years). There were 5 femoral and 4 tibial procedures. The causes of LLD were posttraumatic (n = 5) and congenital (n = 4). The mean LLD was 36.4 ± 11.4 mm. The minimum follow-ups were 2 months (average, 5 months; range, 2 - 9 months). Results: The mean distraction rate was 0.5 ± 0.1 mm/day. We observed in 7 patients differences in achieving the lengthening goals (average, 1.6 mm; range, -20.0 - 5.0 mm). Average lengthening was 34.7 ± 10.7 mm. All patients reached normal alignment and normal joint orientation. An unintentional loss of the achieved length during the consolidation phase was noticed in patients with delayed bone healing in two cases. In the first case (loss of 20mm distraction) the nail could be redistracted and the goal length was achieved. In the second case (loss of 10mm distraction) the nail broke shortly after the diagnosis and the nail was exchanged. Conclusions: We report of loss of achieved length after lengthening with a telescopic nail. Weight bearing before complete consolidation of the regenerate might be a risk factor for that. Thorough examination of the limb length and careful evaluation of the radiographs are required in the follow-up period. The PRECICE nail system requires the same vigilance like the other intramedullary systems too

An Improved Isomorphism Test for Bounded-Tree-Width Graphs

We give a new fpt algorithm testing isomorphism of n-vertex graphs of tree width k in time 2^{k polylog(k)} poly n, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2^{O(k^5 log k)}poly n. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width k. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai\u27s algorithm as a black box in one place. We give a second algorithm which, at the price of a slightly worse run time 2^{O(k^2 log k)}poly n, avoids the use of Babai\u27s algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm

Interoception in insula subregions as a possible state marker for depression - an exploratory fMRI study investigating healthy, depressed and remitted participants

Background: Interoceptive awareness (iA), the awareness of stimuli originating inside the body, plays an important role in human emotions and psychopathology. The insula is particularly involved in neural processes underlying iA. However, iA-related neural activity in the insula during the acute state of major depressive disorder (MDD) and in remission from depression has not been explored. Methods: A well-established fMRI paradigm for studying (iA; heartbeat counting) and exteroceptive awareness (eA; tone counting) was used. Study participants formed three independent groups: patients suffering from MDD, patients in remission from MDD or healthy controls. Task-induced neural activity in three functional subdivisions of the insula was compared between these groups. Results: Depressed participants showed neural hypo-responses during iA in anterior insula regions, as compared to both healthy and remitted participants. The right dorsal anterior insula showed the strongest response to iA across all participant groups. In depressed participants there was no differentiation between different stimuli types in this region (i.e., between iA, eA and noTask). Healthy and remitted participants in contrast showed clear activity differences. Conclusions: This is the first study comparing iA and eA-related activity in the insula in depressed participants to that in healthy and remitted individuals. The preliminary results suggest that these groups differ in there being hypo- responses across insula regions in the depressed participants, whilst non- psychiatric participants and patients in remission from MDD show the same neural activity during iA in insula subregions implying a possible state marker for MDD. The lack of activity differences between different stimulus types in the depressed group may account for their symptoms of altered external and internal focus

Isomorphism Testing for Graphs Excluding Small Minors

We prove that there is a graph isomorphism test running in time n^{polylog(h)} on n-vertex graphs excluding some h-vertex graph as a minor. Previously known bounds were n^{poly(h)} (Ponomarenko, 1988) and n^{polylog(n)} (Babai, STOC 2016). For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments