A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is $\Theta(\sqrt 2^k)$ and used this to give an $O^*((2+\sqrt{2})^{\mathsf{pw}})$ time algorithm for counting Hamiltonian cycles modulo $2$ on graphs of pathwidth $\mathsf{pw}$. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within $\mathbf{M}_k$, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of $\mathbf{M}_k$ is given; no stronger structural insights such as the existence of large permutation submatrices in $\mathbf{M}_k$ are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes $p$) parameterized by pathwidth. To apply this technique, we prove that the rank of $\mathbf{M}_k$ over the rationals is $4^k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_p$ is $\Omega(1.97^k)$ for any prime $p\neq 2$ and even $\Omega(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-\epsilon)^{\mathsf{pw}})$ for any $\epsilon>0$ unless SETH fails. This bound is tight due to a $O^*(6^{\mathsf{pw}})$ time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes $p\neq 2$ in time $O^*(3.97^\mathsf{pw})$, indicating that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in SODA 201

Abnormal connectional fingerprint in schizophrenia: a novel network analysis of diffusion tensor imaging data

The graph theoretical analysis of structural magnetic resonance imaging (MRI) data has received a great deal of interest in recent years to characterize the organizational principles of brain networks and their alterations in psychiatric disorders, such as schizophrenia. However, the characterization of networks in clinical populations can be challenging, since the comparison of connectivity between groups is influenced by several factors, such as the overall number of connections and the structural abnormalities of the seed regions. To overcome these limitations, the current study employed the whole-brain analysis of connectional fingerprints in diffusion tensor imaging data obtained at 3 T of chronic schizophrenia patients (n = 16) and healthy, age-matched control participants (n = 17). Probabilistic tractography was performed to quantify the connectivity of 110 brain areas. The connectional fingerprint of a brain area represents the set of relative connection probabilities to all its target areas and is, hence, less affected by overall white and gray matter changes than absolute connectivity measures. After detecting brain regions with abnormal connectional fingerprints through similarity measures, we tested each of its relative connection probability between groups. We found altered connectional fingerprints in schizophrenia patients consistent with a dysconnectivity syndrome. While the medial frontal gyrus showed only reduced connectivity, the connectional fingerprints of the inferior frontal gyrus and the putamen mainly contained relatively increased connection probabilities to areas in the frontal, limbic, and subcortical areas. These findings are in line with previous studies that reported abnormalities in striatal–frontal circuits in the pathophysiology of schizophrenia, highlighting the potential utility of connectional fingerprints for the analysis of anatomical networks in the disorder

The Joint European Compound Library:boosting precompetitive research

The Joint European Compound Library (JECL) is a new high-throughput screening collection aimed at driving precompetitive drug discovery and target validation. The JECL has been established with a core of over 321000 compounds from the proprietary collections of seven pharmaceutical companies and will expand to around 500000 compounds. Here, we analyse the physicochemical profile and chemical diversity of the core collection, showing that the collection is diverse and has a broad spectrum of predicted biological activity. We also describe a model for sharing compound information from multiple proprietary collections, enabling diversity and quality analysis without disclosing structures. The JECL is available for screening at no cost to European academic laboratories and SMEs through the IMI European Lead Factory (http://www.europeanleadfactory.eu/)

Fast and simple connectivity in graph timelines

In this paper we study the problem of answering connectivity queries about a \emph{graph timeline}. A graph timeline is a sequence of undirected graphs $G_1,\ldots,G_t$ on a common set of vertices of size $n$ such that each graph is obtained from the previous one by an addition or a deletion of a single edge. We present data structures, which preprocess the timeline and can answer the following queries: - forall$(u,v,a,b)$ -- does the path $u\to v$ exist in each of $G_a,\ldots,G_b$? - exists$(u,v,a,b)$ -- does the path $u\to v$ exist in any of $G_a,\ldots,G_b$? - forall2$(u,v,a,b)$ -- do there exist two edge-disjoint paths connecting $u$ and $v$ in each of $G_a,\ldots,G_b$ We show data structures that can answer forall and forall2 queries in $O(\log n)$ time after preprocessing in $O(m+t\log n)$ time. Here by $m$ we denote the number of edges that remain unchanged in each graph of the timeline. For the case of exists queries, we show how to extend an existing data structure to obtain a preprocessing/query trade-off of $\langle O(m+\min(nt, t^{2-\alpha})), O(t^\alpha)\rangle$ and show a matching conditional lower bound.Comment: 21 pages, extended abstract to appear in WADS'1