Towards optimal kernel for connected vertex cover in planar graphs

We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP is a subset of coNP/poly), for planar graphs Guo and Niedermeier [ICALP'08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS'11] to 4k. The constant 4 here is so small that a natural question arises: could it be already an optimal value for this problem? In this paper we answer this quesion in negative: we show a (11/3)k-vertex kernel for Connected Vertex Cover in planar graphs. We believe that this result will motivate further study in search for an optimal kernel

Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP

The Traveling Salesman Problem asks to find a minimum-weight Hamiltonian cycle in an edge-weighted complete graph. Local search is a widely-employed strategy for finding good solutions to TSP. A popular neighborhood operator for local search is k-opt, which turns a Hamiltonian cycle C into a new Hamiltonian cycle C\u27 by replacing k edges. We analyze the problem of determining whether the weight of a given cycle can be decreased by a k-opt move. Earlier work has shown that (i) assuming the Exponential Time Hypothesis, there is no algorithm that can detect whether or not a given Hamiltonian cycle C in an n-vertex input can be improved by a k-opt move in time f(k) n^o(k / log k) for any function f, while (ii) it is possible to improve on the brute-force running time of O(n^k) and save linear factors in the exponent. Modern TSP heuristics are very successful at identifying the most promising edges to be used in k-opt moves, and experiments show that very good global solutions can already be reached using only the top-O(1) most promising edges incident to each vertex. This leads to the following question: can improving k-opt moves be found efficiently in graphs of bounded degree? We answer this question in various regimes, presenting new algorithms and conditional lower bounds. We show that the aforementioned ETH lower bound also holds for graphs of maximum degree three, but that in bounded-degree graphs the best improving k-move can be found in time O(n^((23/135+epsilon_k)k)), where lim_{k -> infty} epsilon_k = 0. This improves upon the best-known bounds for general graphs. Due to its practical importance, we devote special attention to the range of k in which improving k-moves in bounded-degree graphs can be found in quasi-linear time. For k <= 7, we give quasi-linear time algorithms for general weights. For k=8 we obtain a quasi-linear time algorithm when the weights are bounded by O(polylog n). On the other hand, based on established fine-grained complexity hypotheses about the impossibility of detecting a triangle in edge-linear time, we prove that the k = 9 case does not admit quasi-linear time algorithms. Hence we fully characterize the values of k for which quasi-linear time algorithms exist for polylogarithmic weights on bounded-degree graphs

A search for factors specifying tonotopy implicates DNER in hair-cell development in the chick's cochlea

AbstractThe accurate perception of sound frequency by vertebrates relies upon the tuning of hair cells, which are arranged along auditory organs according to frequency. This arrangement, which is termed a tonotopic gradient, results from the coordination of many cellular and extracellular features. Seeking the mechanisms that orchestrate those features and govern the tonotopic gradient, we used expression microarrays to identify genes differentially expressed between the high- and low-frequency cochlear regions of the chick (Gallus gallus). Of the three signaling systems that were represented extensively in the results, we focused on the notch pathway and particularly on DNER, a putative notch ligand, and PTPζ, a receptor phosphatase that controls DNER trafficking. Immunohistochemistry confirmed that both proteins are expressed more strongly in hair cells at the cochlear apex than in those at the base. At the apical surface of each hair cell, the proteins display polarized, mutually exclusive localization patterns. Using morpholinos to decrease the expression of DNER or PTPζ as well as a retroviral vector to overexpress DNER, we observed disturbances of hair-bundle morphology and orientation. Our results suggest a role for DNER and PTPζ in hair-cell development and possibly in the specification of tonotopy

Tight Lower Bounds for List Edge Coloring

The fastest algorithms for edge coloring run in time 2^m n^{O(1)}, where m and n are the number of edges and vertices of the input graph, respectively. For dense graphs, this bound becomes 2^{Theta(n^2)}. This is a somewhat unique situation, since most of the studied graph problems admit algorithms running in time 2^{O(n log n)}. It is a notorious open problem to either show an algorithm for edge coloring running in time 2^{o(n^2)} or to refute it, assuming the Exponential Time Hypothesis (ETH) or other well established assumptions. We notice that the same question can be asked for list edge coloring, a well-studied generalization of edge coloring where every edge comes with a set (often called a list) of allowed colors. Our main result states that list edge coloring for simple graphs does not admit an algorithm running in time 2^{o(n^2)}, unless ETH fails. Interestingly, the algorithm for edge coloring running in time 2^m n^{O(1)} generalizes to the list version without any asymptotic slow-down. Thus, our lower bound is essentially tight. This also means that in order to design an algorithm running in time 2^{o(n^2)} for edge coloring, one has to exploit its special features compared to the list version

Improving TSP Tours Using Dynamic Programming over Tree Decompositions

Given a traveling salesman problem (TSP) tour H in graph G, a k-move is an operation which removes k edges from H, and adds k edges of G so that a new tour H\u27 is formed. The popular k-opt heuristic for TSP finds a local optimum by starting from an arbitrary tour H and then improving it by a sequence of k-moves. Until 2016, the only known algorithm to find an improving k-move for a given tour was the naive solution in time O(n^k). At ICALP\u2716 de Berg, Buchin, Jansen and Woeginger showed an O(n^{floor(2/3k)+1})-time algorithm. We show an algorithm which runs in O(n^{(1/4 + epsilon_k)k}) time, where lim_{k -> infinity} epsilon_k = 0. It improves over the state of the art for every k >= 5. For the most practically relevant case k=5 we provide a slightly refined algorithm running in O(n^{3.4}) time. We also show that for the k=4 case, improving over the O(n^3)-time algorithm of de Berg et al. would be a major breakthrough: an O(n^{3 - epsilon})-time algorithm for any epsilon > 0 would imply an O(n^{3 - delta})-time algorithm for the All Pairs Shortest Paths problem, for some delta>0

H-index in medicine is driven by original research

Aim To investigate the contribution of selected types of articles to h-indices of medical researchers. Methods We used the Web of Science to export the publication records of various members from 26 scientific medical societies (13 European, 13 North American) associated with 13 medical specialties. Those included were presidents (n = 26), heads of randomly chosen committees (n = 52), and randomly selected members of those committees (n = 52). Publications contributing to h-index were categorized as research articles, reviews, guidelines, metaanalyses, or other published work. Results Overall, 3259 items authored by 129 scholars were analyzed. The median h-index was 19.5. The median contribution of research articles to h-index was 84.4%. Researchers in the upper h-index tercile (≥28.5) had a larger share of research articles that contributed to h-index in comparison with those in the lower h-index tercile (≤12.5) (median 87.3% [1st-3rd quartile: 80.0%-93.1%] vs 80.0% [50.0%- 88.9%], P = 0.015). We observed an analogous difference with regard to guidelines (1.1% [0%-3.7%] vs 0% [0%-0%], P = 0.007). Conclusions Original research drives h-indices in medicine. Although guidelines contribute to h-indices in medicine, their influence is low. The specific role of randomized controlled trials in building h-index in medicine remains to be assessed