Local resilience of an almost spanning kk-cycle in random graphs

The famous P\'{o}sa-Seymour conjecture, confirmed in 1998 by Koml\'{o}s, S\'{a}rk\"{o}zy, and Szemer\'{e}di, states that for any $k \geq 2$, every graph on $n$ vertices with minimum degree $kn/(k + 1)$ contains the $k$-th power of a Hamilton cycle. We extend this result to a sparse random setting. We show that for every $k \geq 2$ there exists $C > 0$ such that if $p \geq C(\log n/n)^{1/k}$ then w.h.p. every subgraph of a random graph $G_{n, p}$ with minimum degree at least $(k/(k + 1) + o(1))np$, contains the $k$-th power of a cycle on at least $(1 - o(1))n$ vertices, improving upon the recent results of Noever and Steger for $k = 2$, as well as Allen et al. for $k \geq 3$. Our result is almost best possible in three ways: for $p \ll n^{-1/k}$ the random graph $G_{n, p}$ w.h.p. does not contain the $k$-th power of any long cycle; there exist subgraphs of $G_{n, p}$ with minimum degree $(k/(k + 1) + o(1))np$ and $\Omega(p^{-2})$ vertices not belonging to triangles; there exist subgraphs of $G_{n, p}$ with minimum degree $(k/(k + 1) - o(1))np$ which do not contain the $k$-th power of a cycle on $(1 - o(1))n$ vertices.Comment: 24 pages; small updates to the paper after anonymous reviewers' report

Covering Points by Disjoint Boxes with Outliers

For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p} log^{p-1} k) time for p = 2,3. In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to 'cover at least n-k points' to avoid having non-feasible solutions. Results are unchanged. - added Proof to Lemma 11, clarified some sections - corrected typos and small errors - updated affiliations of two author

Parameterized Algorithms for Graph Partitioning Problems

We study a broad class of graph partitioning problems, where each problem is specified by a graph $G=(V,E)$, and parameters $k$ and $p$. We seek a subset $U\subseteq V$ of size $k$, such that $\alpha_1m_1 + \alpha_2m_2$ is at most (or at least) $p$, where $\alpha_1,\alpha_2\in\mathbb{R}$ are constants defining the problem, and $m_1, m_2$ are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in $U$, respectively. This class of fixed cardinality graph partitioning problems (FGPP) encompasses Max $(k,n-k)$-Cut, Min $k$-Vertex Cover, $k$-Densest Subgraph, and $k$-Sparsest Subgraph. Our main result is an $O^*(4^{k+o(k)}\Delta^k)$ algorithm for any problem in this class, where $\Delta \geq 1$ is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain faster algorithms for certain subclasses of FGPPs, parameterized by $p$, or by $(k+p)$. In particular, we give an $O^*(4^{p+o(p)})$ time algorithm for Max $(k,n-k)$-Cut, thus improving significantly the best known $O^*(p^p)$ time algorithm

Propofol-Based Procedural Sedation with or without Low-Dose Ketamine in Children

Objective Examine comparative dosing, efficacy, and safety of propofol alone or with an initial, subdissociative dose of ketamine approach for deep sedation. Background Propofol is a sedative-hypnotic agent used increasingly in children for deep sedation. As a nonanalgesic agent, use in procedures (e.g., bone marrow biopsies/aspirations, renal biopsies) is debated. Our intensivist procedural sedation team sedates using one of two protocols: propofol-only (P-O) approach or age-adjusted dose of 0.25 or 0.5 mg/kg intravenous ketamine (K + P) prior to propofol. With either approach, an initial induction dose of 1 mg/kg propofol is recommended and then intermittent dosing throughout the procedure to achieve adequate sedation to safely and effectively perform the procedure. Approach: Retrospective evaluation of 754 patients receiving either the P-O or K + P approach to sedation. Results A total of 372 P-O group patients and 382 K + P group. Mean age (7.3 ± 5.5 years for P-O; 7.3 ± 5.4 years for K + P) and weight (30.09 ± 23.18 kg for P-O; 30.14 ± 24.45 kg for K + P) were similar in both groups (p = NS). All patients successfully completed procedures with a 16% combined incidence of hypoxia (SPO2 < 90%). Procedure time was 3 minutes longer for K + P group than P-O group (18.68 ± 15.13 minutes for K + P; 15.11 ± 12.77 minutes for P-O; p < 0.01), yet recovery times were 5 minutes shorter (17.04 ± 9.36 minutes for K + P; 22.17 ± 12.84 minutes for P-O; p < 0.01). Mean total dose of propofol was significantly greater in P-O than in K + P group (0.28 ± 0.20 mg/kg/min for K + P; 0.40 ± 0.26 mg/kg/min for P-O; p < 0.0001), and might explain the shorter recovery time. Conclusion Both sedation approaches proved to be well tolerated and equally effective. Addition of ketamine was associated with reduction in the recovery time, probably explained by the statistically significant decrease in the propofol dose

Erd\H{o}s-Ko-Rado for random hypergraphs: asymptotics and stability

We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as $n\to\infty$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $(1+o(1))p\frac kn N$, for any $p\gg \frac nk\ln^2\!\left(\frac nk\right)D^{-1}$. This lower bound on $p$ is asymptotically best possible for $k=\Theta(n)$. For this range of $k$ and $p$, we are able to show stability as well. A different behavior occurs when $k = o(n)$. In this case, the lower bound on $p$ is almost optimal. Further, for the small interval $D^{-1}\ll p \leq (n/k)^{1-\varepsilon}D^{-1}$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $\Theta(\ln (pD)N D^{-1})$, provided that $k \gg \sqrt{n \ln n}$. Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}^k(n,p)$, for essentially all values of $p$ and $k$

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomass\'e, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O(k^5)} \cdot n^9$. In this article we improve this running time to $2^{O(k^2)} \cdot n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O(k^5)$ to $O(k^2)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p-1$ for $p \geq 3$, we speed up the running time to $2^{O(k \cdot k^{\frac{1}{p-1}})} \cdot n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p \geq 3$, Weighted Independent Set cannot be solved in time $2^{o(k)} \cdot n^{O(1)}$ in the class of $\{bull,C_4,\ldots,C_{2p-1}\}$-free graphs unless the ETH fails.Comment: 15 page

On the Benefit of Merging Suffix Array Intervals for Parallel Pattern Matching

We present parallel algorithms for exact and approximate pattern matching with suffix arrays, using a CREW-PRAM with $p$ processors. Given a static text of length $n$, we first show how to compute the suffix array interval of a given pattern of length $m$ in $O(\frac{m}{p}+ \lg p + \lg\lg p\cdot\lg\lg n)$ time for $p \le m$. For approximate pattern matching with $k$ differences or mismatches, we show how to compute all occurrences of a given pattern in $O(\frac{m^k\sigma^k}{p}\max\left(k,\lg\lg n\right)\!+\!(1+\frac{m}{p}) \lg p\cdot \lg\lg n + \text{occ})$ time, where $\sigma$ is the size of the alphabet and $p \le \sigma^k m^k$. The workhorse of our algorithms is a data structure for merging suffix array intervals quickly: Given the suffix array intervals for two patterns $P$ and $P'$, we present a data structure for computing the interval of $PP'$ in $O(\lg\lg n)$ sequential time, or in $O(1+\lg_p\lg n)$ parallel time. All our data structures are of size $O(n)$ bits (in addition to the suffix array)