Light Spanners

A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -- in this work we focus on the latter. Specifically, it is shown that for any parameters $k\ge 1$ and $\epsilon>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+\epsilon)$-stretch spanner of weight at most $w(MST(G))\cdot O_\epsilon(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$. Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of $O(\log k)$.Comment: 10 pages, 1 figure, to appear in ICALP 201

Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

In this paper we consider kernelization for problems on d-degenerate graphs, i.e. graphs such that any subgraph contains a vertex of degree at most $d$. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and H-topological-minor free graphs. We show that for several natural problems on d-degenerate graphs the best known kernelization upper bounds are essentially tight.Comment: Full version of ESA 201

Subset feedback vertex set is fixed parameter tractable

The classical Feedback Vertex Set problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (SUBSET-FVS in short) where an instance comes additionally with a set S ? V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00]. The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2^k n^O(1) instances with the size of S bounded by O(k^3), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2^O(k log k) n^O(1) time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.Comment: full version of a paper presented at ICALP'1

International nosocomial infection control consortium (INICC) report, data summary of 36 countries, for 2004-2009

The results of a surveillance study conducted by the International Nosocomial Infection Control Consortium (INICC) from January 2004 through December 2009 in 422 intensive care units (ICUs) of 36 countries in Latin America, Asia, Africa, and Europe are reported. During the 6-year study period, using Centers for Disease Control and Prevention (CDC) National Healthcare Safety Network (NHSN; formerly the National Nosocomial Infection Surveillance system [NNIS]) definitions for device-associated health care-associated infections, we gathered prospective data from 313,008 patients hospitalized in the consortium's ICUs for an aggregate of 2,194,897 ICU bed-days. Despite the fact that the use of devices in the developing countries' ICUs was remarkably similar to that reported in US ICUs in the CDC's NHSN, rates of device-associated nosocomial infection were significantly higher in the ICUs of the INICC hospitals; the pooled rate of central line-associated bloodstream infection in the INICC ICUs of 6.8 per 1,000 central line-days was more than 3-fold higher than the 2.0 per 1,000 central line-days reported in comparable US ICUs. The overall rate of ventilator-associated pneumonia also was far higher (15.8 vs 3.3 per 1,000 ventilator-days), as was the rate of catheter-associated urinary tract infection (6.3 vs. 3.3 per 1,000 catheter-days). Notably, the frequencies of resistance of Pseudomonas aeruginosa isolates to imipenem (47.2% vs 23.0%), Klebsiella pneumoniae isolates to ceftazidime (76.3% vs 27.1%), Escherichia coli isolates to ceftazidime (66.7% vs 8.1%), Staphylococcus aureus isolates to methicillin (84.4% vs 56.8%), were also higher in the consortium's ICUs, and the crude unadjusted excess mortalities of device-related infections ranged from 7.3% (for catheter-associated urinary tract infection) to 15.2% (for ventilator-associated pneumonia). Copyright © 2012 by the Association for Professionals in Infection Control and Epidemiology, Inc. Published by Elsevier Inc. All rights reserved

Parameterized Algorithms for the Matrix Completion Problem

We consider two matrix completion problems, in which we are given a matrix with missing entries and the task is to complete the matrix in a way that (1) minimizes the rank, or (2) minimizes the number of distinct rows. We study the parameterized complexity of the two aforementioned problems with respect to several parameters of interest, including the minimum number of matrix rows, columns, and rows plus columns needed to cover all missing entries. We obtain new algorithmic results showing that, for the bounded domain case, both problems are fixed-parameter tractable with respect to all aforementioned parameters. We complement these results with a lower-bound result for the unbounded domain case that rules out fixed-parameter tractability w.r.t. some of the parameters under consideration

Convex Hulls and Metric Gabriel Graphs

International audienceThe convex hulls construction is mostly known from the point of view of 2D Euclidean geometry where it associates to a given set of points called seeds, the smallest convex polygon containing these seeds. For the cellular automata case, different adaptations of the definition and associated constructions have been proposed to fit with the discreteness of the cellular spaces. We review some of these propositions and show the link with the famous majority and voting rules. We then unify all these definitions in a unique framework using metric spaces and provide a general solution to the problem. This will lead us to an understanding of the convex hull construction as a chase for shortest paths. This emphases the importance of Voronoï diagrams and its related proximity graphs: Delaunay and Gabriel graphs. Indeed, the central problem to be solved is that of connecting arbitrary sets of seeds, in a local and finite-state way, while remaining inside the desired convex hull, i.e by shortest paths. This is exactly what will be made possible by a suitable generalization of Gabriel graphs from Euclidean to arbitrary metric spaces and the study of its construction by cellular automata. The general solution therefore consists of two levels: a connecting level using the metric Gabriel graphs and a level completing the convex hull locally as the majority rule does. Both levels can be generalized to compute the convex hull, when the seeds are moving

Geosphere

It is known that the treewidth of a planar graph with a dominating set of size d is O( # d) and this fact is used as the basis for several fixed parameter algorithms on planar graphs. An interesting question motivating our study is if similar bounds can be obtained for larger minor closed graph families. We say that a graph family has the domination-treewidth property if there is some function f(d) such that every graph G f(d). We show that a minor-closed graph family has bounded local treewidth. This result has important algorithmic consequences

Polynomial Kernels and Faster Algorithms for the Dominating Set Problem on Graphs with an Excluded Minor

Abstract. The domination number of a graph G = (V, E) is the minimum size of a dominating set U ⊆ V, which satisfies that every vertex in V \ U is adjacent to at least one vertex in U. The notion of a problem kernel refers to a polynomial time algorithm that achieves some provable reduction of the input size. Given a graph G whose domination number is k, the objective is to design a polynomial time algorithm that produces a graph G ′ whose size depends only on k, and also has domination number equal to k. Note that the graph G ′ is constructed without knowing the value of k. Problem kernels can be used to obtain efficient approximation and exact algorithms for the domination number, and are also useful in practical settings. In this paper, we present the first nontrivial result for the general case of graphs with an excluded minor, as follows. For every fixed h, given a graph G with n vertices that does not contain Kh as a topological minor, our O(n 3.5 + k O(1) ) time algorithm constructs a subgraph G ′ of G, such that if the domination number of G is k, then the domination number of G ′ is also k and G ′ has at most k c vertices, where c is a constant that depends only on h. This result is improved for graphs that do not contain K3,h as a topological minor, using a simpler algorithm that constructs a subgraph with at most ck vertices, where c is a constant that depends only on h. Our results imply that there is a problem kernel of polynomial size for graphs with an excluded minor and a linear kernel for graphs that are K3,h-minor-free. The only previous kernel results known for the dominating set problem are the existence of a linear kernel for the planar case as well as for graphs of bounded genus. Using the polynomial kernel construction, we give an O(n 3.5 + 2 O( √ k)) time algorithm for finding a dominating set of size at most k in an H-minor-free graph with n vertices. This improves the running time of the previously best known algorithm. Key words: H-minor-free graphs, degenerated graphs, dominating set problem, fixed-parameter tractable algorithms, problem kernel