General dd-position sets

The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to the suitable values of $d$. We show that the decision problem concerning finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural characterization of general $d$-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that ${\rm gp}_d(G)$ is infinite whenever $G$ is an infinite graph and $d$ is a finite integer.Comment: 16 page

On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion

In this paper, we investigate the approximability of two node deletion problems. Given a vertex weighted graph $G=(V,E)$ and a specified, or "distinguished" vertex $p \in V$, MDD(min) is the problem of finding a minimum weight vertex set $S \subseteq V\setminus \{p\}$ such that $p$ becomes the minimum degree vertex in $G[V \setminus S]$; and MDD(max) is the problem of finding a minimum weight vertex set $S \subseteq V\setminus \{p\}$ such that $p$ becomes the maximum degree vertex in $G[V \setminus S]$. These are known $NP$-complete problems and have been studied from the parameterized complexity point of view in previous work. Here, we prove that for any $\epsilon > 0$, both the problems cannot be approximated within a factor $(1 - \epsilon)\log n$, unless $NP \subseteq DTIME(n^{\log\log n})$. We also show that for any $\epsilon > 0$, MDD(min) cannot be approximated within a factor $(1 -\epsilon)\log n$ on bipartite graphs, unless $NP \subseteq DTIME(n^{\log\log n})$, and that for any $\epsilon > 0$, MDD(max) cannot be approximated within a factor $(1/2 - \epsilon)\log n$ on bipartite graphs, unless $NP \subseteq DTIME(n^{\log\log n})$. We give an $O(\log n)$ factor approximation algorithm for MDD(max) on general graphs, provided the degree of $p$ is $O(\log n)$. We then show that if the degree of $p$ is $n-O(\log n)$, a similar result holds for MDD(min). We prove that MDD(max) is $APX$-complete on 3-regular unweighted graphs and provide an approximation algorithm with ratio $1.583$ when $G$ is a 3-regular unweighted graph. In addition, we show that MDD(min) can be solved in polynomial time when $G$ is a regular graph of constant degree.Comment: 16 pages, 4 figures, submitted to Elsevier's Journal of Discrete Algorithm

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

A 3-path vertex cover in a graph is a vertex subset $C$ such that every path of three vertices contains at least one vertex from $C$. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most $k$. In this paper, we give a kernel of $5k$ vertices and an $O^*(1.7485^k)$-time and polynomial-space algorithm for this problem, both new results improve previous known bounds.Comment: in TAMC 2016, LNCS 9796, 201

Oblivious Bounds on the Probability of Boolean Functions

This paper develops upper and lower bounds for the probability of Boolean functions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. We call this approach dissociation and give an exact characterization of optimal oblivious bounds, i.e. when the new probabilities are chosen independent of the probabilities of all other variables. Our motivation comes from the weighted model counting problem (or, equivalently, the problem of computing the probability of a Boolean function), which is #P-hard in general. By performing several dissociations, one can transform a Boolean formula whose probability is difficult to compute, into one whose probability is easy to compute, and which is guaranteed to provide an upper or lower bound on the probability of the original formula by choosing appropriate probabilities for the dissociated variables. Our new bounds shed light on the connection between previous relaxation-based and model-based approximations and unify them as concrete choices in a larger design space. We also show how our theory allows a standard relational database management system (DBMS) to both upper and lower bound hard probabilistic queries in guaranteed polynomial time.Comment: 34 pages, 14 figures, supersedes: http://arxiv.org/abs/1105.281

Clustering by soft-constraint affinity propagation: Applications to gene-expression data

Motivation: Similarity-measure based clustering is a crucial problem appearing throughout scientific data analysis. Recently, a powerful new algorithm called Affinity Propagation (AP) based on message-passing techniques was proposed by Frey and Dueck \cite{Frey07}. In AP, each cluster is identified by a common exemplar all other data points of the same cluster refer to, and exemplars have to refer to themselves. Albeit its proved power, AP in its present form suffers from a number of drawbacks. The hard constraint of having exactly one exemplar per cluster restricts AP to classes of regularly shaped clusters, and leads to suboptimal performance, {\it e.g.}, in analyzing gene expression data. Results: This limitation can be overcome by relaxing the AP hard constraints. A new parameter controls the importance of the constraints compared to the aim of maximizing the overall similarity, and allows to interpolate between the simple case where each data point selects its closest neighbor as an exemplar and the original AP. The resulting soft-constraint affinity propagation (SCAP) becomes more informative, accurate and leads to more stable clustering. Even though a new {\it a priori} free-parameter is introduced, the overall dependence of the algorithm on external tuning is reduced, as robustness is increased and an optimal strategy for parameter selection emerges more naturally. SCAP is tested on biological benchmark data, including in particular microarray data related to various cancer types. We show that the algorithm efficiently unveils the hierarchical cluster structure present in the data sets. Further on, it allows to extract sparse gene expression signatures for each cluster.Comment: 11 pages, supplementary material: http://isiosf.isi.it/~weigt/scap_supplement.pd