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

Potential Applications of Active Antenna Technologies for Emerging NASA Space Communications Scenarios

AbstractThe National Aeronautics and Space Administration (NASA) is presently embarking on the implementation of far-reaching changes within the framework of both space and aeronautics communications architectures. For example, near earth relays are looking to transition from the traditional few large geostationary satellites to satellite constellations consisting of thousands of small low earth orbiting satellites while lunar space communications will require the need to relay data from many assets distributed on the lunar surface back to earth. Furthermore, within the aeronautics realm, satellite communications for beyond line of sight (BLOS) links are being investigated in tandem with the proliferation of unmanned aerial systems (UAS) within the urban air mobility (UAM) environment. In all of these scenarios, future communications architectures will demand the need to connect and quickly transition between many nodes for large data volume transport. As such, NASA Glenn Research Center (GRC) has been heavily investigating the development of low cost phased array technologies that can readily address these various scenario conditions. In particular, GRC is presently exploring 5G-based beamformer technologies to leverage commercial timescale and volume production cycles which have heretofore not existed within the frequency allocations utilized for NASA applications. In this paper, an overview of the potential future applications of phased arrays being envisioned by NASA are discussed, along with technology feasibility demonstrations being conducted by GRC implementing low cost, 5G based beamformer technologies

Impossibility of independence amplification in Kolmogorov complexity theory

The paper studies randomness extraction from sources with bounded independence and the issue of independence amplification of sources, using the framework of Kolmogorov complexity. The dependency of strings $x$ and $y$ is ${\rm dep}(x,y) = \max\{C(x) - C(x \mid y), C(y) - C(y\mid x)\}$, where $C(\cdot)$ denotes the Kolmogorov complexity. It is shown that there exists a computable Kolmogorov extractor $f$ such that, for any two $n$-bit strings with complexity $s(n)$ and dependency $\alpha(n)$, it outputs a string of length $s(n)$ with complexity $s(n)- \alpha(n)$ conditioned by any one of the input strings. It is proven that the above are the optimal parameters a Kolmogorov extractor can achieve. It is shown that independence amplification cannot be effectively realized. Specifically, if (after excluding a trivial case) there exist computable functions $f_1$ and $f_2$ such that ${\rm dep}(f_1(x,y), f_2(x,y)) \leq \beta(n)$ for all $n$-bit strings $x$ and $y$ with ${\rm dep}(x,y) \leq \alpha(n)$, then $\beta(n) \geq \alpha(n) - O(\log n)$

Counting dependent and independent strings

The paper gives estimations for the sizes of the the following sets: (1) the set of strings that have a given dependency with a fixed string, (2) the set of strings that are pairwise \alpha independent, (3) the set of strings that are mutually \alpha independent. The relevant definitions are as follows: C(x) is the Kolmogorov complexity of the string x. A string y has \alpha -dependency with a string x if C(y) - C(y|x) \geq \alpha. A set of strings {x_1, \ldots, x_t} is pairwise \alpha-independent if for all i different from j, C(x_i) - C(x_i | x_j) \leq \alpha. A tuple of strings (x_1, \ldots, x_t) is mutually \alpha-independent if C(x_{\pi(1)} \ldots x_{\pi(t)}) \geq C(x_1) + \ldots + C(x_t) - \alpha, for every permutation \pi of [t]

Seven exercises planned to stimulate the flow of ideas in creative composition

Thesis (Ed.M.)--Boston Universit

Parameterized Complexity of the k-anonymity Problem

The problem of publishing personal data without giving up privacy is becoming increasingly important. An interesting formalization that has been recently proposed is the $k$-anonymity. This approach requires that the rows of a table are partitioned in clusters of size at least $k$ and that all the rows in a cluster become the same tuple, after the suppression of some entries. The natural optimization problem, where the goal is to minimize the number of suppressed entries, is known to be APX-hard even when the records values are over a binary alphabet and $k=3$, and when the records have length at most 8 and $k=4$ . In this paper we study how the complexity of the problem is influenced by different parameters. In this paper we follow this direction of research, first showing that the problem is W[1]-hard when parameterized by the size of the solution (and the value $k$). Then we exhibit a fixed parameter algorithm, when the problem is parameterized by the size of the alphabet and the number of columns. Finally, we investigate the computational (and approximation) complexity of the $k$-anonymity problem, when restricting the instance to records having length bounded by 3 and $k=3$. We show that such a restriction is APX-hard.Comment: 22 pages, 2 figure

Messenger RNA coding for only the alpha subunit of the rat brain Na channel is sufficient for expression of functional channels in Xenopus oocytes

Several cDNA clones coding for the high molecular weight (alpha) subunit of the voltage-sensitive Na channel have been selected by immunoscreening a rat brain cDNA library constructed in the expression vector lambda gt11. As will be reported elsewhere, the amino acid sequence translated from the DNA sequence shows considerable homology to that reported for the Electrophorus electricus electroplax Na channel. Several of the cDNA inserts hybridized with a low-abundance 9-kilobase RNA species from rat brain, muscle, and heart. Sucrose-gradient fractionation of rat brain poly(A) RNA yielded a high molecular weight fraction containing this mRNA, which resulted in functional Na channels when injected into oocytes. This fraction contained undetectable amounts of low molecular weight RNA. The high molecular weight Na channel RNA was selected from rat brain poly(A) RNA by hybridization to a single-strand antisense cDNA clone. Translation of this RNA in Xenopus oocytes resulted in the appearance of tetrodotoxin-sensitive voltage-sensitive Na channels in the oocyte membrane. These results demonstrate that mRNA encoding the alpha subunit of the rat brain Na channel, in the absence of any beta-subunit mRNA, is sufficient for translation to give functional channels in oocytes

Parameterized Inapproximability of Target Set Selection and Generalizations

In this paper, we consider the Target Set Selection problem: given a graph and a threshold value $thr(v)$ for any vertex $v$ of the graph, find a minimum size vertex-subset to "activate" s.t. all the vertices of the graph are activated at the end of the propagation process. A vertex $v$ is activated during the propagation process if at least $thr(v)$ of its neighbors are activated. This problem models several practical issues like faults in distributed networks or word-to-mouth recommendations in social networks. We show that for any functions $f$ and $\rho$ this problem cannot be approximated within a factor of $\rho(k)$ in $f(k) \cdot n^{O(1)}$ time, unless FPT = W[P], even for restricted thresholds (namely constant and majority thresholds). We also study the cardinality constraint maximization and minimization versions of the problem for which we prove similar hardness results

Structural parameterizations for boxicity

The boxicity of a graph $G$ is the least integer $d$ such that $G$ has an intersection model of axis-aligned $d$-dimensional boxes. Boxicity, the problem of deciding whether a given graph $G$ has boxicity at most $d$, is NP-complete for every fixed $d \ge 2$. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive $1$-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. that boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page

Parametrised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

In this paper, we initiate a systematic study of the parametrised complexity in the field of Dependence Logics which finds its origin in the Dependence Logic of V\"a\"an\"anen from 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parametrisations with respect to the central decision problems. The model checking problem (MC) of PDL is NP-complete. The subject of this research is to identify a list of parametrisations (formula-size, treewidth, treedepth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) showing a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is in FPT. Finally, we introduce a variant of the satisfiability problem, asking for teams of a given size, and show for this problem an almost complete picture.Comment: Update includes refined result