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

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

Thesis (Ed.M.)--Boston Universit

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

A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and its Algorithmic Applications

A pair of unit clauses is called conflicting if it is of the form $(x)$, $(\bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pair of conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed that for each UCF CNF formula with $m$ clauses we can simultaneously satisfy at least \pp m clauses, where \pp =(\sqrt{5}-1)/2. We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula $F$ with $m$ clauses we can find, in polynomial time, a subformula $F'$ with $m'$ clauses such that we can simultaneously satisfy at least \pp m+(1-\pp)m'+(2-3\pp)n"/2 clauses (in $F$), where $n"$ is the number of variables in $F$ which are not in $F'$. We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds $m/2$ and $m(\sqrt{5}-1)/2$. The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most $6k+3$ variables and $10k$ clauses. We improve this to $4k$ variables and $(2\sqrt{5}+4)k$ clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $(7+3\sqrt{5})k$ variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above

One-Pot Enol Silane Formation-Mukaiyama–Mannich Addition of Ketones, Amides, and Thioesters to Nitrones in the Presence of Trialkylsilyl Trifluoromethanesulfonates

Ketones, amides, and thioesters form enol silanes and add to N-phenylnitrones in one pot in the presence of trimethylsilyl trifluoromethanesulfonate and trialkylamine. The reaction is general to a range of silyl trifluoromethanesulfonates and N-phenylnitrones. The b-(silyloxy)amino carbonyl products are stable to chromatography and can be isolated in 63-99% yield

One-pot silyl ketene imine formation-nucleophilic addition reactions of acetonitrile with acetals and nitrones

Trimethylsilyl trifluoromethanesulfonate (TMSOTf) and a trialkylamine base promote the conversion of acetonitrile to its silyl ketene imine in situ when acetonitrile is employed as solvent. Residual TMSOTf acts as a Lewis acid catalyst to activate acetals and nitrones in the reaction mixture, yielding β-methoxynitriles and β-(silyloxy)aminonitriles, respectively. Some reaction products undergo elimination under the reaction conditions to provide the α,β-unsaturated nitrile directly

Friedel–Crafts addition of indoles to nitrones promoted by trimethylsilyl trifluoromethanesulfonate

N-alkylindoles undergo Friedel–Crafts addition to aryl and secondary alkyl nitrones in the presence of trimethylsilyl trifluoromethanesulfonate and a trialkylamine to produce 3-(1- (silyloxyamino)alkyl)indoles. Spontaneous conversion to the bisindolyl(aryl)methanes, which is thermodynamically favored for nitrones derived from aromatic aldehydes, is suppressed under the reaction conditions. The silyloxyamino group can be deprotected with tetrabutylammonium fluoride to yield the hydroxylamine

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

Polynomial kernelization for removing induced claws and diamonds

A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fai