Families with infants: a general approach to solve hard partition problems

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with $O^*((2-\varepsilon(d))^n)$ time and exponential space for graphs of average degree $d$. This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in $O^*((2-\varepsilon(d))^n)$ time and polynomial space for graphs of average degree $d$. The previously known results of this kind is a polyspace algorithm by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~$d$ we present an algorithm with running time $O^*((2-\varepsilon(d))^{n/2})$ and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at http://arxiv.org/abs/1410.220

Fast algorithms for min independent dominating set

We first devise a branching algorithm that computes a minimum independent dominating set on any graph with running time O*(2^0.424n) and polynomial space. This improves the O*(2^0.441n) result by (S. Gaspers and M. Liedloff, A branch-and-reduce algorithm for finding a minimum independent dominating set in graphs, Proc. WG'06). We then show that, for every r>3, it is possible to compute an r-((r-1)/r)log_2(r)-approximate solution for min independent dominating set within time O*(2^(nlog_2(r)/r))

Approximation Algorithms for Connected Maximum Cut and Related Problems

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut \delta(S) such that the induced graph G[S] is connected. We present the first non-trivial \Omega(1/log n) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark contrast to the classical max-cut problem, we show that the connected maximum cut problem remains NP-hard even on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus.Comment: 17 pages, Conference version to appear in ESA 201

CSNE: Conditional Signed Network Embedding

Signed networks are mathematical structures that encode positive and negative relations between entities such as friend/foe or trust/distrust. Recently, several papers studied the construction of useful low-dimensional representations (embeddings) of these networks for the prediction of missing relations or signs. Existing embedding methods for sign prediction generally enforce different notions of status or balance theories in their optimization function. These theories, however, are often inaccurate or incomplete, which negatively impacts method performance. In this context, we introduce conditional signed network embedding (CSNE). Our probabilistic approach models structural information about the signs in the network separately from fine-grained detail. Structural information is represented in the form of a prior, while the embedding itself is used for capturing fine-grained information. These components are then integrated in a rigorous manner. CSNE's accuracy depends on the existence of sufficiently powerful structural priors for modelling signed networks, currently unavailable in the literature. Thus, as a second main contribution, which we find to be highly valuable in its own right, we also introduce a novel approach to construct priors based on the Maximum Entropy (MaxEnt) principle. These priors can model the \emph{polarity} of nodes (degree to which their links are positive) as well as signed \emph{triangle counts} (a measure of the degree structural balance holds to in a network). Experiments on a variety of real-world networks confirm that CSNE outperforms the state-of-the-art on the task of sign prediction. Moreover, the MaxEnt priors on their own, while less accurate than full CSNE, achieve accuracies competitive with the state-of-the-art at very limited computational cost, thus providing an excellent runtime-accuracy trade-off in resource-constrained situations

Minimum Common String Partition: Exact Algorithms

In the minimum common string partition problem (MCSP), one gets two strings and is asked to find the minimum number of cuts in the first string such that the second string can be obtained by rearranging the resulting pieces. It is a difficult algorithmic problem having applications in computational biology, text processing, and data compression. MCSP has been studied extensively from various algorithmic angles: there are many papers studying approximation, heuristic, and parameterized algorithms. At the same time, almost nothing is known about its exact complexity. In this paper, we present new results in this direction. We improve the known 2? upper bound (where n is the length of input strings) to ?? where ? ? 1.618... is the golden ratio. The algorithm uses Fibonacci numbers to encode subsets as monomials of a certain implicit polynomial and extracts one of its coefficients using the fast Fourier transform. Then, we show that the case of constant size alphabet can be solved in subexponential time 2^{O(nlog log n/log n)} by a hybrid strategy: enumerate all long pieces and use dynamic programming over histograms of short pieces. Finally, we prove almost matching lower bounds assuming the Exponential Time Hypothesis

Cyanoresin, cyanoresin/cellulose triacetate blends for thin film, dielectric capacitors

Non brittle dielectric films are formed by blending a cyanoresin such as cyanoethyl, hydroxyethyl cellulose (CRE) with a compatible, more crystalline resin such as cellulose triacetate. The electrical breakdown strength of the blend is increased by orienting the films by uniaxial or biaxial stretching. Blends of high molecular weight CRE with high molecular weight cyanoethyl cellulose (CRC) provide films with high dielectric constants

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

Thermal resistance of PCD materials with borides bonding phase

In these studies, one group of PCD materials was prepared using diamond powder and 10 wt % of TiB₂ and the second batch of the PCD material was prepared using a mixture of diamond powder with 5 wt % of TiB₂ and 2 wt % of Co. The materials have been sintered using a Bridgman-type high-pressure apparatus at 8.0±0.2 GPa, at a temperature of 2000±50 °C. Thermogravimetric (TG) measurements and Differential Thermal Analysis (DTA) have been carried out for diamond micropowders, TiB₂ bonding phase, and sintered composites. The coefficients of friction for diamond composites in a sliding contact with an Al₂O₃ ceramic ball have been determined from the room temperature up to 800 °C. Material phase compositions were analyzed for initial samples and after wear tests, at the temperature of 800 °C. Raman spectra of diamond composites with borides bonding phases, observed for the first-order zone centre modes of diamond and graphite during the heating up to 800 °C in air have been presented. Thermal properties have been compared with the commercial diamond-cobalt PCD. It has been found that diamond with TiB₂ and Co is the most resistant to the hardness changes at elevated temperatures and this material maintains the high hardness value up to 800 °C but it has a high coefficient of friction.Досліджено полікристалічні алмазні композити – одну групу матеріалів було приготовано з використанням алмазного порошку і 10 % (за масою) TiB₂, а другу – з алмазного порошку, 5 % (за масою) TiB₂ і 2 % (за масою) Co. Матеріали було спечено в апараті високого тиску типу Бріджмена при тиску 8,0±0,2 ГПа і температурі 2000±50 °С. Термогравіметричні вимірювання та диференційний термічний аналіз було проведено для алмазних мікропорошків, зв’язуючої фази TiB₂ і спечених композітов. Визначено коефіцієнти тертя для алмазних композитів при ковзному контакті з кулькою з кераміки Al₂O₃ при температурі від кімнатної до 800 °С. Фазові склади матеріалів проаналізовано для вихідних зразків і після їх випробування на знос при температурі 800 °С. Представлено спектри комбінаційного розсіювання алмазних композитів зі зв’язуючими фазами боридів, що спостерігаються в центрі зони першого порядку алмазу і графіту в процесі нагрівання до 800 °С на повітрі. Порівнювали термічні властивості отриманих полікристалічних алмазних композитів і промислового композита алмаз–кобальт. Було виявлено, що алмаз з TiB₂ і Co є найбільш стійким до змін твердості при підвищених температурах і зберігає високу твердість до 800 °С, але має високий коефіцієнт тертя.Исследованы поликристаллические алмазных композиты – одна группа материалов была приготовлена с использованием алмазного порошка и 10 % (по массе) TiB₂, а вторая – из алмазного порошка, 5 % (по массе) TiB₂ и 2 % (по массе) Co. Материалы были спечены в аппарате высокого давления типа Бриджмена при давлении 8,0±0,2 ГПа и температуре 2000±50 °С. Термогравиметрические измерения и дифференциальный термический анализ были проведены для алмазных микропорошков, связующей фазы TiB₂ и спеченных композитов. Определены коэффициенты трения для алмазных композитов при скользящем контакте с шариком из керамики Al₂O₃ при температуре от комнатной до 800 °С. Фазовые составы материалов проанализированы для исходных образцов и после их испытания на износ при температуре 800 °С. Представлены спектры комбинационного рассеяния алмазных композитов со связующими фазами боридов, наблюдаемые в центре зоны первого порядка алмаза и графита в процессе нагрева до 800 °С на воздухе. Сравнивали термические свойства полученных поликристаллических алмазных композитов и промышленного поликристаллического композита алмаз–кобальт. Было обнаружено, что алмаз с TiB₂ и Co является наиболее устойчивым к изменениям твердости при повышенных температурах и сохраняет высокую твердость до 800 °С, но имеет высокий коэффициент трения

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.Comment: 20