Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time

It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in terms of the weights). We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms, also for weighted and counting versions. For example, in this time we can solve the traveling salesman problem or count the number of Hamiltonian cycles. The rank-based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the matrix tree theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page

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

Technical college choice: an action research, mixed methods study of influences on the enrollment decisions of students admitted to a technical college

Many colleges and universities struggle with enrollment management challenges, which in turn create escalating financial pressures. One important aspect of this multifaceted problem that is worthy of closer inspection involves learning how students' college choice factors impact enrollment. Though prior research on this issue is readily available, none consider the matter from the perspective of a technical college; an often overlooked sector of the higher education community. The purpose of this study was to examine the relationship between admitted students' reported perceptions of institutional characteristics and the subsequent enrollment decisions of these matriculating students at a technical college. The central research question in this study asked the following: What actions will motivate a greater proportion of admitted students to enroll at a technical college? Employing a post-positivist approach, the study's conceptual framework placed students' perceptions of institutional characteristics in the context of the college choice process. A mixed research methodology was utilized to collect and analyze data on influential college choice factors for students admitted to a mid-sized technical college in the Mid-Atlantic region of the United States. The quantitative research method involved analysis of pre-existing survey data, previously collected by the technical college via the Admitted Student Questionnaire®. Two qualitative research methods were necessary to capture in-depth, follow-up information from sample drawn from two (2) sub-populations who participated in the survey: telephone interviews with non-enrolling students and a focus group discussion with students who enrolled. The findings revealed that five (5) out of six (6) general, college choice factor categories (Academic, Cost, Location of Campus, Service Expectations, and Student Life) undoubtedly contained one or more key aspects influential to college choice decisions. Athletics was the only category with questionable evidence of enrollment impact. Ultimately, the study concluded that eight (8) actions were judicious to address the problem. Prominent among the recommended actions were academic items regarding a robust and diverse academic program portfolio, showcasing academic facilities, and building upon a solid academic reputation. Other important, recommended actions involved reducing cost for students, providing a learned and supportive faculty, continuing campus beautification efforts, optimizing on-campus housing, and expanding extracurricular activities.Ed.D., Educational Leadership and Management -- Drexel University, 201

Parameterized lower bound and NP-completeness of some HH-free Edge Deletion problems

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. We prove that $H$-free Edge Deletion is NP-complete if $H$ is a graph with at least two edges and $H$ has a component with maximum number of vertices which is a tree or a regular graph. Furthermore, we obtain that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cdot |G|^{O(1)}$, unless Exponential Time Hypothesis fails.Comment: 15 pages, COCOA 15 accepted pape

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

Relaxing the Irrevocability Requirement for Online Graph Algorithms

Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is 3/2

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

Graph editing to a given degree sequence

We investigate the parameterized complexity of the graph editing problem called Editing to a Graph with a Given Degree Sequence where the aim is to obtain a graph with a given degree sequence σ by at most k vertex deletions, edge deletions and edge additions. We show that the problem is W[1]-hard when parameterized by k for any combination of the allowed editing operations. From the positive side, we show that the problem can be solved in time 2O(k(Δ⁎+k)2)n2log⁡n for n -vertex graphs, where Δ⁎=max⁡σ, i.e., the problem is FPT when parameterized by k+Δ⁎. We also show that Editing to a Graph with a Given Degree Sequence has a polynomial kernel when parameterized by k+Δ⁎ if only edge additions are allowed, and there is no polynomial kernel unless NP⊆co-NP/poly for all other combinations of the allowed editing operations

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