Sliding window temporal graph coloring

Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static graphs, which often stand in contrast to practice where data is inherently dynamic. A temporal graph has an edge set that changes over time. We present a natural temporal extension of the classical graph coloring problem. Given a temporal graph and integers k and Δ, we ask for a coloring sequence with at most k colors for each vertex such that in every time window of Δ consecutive time steps, in which an edge is present, this edge is properly colored at least once. We thoroughly investigate the computational complexity of this temporal coloring problem. More specifically, we prove strong computational hardness results, complemented by efficient exact and approximation algorithms

Temporal vertex cover with a sliding time window.

Modern, inherently dynamic systems are usually characterized by a network structure which is subject to discrete changes over time. Given a static underlying graph, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs focused on temporal paths and other “path-related” temporal notions, only few attempts have been made to investigate “non-path” temporal problems. In this paper we introduce and study two natural temporal extensions of the classical problem VERTEX COVER. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. We provide strong hardness results, complemented by approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions

Faster Exploration of Some Temporal Graphs

A temporal graph G = (G_1, G_2, ..., G_T) is a graph represented by a sequence of T graphs over a common set of vertices, such that at the i-th time step only the edge set E_i is active. The temporal graph exploration problem asks for a shortest temporal walk on some temporal graph visiting every vertex. We show that temporal graphs with n vertices can be explored in O(k n^{1.5} log n) days if the underlying graph has treewidth k and in O(n^{1.75} log n) days if the underlying graph is planar. Furthermore, we show that any temporal graph whose underlying graph is a cycle with k chords can be explored in at most 6kn days. Finally, we demonstrate that there are temporal realisations of sub cubic planar graphs that cannot be explored faster than in ?(n log n) days. All these improve best known results in the literature

Union-closed sets and Horn Boolean functions

A family of sets is union-closed if the union of any two sets from belongs to. The union-closed sets conjecture states that if is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality the conjecture remains wide open, but it was verified for various important classes of lattices, such as lower semimodular lattices, and graphs, such as chordal bipartite graphs. In the present paper we develop a Boolean approach to the conjecture and verify it for several classes of Boolean functions, such as submodular functions and double Horn functions

FEATURES OF CU-AG BANDS FOR GENERATORS

Проведенные в условиях производства исследования с целью повышения уровня физико-механических характеристик полос из медносеребряного сплава – удельного электросопротивления, временного сопротивления разрыву, предела текучести, относительного удлинения – явились основой для разработки и реализации нетрадиционного технологического регламента при выпуске промышленных партий полос.Held in conditions of research production with the aim of increasing the level of physical-mechanical characteristics of strips of magnocurarine alloy – specific resistance of the temporary rupture strength, yield strength, elongation – the basis for the development and implementation of non-traditional technological regulations for production of industrial batches of strips

Exact and approximate algorithms for computing a second Hamiltonian cycle.

A classic result by Stockmeyer [Stockmeyer, 1974] gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of the chop operator under the homogeneity assumption [Halpern et al., 1983]. In this paper, we study the complexity of the satisfiability problem for a suitable weakening of the chop interval temporal logic, that can be equivalently viewed as a fragment of Halpern and Shoham interval logic featuring the operators B, for "begins", corresponding to the prefix relation on pairs of intervals, and D, for "during", corresponding to the infix relation. The homogeneous models of the considered logic naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations

THE INFLUENCE OF BACKPRESSURE ON THE DUCTILITY OF MG-ALLOY MA2-1 DURING BACKWARD EXTRUSION

На прессе номинальным усилием 10 МН выполнены эксперименты по получению полых заготовок из сплава системы магний-алюминий. Применен метод обратного выдавливания с созданием противодавления и без него. Выявлено, что при отсутствии противодавления получаемые заготовки покрываются сеткой трещин, что говорит о недостаточной пластичности металла. Применение противодавления позволило улучшить качество заготовок.Experiments were performed to obtain hollow billets of magnesium - aluminum alloy on the press of nominal force 10 MN. Backward extrusion method with counter and without is applied. It was revealed that in the absence of backpressure blank grid covered cracks, indicating that the lack of ductility of the metal. Application of back pressure has improved the quality of the workpieces

Computing maximum matchings in temporal graphs

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ ℕ is given. The requirement that a vertex cannot be matched twice in any Δ-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms

Computing maximum matchings in temporal graphs.

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ ℕ is given. The requirement that a vertex cannot be matched twice in any Δ-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms