scholarly journals Tangles are Decided by Weighted Vertex Sets

2020 ◽  
Author(s):  
Christian Elbracht ◽  
Jakob Kneip ◽  
Maximilian Teegen
Keyword(s):  
Algorithm Design ◽  
Mathematical Optimization ◽  
Strong Duality ◽  
Combinatorial Structures ◽  
Maximum Order ◽  
Minimum Width ◽  
Vertex Set ◽  
Weighted Vertex ◽  
Connected Subgraphs ◽  
The One

Tree-like decompositions play an important role both in structural combinatorics and algorithm design. The most well-known are [tree decompositions](https://en.wikipedia.org/wiki/Tree_decomposition) of graphs, which are of key importance in the Graph Minor Project of Robertson and Seymour as well as in algorithm design. The minimum width of a tree decomposition is the [treewidth](https://en.wikipedia.org/wiki/Treewidth) of a graph; this parameter captures how closely the global structure of a graph resembles a tree. In particular, a graph has treewidth one if and only if each of its components is a tree. Treewidth has a strong [duality property](https://en.wikipedia.org/wiki/Duality_(optimization)) in the sense of mathematical optimization: the minimum treewidth of a graph is equal to the maximum order of a [bramble](https://en.wikipedia.org/wiki/Bramble_(graph_theory)) increased by one; a bramble is a collection of touching connected subgraphs that cannot be hit by a small set of vertices. [Branchwidth](https://en.wikipedia.org/wiki/Branch-decomposition), which is linearly related to treewidth, is another important width parameter, and its dual parameter is the maximum order of a *tangle*, an object that is more amenable to generalizations to other combinatorial structures than a bramble. Informally speaking, a tangle picks for every separation a small part in a way that small parts of any three separations do not cover the whole structure. In the case of graphs, a tangle of order $k$ concerns vertex $k$-separations, i.e., pairs of subsets of the vertex set such that the union of the two subsets is the whole vertex set, the intersection has size at most $k$ and every edge is in one of the subsets. The main result of the article asserts that every tangle of a graph can be obtained by assigning weights to the vertices so that the small part of every $k$-separation is the one of smaller weight.

scholarly journals C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Algorithmica ◽  
2021 ◽  
Author(s):  
Giordano Da Lozzo ◽  
David Eppstein ◽  
Michael T. Goodrich ◽  
Siddharth Gupta
Keyword(s):  
Dual Graph ◽  
Graph Visualization ◽  
Closed Disk ◽  
Running Time ◽  
Planar Embedding ◽  
Bounded Treewidth ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
The One ◽  
Clustered Planarity

AbstractFor a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances.

scholarly journals On implicit variables in optimization theory

2021 ◽  
Vol Volume 2 (Original research articles) ◽  
Author(s):  
Matúš Benko ◽  
Patrick Mehlitz
Keyword(s):  
Optimization Problems ◽  
Constraint Qualifications ◽  
Optimization Theory ◽  
Mathematical Optimization ◽  
General Setting ◽  
Necessary Optimality Conditions ◽  
Detailed Comparison ◽  
Mordukhovich Stationarity ◽  
Stationarity Conditions ◽  
The One

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory. Comment: 34 pages

Selective withdrawal from a stably stratified fluid

1968 ◽  
Vol 32 (2) ◽  
pp. 209-223 ◽  
Author(s):  
I. R. Wood
Keyword(s):  
Single Layer ◽  
Layered System ◽  
Volume Discharge ◽  
Layer System ◽  
Minimum Width ◽  
Theoretical Predictions ◽  
Small Rise ◽  
Self Similar ◽  
The One ◽  
Dead Water

In this paper a reservoir connected through a horizontal contraction to a channel is considered. Both the reservoir and the channel are considered to contain a stable multi-layered system of fluids. The conditions under which there is a flow in only one layer, and the depth in this flowing layer decreases continuously from its depth in the reservoir to its depth in the channel, give the maximum discharge that can be obtained with a flow only from this single layer. For this case the volume discharge calculations are carried out at a single section (the section of minimum width). Where there are velocities in only two layers and the depth in each of these layers decreases continuously from their depths in the reservoir to their depths in the channel, the theory involves computations at two sections in the flow. These are the section of minimum width and a section upstream of the position of minimum width (the virtual point of control). For this flow it is shown that the solution is the one in which the velocity and density distributions are self similar and that the depths of the layers at the point of maximum contraction are two-thirds of those far upstream. It is then shown that for any stable continuous or discrete density stratification in the reservoir a self similar solution will satisfy the conditions for the depths of the flowing layers to decrease smoothly from the reservoir to downstream of the contraction. Again the ratio of the depth at the contraction to that far upstream is two-thirds.When there is a very large density difference between the fluid in the lower dead water and that in the lowest flowing streamline then this streamline becomes horizontal and may be considered as a frictionless bed. The flow when the bed is not horizontal but where there is a small rise in the channel at the position of maximum contraction is considered for the case where two discrete layers flow under a volume of dead water. In this case the velocity and density profiles are not self similar.Experiments have been carried out with a contraction in a flume for the withdrawal of two discrete layers from a three layer system and the withdrawal from a fluid with a linear density gradient. In both cases the reservoir and channel bed and hence the lowest streamline was effectively horizontal. These experiments confirmed the theoretical predictions.

Automorphism group of the one-divisor graph over the semigroup of upper triangular matrices

2020 ◽  
pp. 2150214
Author(s):  
Xinlei Wang ◽  
Dein Wong ◽  
Fenglei Tian
Keyword(s):  
Automorphism Group ◽  
Zero Divisor ◽  
Symmetric Groups ◽  
Directed Edge ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisor Graph ◽  
Vertex Set ◽  
The One ◽  
Definition Of

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

Dominating a Family of Graphs with Small Connected Subgraphs

2000 ◽  
Vol 9 (4) ◽  
pp. 309-313 ◽  
Author(s):  
YAIR CARO ◽  
RAPHAEL YUSTER
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Minimum Size ◽  
Dominating Sets ◽  
Asymptotic Results ◽  
Connected Dominating Sets ◽  
Vertex Set ◽  
Connected Subgraphs

Let F = {G1, …, Gt} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F, k)-core if, for each v ∈ V and for each i = 1, …, t, there are at least k neighbours of v in Gi that belong to D. The subset D is called a connected (F, k)-core if the subgraph induced by D in each Gi is connected. Let δi be the minimum degree of Gi and let δ(F) = minti=1δi. Clearly, an (F, k)-core exists if and only if δ(F) [ges ] k, and a connected (F, k)-core exists if and only if δ(F) [ges ] k and each Gi is connected. Let c(k, F) and cc(k, F) be the minimum size of an (F, k)-core and a connected (F, k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √lnδ:formula hereThe results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.

scholarly journals Study on the Influence of Diversity and Quality in Entropy Based Collaborative Clustering

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 951
Author(s):  
Jérémie Sublime ◽  
Guénaël Cabanes ◽  
Basarab Matei
Keyword(s):  
Clustering Algorithms ◽  
Mathematical Optimization ◽  
Data Sets ◽  
Distributed Data ◽  
Clustering Methods ◽  
Local Structures ◽  
Collaborative Clustering ◽  
Privacy Constraints ◽  
The Stability ◽  
The One

The aim of collaborative clustering is to enhance the performances of clustering algorithms by enabling them to work together and exchange their information to tackle difficult data sets. The fundamental concept of collaboration is that clustering algorithms operate locally but collaborate by exchanging information about the local structures found by each algorithm. This kind of collaborative learning can be beneficial to a wide number of tasks including multi-view clustering, clustering of distributed data with privacy constraints, multi-expert clustering and multi-scale analysis. Within this context, the main difficulty of collaborative clustering is to determine how to weight the influence of the different clustering methods with the goal of maximizing the final results and minimizing the risk of negative collaborations—where the results are worse after collaboration than before. In this paper, we study how the quality and diversity of the different collaborators, but also the stability of the partitions can influence the final results. We propose both a theoretical analysis based on mathematical optimization, and a second study based on empirical results. Our findings show that on the one hand, in the absence of a clear criterion to optimize, a low diversity pool of solution with a high stability are the best option to ensure good performances. And on the other hand, if there is a known criterion to maximize, it is best to rely on a higher diversity pool of solution with a high quality on the said criterion. While our approach focuses on entropy based collaborative clustering, we believe that most of our results could be extended to other collaborative algorithms.

An Efficient Algorithm Design for the One-Dimensional Cutting-Stock Problem

2012 ◽  
Vol 602-604 ◽  
pp. 1753-1756 ◽  
Author(s):  
Gui Cong Wang ◽  
Chuan Peng Li ◽  
Jie Lv ◽  
Xiu Xia Zhao ◽  
Huan Yong Cui
Keyword(s):  
Programming Model ◽  
Algorithm Design ◽  
Cutting Stock Problem ◽  
Cutting Stock ◽  
Utilization Rate ◽  
Branch And Bound Method ◽  
Integer Linear Programming Model ◽  
One Dimensional ◽  
Level Method ◽  
The One

A Two-level method is proposed to solve the one-dimensional cutting-stock problem during the production process in this paper. First, nested layer cycle is designed to enumerate all the feasible cutting patterns, then a integer linear programming model is established with quantity demands as the constraint. The best cutting scheme is obtained finally according to the branch and bound method. The effectiveness of the proposed method is proved through a real world example, the calculations demonstrate that scheme obviously improves the utilization rate of stock

scholarly journals A comparison of constraint qualifications in infinite-dimensional convex programming revisited

1999 ◽  
Vol 40 (3) ◽  
pp. 353-378 ◽  
Author(s):  
C. Zặlinescu
Keyword(s):  
Convex Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Strong Duality ◽  
Sufficient Condition ◽  
Subdifferential Calculus ◽  
Infinite Dimensional ◽  
Qualification Conditions ◽  
The One

In 1990 Gowda and Teboulle published the paper [16], making a comparison of several conditions ensuring the Fenchel-Rockafellar duality formulainf{f(x) + g(Ax) | x ∈ X} = max{−f*(A*y*) − g*(− y*) | y* ∈ Y*}.Probably the first comparison of different constraint qualification conditions was made by Hiriart-Urruty [17] in connection with ε-subdifferential calculus. Among them appears, as the basic sufficient condition, the formula for the conjugate of the corresponding function; such functions are: f1 + f2, g o A, max{fl,…, fn}, etc. In fact strong duality formulae (like the one above) and good formulae for conjugates are equivalent and they can be used to obtain formulae for ε-subdifferentials, using a technique developed in [17] and extensively used in [46].

scholarly journals The Noncrossing Bond Poset of a Graph

10.37236/9253 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
C. Matthew Farmer ◽  
Joshua Hallam ◽  
Clifford Smyth
Keyword(s):  
Sufficient Conditions ◽  
Partition Lattice ◽  
Open Problems ◽  
Combinatorial Properties ◽  
Noncrossing Partition ◽  
Vertex Set ◽  
Connected Subgraphs ◽  
Necessary And Sufficient ◽  
Broken Circuit ◽  
Noncrossing Partition Lattice

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$. Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice.  Additionally, for several families of graphs, we give combinatorial descriptions of the Möbius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems. 

Transitivity of trees

2021 ◽  
Author(s):  
Libin Chacko Samuel ◽  
Mayamma Joseph
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Maximum Order ◽  
Vertex Set

For a graph [Formula: see text], a partition [Formula: see text] of the vertex set [Formula: see text] is a transitive partition if [Formula: see text] dominates [Formula: see text] whenever [Formula: see text]. The transitivity [Formula: see text] of a graph [Formula: see text] is the maximum order of a transitive partition of [Formula: see text]. For any positive integer [Formula: see text], we characterize the smallest tree with transitivity [Formula: see text] and obtain an algorithm to determine the transitivity of any tree of finite order.

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