Disjoint Sets Research Articles

A minimum semi-degree sufficient condition for one-to-many disjoint path covers in semicomplete digraphs

Let D be a digraph. We define the minimum semi-degree of D as δ0(D):=min⁡{δ+(D),δ−(D)}. Let k be a positive integer, and let S={s} and T={t1,…,tk} be any two disjoint subsets of V(D). A set of k internally disjoint paths joining source set S and sink set T that cover all vertices of D are called a one-to-many k-disjoint directed path cover (k-DDPC for short) of D. A digraph D is semicomplete if for every pair x,y of vertices of it, there is at least one arc between x and y.In this paper, we prove that every semicomplete digraph D of sufficiently large order n with δ0(D)≥⌈(n+k−1)/2⌉ has a one-to-many k-DDPC joining any disjoint source set S and sink set T, where S={s},T={t1,…,tk}.

Adversarial co-training for semantic segmentation over medical images

Background and objective:Abundant labeled data drives the model training for better performance, but collecting sufficient labels is still challenging. To alleviate the pressure of label collection, semi-supervised learning merges unlabeled data into training process. However, the joining of unlabeled data (e.g., data from different hospitals with different acquisition parameters) will change the original distribution. Such a distribution shift leads to a perturbation in the training process, potentially leading to a confirmation bias. In this paper, we investigate distribution shift and develop methods to increase the robustness of our models, with the goal of improving performance in semi-supervised semantic segmentation of medical images. We study distribution shift and increase model robustness to it, for improving practical performance in semi-supervised segmentation over medical images. Methods:To alleviate the issue of distribution shift, we introduce adversarial training into the co-training process. We simulate perturbations caused by the distribution shift via adversarial perturbations and introduce the adversarial perturbation to attack the supervised training to improve the robustness against the distribution shift. Benefiting from label guidance, supervised training does not collapse under adversarial attacks. For co-training, two sub-models are trained from two views (over two disjoint subsets of the dataset) to extract different kinds of knowledge independently. Co-training outperforms single-model by integrating both views of knowledge to avoid confirmation bias. Results:For practicality, we conduct extensive experiments on challenging medical datasets. Experimental results show desirable improvements to state-of-the-art counterparts (Yu and Wang, 2019; Peng et al., 2020; Perone et al., 2019). We achieve a DSC score of 87.37% with only 20% of labels on the ACDC dataset, almost same to using 100% of labels. On the SCGM dataset with more distribution shift, we achieve a DSC score of 78.65% with 6.5% of labels, surpassing 10.30% over Peng et al. (2020). Our evaluative results show superior robustness against distribution shifts in medical scenarios. Conclusion:Empirical results show the effectiveness of our work for handling distribution shift in medical scenarios.

Perfectly matched sets in graphs: Parameterized and exact computation

In an undirected graph G=(V,E), we say that (A,B) is a pair of perfectly matched sets if A and B are disjoint subsets of V and every vertex in A (resp. B) has exactly one neighbor in B (resp. A). The size of a pair of perfectly matched sets (A,B) is |A|=|B|. The PERFECTLY MATCHED SETS problem is to decide whether a given graph G has a pair of perfectly matched sets of size k. We show that PMS is W[1]-hard when parameterized by solution size k even when restricted to split graphs and bipartite graphs. We observe that PMS is FPT when parameterized by clique-width, and give FPT algorithms with respect to the parameters distance to cluster, distance to co-cluster and treewidth. Complementing FPT results, we show that PMS does not admit a polynomial kernel when parameterized by vertex cover number unless NP⊆coNP/poly. We also provide an exact exponential algorithm running in time O⁎(1.966n). Finally, considering graphs with structural assumptions, we show that PMS remains NP-hard on planar graphs.

On optimally solving sub‐tree scheduling for wireless sensor networks with partial coverage: A branch‐and‐cut algorithm

AbstractGiven a wireless sensor network, we consider the problem to minimize its total energy consumption over consecutive time slots with respect to communication activities. Nonempty and disjoint subsets of nodes are required to be active and connected under a tree topology configuration in the different time slots, and each network node must be active in a unique time slot. Moreover, the power required by the same pair of network nodes to communicate on the associated direct channel may vary in the different time slots. The problem has been recently introduced in the literature under the name Sub‐Tree Scheduling for Wireless Sensor Networks with Partial Coverage. We focus on the exact solution of the problem. We present a branch‐and‐cut (BC) algorithm based on a novel integer linear programming formulation which allows avoiding the introduction of symmetries in the solution space. In particular, the algorithm relies on an efficient and nontypical separation algorithm for known valid inequalities, and on an easy‐to‐implement primal bound heuristic. The effectiveness of the BC algorithm is empirically shown through an extensive experimental analysis involving 300 newly generated benchmark instances with up to 200 network nodes and 8 time slots. Additionally, the experimental results show that the BC algorithm represents a valid computational tool to benchmark the performance of heuristics addressing the problem, and can be used in practice, as an heuristic solver, to tackle problem instances that are not too large.

Zero-sum partitions of Abelian groups of order $2^n$

The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \leq i \leq t$, such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 \leq i \leq t$. It is easy to check that $m_i\geq 2$ (for every $i$, $1 \leq i \leq t$) and $|I(\Gamma)|\neq 1$ are necessary conditions for the existence of such partitions, where $I(\Gamma)$ is the set of involutions of $\Gamma$. It was proved that the condition $m_i\geq 2$ is sufficient if and only if $|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., for which $|I(\Gamma)|\neq 3$ and $|I(\Gamma)|>1$), only the case of any group $\Gamma$ with $\Gamma\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i\geq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|\Gamma|$ is large enough and $|I(\Gamma)|>1$, then $m_i\geq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i\geq 3$ is sufficient for $\Gamma$ such that $|I(\Gamma)|>1$ and $|\Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.

Thermodynamical u-formalism I: measures of maximal u-entropy for maps that factor over Anosov

AbstractWe construct measures of maximal u-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has finite dimension, and its extreme points are ergodic measures with pairwise disjoint supports.

Clustering dependence on Lyα luminosity from MUSE surveys at 3 < z < 6

We investigate the dependence of Lyα emitter (LAE) clustering on Lyα luminosity and connect the clustering properties of ≈L⋆ LAEs with those of much fainter ones, namely, ≈0.04L⋆. We use 1030 LAEs from the MUSE-Wide survey, 679 LAEs from MUSE-Deep, and 367 LAEs from the to-date deepest ever spectroscopic survey, the MUSE Extremely Deep Field. All objects have spectroscopic redshifts of 3 < z < 6 and cover a large dynamic range of Lyα luminosities: 40.15 < log(LLyα/erg s−1) < 43.35. We apply the Adelberger et al. K-estimator as the clustering statistic and fit the measurements with state-of-the-art halo occupation distribution (HOD) models. We find that the large-scale bias factor increases weakly with an increasing line luminosity. For the low-luminosity (log⟨LLyα/[erg s−1]⟩ = 41.22) and intermediate-luminosity (log⟨LLyα/[erg s−1]⟩ = 41.64) LAEs, we compute consistent bias factors blow = 2.43−0.15+0.15 and binterm. = 2.42−0.09+0.10, whereas for the high-luminosity (log⟨LLyα/[erg s−1]⟩ = 42.34) LAEs we calculated bhigh = 2.65−0.11+0.13. Consequently, high-luminosity LAEs occupy dark matter halos (DMHs) with typical masses of log(Mh/[h−1 M⊙]) = 11.09−0.09+0.10, while low-luminosity LAEs reside in halos of log(Mh/[h−1 M⊙]) = 10.77−0.15+0.13. The minimum masses to host one central LAE, Mmin, and (on average) one satellite LAE, M1, also vary with Lyα luminosity, growing from log(Mmin/[h−1M⊙]) = 10.3−0.3+0.2 and log(M1/[h−1 M⊙]) = 11.7−0.2+0.3 to log(Mmin/[h−1 M⊙]) = 10.7−0.3+0.2 and log(M1/[h−1 M⊙]) = 12.4−0.6+0.4 from low- to high-luminosity samples, respectively. The satellite fractions are ≲10% (≲20%) at 1σ (3σ) confidence level, supporting a scenario in which DMHs typically host one single LAE. We next bisected the three main samples into disjoint subsets to thoroughly explore the dependence of the clustering properties on LLyα. We report a strong (8σ) clustering dependence on Lyα luminosity, not accounting for cosmic variance effects, where the highest luminosity LAE subsample (log(LLyα/erg s−1) ≈ 42.53) clusters more strongly (bhighest = 3.13−0.15+0.08) and resides in more massive DMHs (log(Mh/[h−1M⊙] )= 11.43−0.10+0.04) than the lowest luminosity one (log(LLyα/erg s−1) ≈ 40.97), which presents a bias of blowest = 1.79−0.06+0.08 and occupies log(Mh/[h−1M⊙]) = 10.00−0.09+0.12 halos. We discuss the implications of these results for evolving Lyα luminosity functions, halo mass dependent Lyα escape fractions, and incomplete reionization signatures.

A counterexample to a conjecture on the chromatic number of r $r$‐stable Kneser hypergraphs

AbstractLet be a set of subsets of and be an integer. The ‐colorability defect of is the least number of elements of that need to be removed such that the remaining ground set can be ‐colored without inducing monochromatic sets in . The set is a subset of containing all elements of such that any two members of are at least at distance apart on the ‐cycle. The main purpose of this note is to give a counterexample to a conjecture raised by Florian Frick, which says if is a set of subsets of and , then the number of blocks needed to partition the elements of such that no pairwise disjoint sets lie in the same block is at least .

Geometric structure of affine Deligne-Lusztig varieties for GL3

In this paper we study the geometric structure of affine Deligne-Lusztig varieties Xλ(b) for GL3 and b basic. We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.

Permutree sorting

Generalizing stack sorting and $c$-sorting for permutations, we define the permutree sorting algorithm. Given two disjoint subsets $U$ and $D$ of $\{2, \dots, n-1\}$, the $(U,D)$-permutree sorting tries to sort the permutation $\pi \in \mathfrak{S}_n$ and fails if and only if there are $1 \le i < j < k \le n$ such that $\pi$ contains the subword $jki$ if $j \in U$ and $kij$ if $j \in D$. This algorithm is seen as a way to explore an automaton which either rejects all reduced expressions of $\pi$, or accepts those reduced expressions for $\pi$ whose prefixes are all $(U,D)$-permutree sortable.

Recovery of disrupted airline operations using k-maximum matching in graphs

When an aircraft is approaching an airport, it gets a short time interval called a slot by Air Traffic Control that it can use to land. If the landing of the aircraft is delayed for some reason, it loses its slot and Airline Operation Controllers have to assign it a new one. However, landing slots are a scarce resource of the airports and, in order to avoid that an aircraft waits too long, Airline Operation Controllers have to regularly modify the assignment of slots to the aircraft. Due to the system implemented to exchange slots, controllers can modify the slot-assignment using only two kinds of operations: either assign a slot that was free to aircraft A, or give the slot of another aircraft B to A and assign a free slot to B. The problem can then be modelled as follows.Let k≥1 be an odd integer, let G be a graph and M be a matching in G, i.e., a set of pairwise disjoint edges. A k-maximum matching is a largest possible matching that can be obtained from M by using only augmenting paths of length at most k. This problem has already been studied in the context of wireless networks, mainly because it provides a simple approximation for the maximum matching problem. This paper provides a polynomial-time algorithm when k≤3. We then prove that the problem is NP-hard in planar bipartite graphs with maximum degree at most 3 for any odd integer k≥5.

An improved kernel for the flip distance problem on simple convex polygons

The complexity of computing the flip distance between two triangulations of a simple convex polygon is unknown. Here we approach the problem from a parameterized complexity perspective and improve upon the 2k kernel of Lucas [12]. Specifically, we describe a kernel of size 4k3 and then show how it can be improved to (1+ϵ)k for every constant ϵ>0. By ensuring that the kernel consists of a single instance our result yields a kernel of the same magnitude (up to additive terms) for the almost equivalent rotation distance problem on rooted, ordered binary trees. The earlier work of Lucas left the kernel as a disjoint set of instances, potentially allowing very minor differences in the definition of the size of instances to accumulate, causing a constant-factor distortion in the kernel size when switching between flip distance and rotation distance formulations. Our approach avoids this sensitivity. We have also undertaken experiments to understand how much reduction is achieved by our kernel in practice.

Routing Strategies for Isochronal-Evolution Random Matching Network.

In order to abstract away a network model from some real-world networks, such as navigation satellite networks and mobile call networks, we proposed an Isochronal-Evolution Random Matching Network (IERMN) model. An IERMN is a dynamic network that evolves isochronally and has a collection of edges that are pairwise disjoint at any point in time. We then investigated the traffic dynamics in IERMNs whose main research topic is packet transmission. When a vertex of an IERMN plans a path for a packet, it is permitted to delay the sending of the packet to make the path shorter. We designed a routing decision-making algorithm for vertices based on replanning. Since the IERMN has a specific topology, we developed two suitable routing strategies: the Least Delay Path with Minimum Hop (LDPMH) routing strategy and the Least Hop Path with Minimum Delay (LHPMD) routing strategy. An LDPMH is planned by a binary search tree and an LHPMD is planned by an ordered tree. The simulation results show that the LHPMD routing strategy outperformed the LDPMH routing strategy in terms of the critical packet generation rate, number of delivered packets, packet delivery ratio, and average posterior path lengths.

Connectedness, local connectedness, and components on bipolar soft generalized topological spaces

Connectedness represents the most significant and fundamental topological property. It highlights the main characteristics of topological spaces and distinguishes one topology from another. There is a constant study of bipolar soft generalized topological spaces (\( \mathcal{BSGTS}s \)) by presenting \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-connected set and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-connected space in \(\mathcal{BSGTS}s\) as well as it is discussing some properties and results for these topics. Additionally, the notion of bipolar soft disjoint sets is put forward, \(\mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-separation set, \(\widetilde{\widetilde{\mathfrak{g}}}\)-separated \(\mathcal{BSS}s\) and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-hereditary property. Moreover, there is an extensive study of \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-locally connected space and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-component with some related properties and theorems following them, such as the concepts of \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-locally connected spaces and \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-connected are independent of each other; also determined the conditions under which the \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-connected subsets are \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-components.

Intertwining Connectivities for Vertex-Minors and Pivot-Minors

We show that for pairs and of disjoint subsets of vertices of a graph , if is sufficiently large, then there exists a vertex in such that there are two ways to reduce by a vertex-minor operation that removes while preserving the connectivity between and and the connectivity between and . Our theorem implies an analogous theorem of Chen and Whittle (SIAM J. Discrete Math., 28 (2014), pp. 1402--1404) for matroids restricted to binary matroids.

Bottleneck Convex Subsets: Finding k Large Convex Sets in a Point Set

Chvátal and Klincsek (1980) gave an [Formula: see text]-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set [Formula: see text] of [Formula: see text] points in the plane. This paper examines a generalization of the problem, the Bottleneck Convex Subsets problem: given a set [Formula: see text] of [Formula: see text] points in the plane and a positive integer [Formula: see text], select [Formula: see text] pairwise disjoint convex subsets of [Formula: see text] such that the cardinality of the smallest subset is maximized. Equivalently, a solution maximizes the cardinality of [Formula: see text] mutually disjoint convex subsets of [Formula: see text] of equal cardinality. We give an algorithm that solves the problem exactly, with running time polynomial in [Formula: see text] when [Formula: see text] is fixed. We then show the problem to be NP-hard when [Formula: see text] is an arbitrary input parameter, even for points in general position. Finally, we give a fixed-parameter tractable algorithm parameterized in terms of the number of points strictly interior to the convex hull.

Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities

Abstract In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem { − Δ u = | u | 2 ∗ ( t ) − 2 u | y | t + μ | u | q − 2 u in Ω , u = 0 on ∂ Ω , \begin{cases}-\Delta u=\frac{|u|^{2^*(t)-2} u}{|y|^t}+\mu|u|^{q-2} u &amp; \text { in } \Omega, \\ u=0 &amp; \text { on } \partial \Omega,\end{cases} where Ω is an open bounded domain in ℝ N , which contains some points (0,z*), μ &gt; 0 , 1 &lt; q &lt; 2 , 2 ∗ ( t ) = 2 ( N − t ) N − 2 \mu&gt;0,1&amp;#x003C;q&amp;#x003C;2,2^*(t)=\frac{2(N-t)}{N-2} , 0 ≤ t &lt; 2, x = (y, z) ∈ ℝ k × ℝ N−k , 2 ≤ k ≤ N. We prove that if N &gt; 2 q + 1 q − 1 + t $N &gt; 2{{q + 1} \over {q - 1}} + t$ , then the above problem has two disjoint and infinite sets of solutions. Here, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in [1] for the case of the critical Hardy-Sobolev-Maz’ya problem.

Differentially Private Vertical Federated Clustering

In many applications, multiple parties have private data regarding the same set of users but on disjoint sets of attributes, and a server wants to leverage the data to train a model. To enable model learning while protecting the privacy of the data subjects, we need vertical federated learning (VFL) techniques, where the data parties share only information for training the model, instead of the private data. However, it is challenging to ensure that the shared information maintains privacy while learning accurate models. To the best of our knowledge, the algorithm proposed in this paper is the first practical solution for differentially private vertical federated k -means clustering, where the server can obtain a set of global centers with a provable differential privacy guarantee. Our algorithm assumes an untrusted central server that aggregates differentially private local centers and membership encodings from local data parties. It builds a weighted grid as the synopsis of the global dataset based on the received information. Final centers are generated by running any k -means algorithm on the weighted grid. Our approach for grid weight estimation uses a novel, light-weight, and differentially private set intersection cardinality estimation algorithm based on the Flajolet-Martin sketch. To improve the estimation accuracy in the setting with more than two data parties, we further propose a refined version of the weights estimation algorithm and a parameter tuning strategy to reduce the final k -means loss to be close to that in the central private setting. We provide theoretical utility analysis and experimental evaluation results for the cluster centers computed by our algorithm and show that our approach performs better both theoretically and empirically than the two baselines based on existing techniques.

Системный подход в оценке динамических состояний технических объектов: элементы концепции динамических инвариантов

The scientific and methodological base is being developed within the framework of systems theory and system analysis in application to the tasks of assessment, control and formation of dynamic states of technical objects, the design schemes of which are mechanical oscillatory systems with concentrated parameters. In the context of the fundamental problems of the theory of system studies of forced movements under deterministic disturbances there are considered the problems, whose solutions are aimed at developing a methodological approach to assessing the dynamic states of technical objects operating under coherent vibration loads. The purpose of the study is to develop a method for evaluating, controlling and forming dynamic states of the objects under harmonic force loads. It is assumed that mechanical oscillatory systems formed by a solid body performing small steady-state oscillations relative to the position of the static equilibrium can be used as calculation schemes of the technical objects. To construct mathematical models, Lagrange formalism, Laplace integral transformations taking into account zero conditions and structural methods are used, which make it possible to consider mechanical oscillatory systems as dynamically equivalent structural schemes of automatic control systems. The dynamic state of an object is assessed by using a transfer function, which in the physical sense represents the interpretation of a dynamic compliance. The amplitude-frequency characteristics used to assess the dynamic state of the object present a parametric family, for which the first parameter is the coefficient of connectivity of external force disturbances, and the second parameter is the coordinate of the point for which the dynamic accuracy is defined. An infinite family of amplitude-frequency characteristics is regularized due to the introduction of dynamic invariants, which can be considered as generalized dynamic states of mechanical oscillatory systems. To determine the dynamic invariants of a family of amplitude-frequency characteristics, taking into account two parameters, a map of dynamic invariants is constructed, which divides the plane of parameters into disjoint sets using boundaries. A theorem is proved to determine the explicit analytic expressions of these boundaries.

Fixed set search applied to the clique partitioning problem

The Clique Partitioning Problem (CPP) seeks to decompose a set of vertices into disjoint subsets (cliques) maximizing the sum of edge weights over all the cliques. The interest for the CPP comes from the fact that it represents well many practical problems from areas like data mining, engineering, bio-informatics, etc. To address this NP-hard combinatorial optimization problem, the novel fixed set search (FSS) metaheuristic is applied. This is performed, firstly, by developing a greedy randomized adaptive search procedure (GRASP) using a new type of neighborhood in the local search. The GRASP is further extended to the FSS by adding a learning mechanism. The FSS application to the CPP provides a new approach for generating fixed sets for problems with solution symmetries. This approach opens the application of the FSS to new families of combinatorial optimization problems. The conducted computational experiments show that the FSS significantly outperforms state-of-the-art metaheuristics for the CPP. The proposed approach obtains a large number of new best solutions for commonly used CPP test instances.