Tree Problem Research Articles

Identificación del estado fitosanitario de árboles mediante aprendizaje automático e imágenes de muy alta resolución espacial

Tree diseases contribute to significant economic and food losses in the agricultural sector. Early detection of phytosanitary problems in trees with non-destructive methods is essential to guarantee sustainable orange production. This study presents the findings of a designed methodology conducted to identify diseased orange trees in an orchard situated in the citrus belt of Mexico, specifically in the Rioverde region of San Luis Potosi. To accomplish this, we captured images using a multispectral camera with very high spatial resolution, which was mounted on an unmanned aerial vehicle. These images were used to construct a georeferenced orthomosaic of the orchard. Six thematic classes were established to distinguish various health levels among the trees. We employed several supervised classification algorithms at the pixel level, including Random Forest (RF), K-Nearest Neighbor (KNN), Spectral Angle Mapper (SAM), Support Vector Machine (SVM), and Maximum Likelihood (ML). Considering the classification accuracy achieved by each algorithm, they can be ranked as follows: Maximum Likelihood (ML) with 88.10%, Support Vector Machine (SVM) with 77.38%, Spectral Angle Mapper (SAM) with 76.19%, K-Nearest Neighbor (KNN) with 64.68%, and Random Forest (RF) with 61.90%. These results successfully identified the phytosanitary status of all the trees in the orchard with an acceptable level of accuracy, providing valuable management information for the grower.

Theoretical Aspects of Generating Instances with Unique Solutions: Pre-assignment Models for Unique Vertex Cover

The uniqueness of an optimal solution to a combinatorial optimization problem attracts many fields of researchers' attention because it has a wide range of applications, it is related to important classes in computational complexity, and the existence of only one solution is often critical for algorithm designs in theory. However, as the authors know, there is no major benchmark set consisting of only instances with unique solutions, and no algorithm generating instances with unique solutions is known; a systematic approach to getting a problem instance guaranteed having a unique solution would be helpful. A possible approach is as follows: Given a problem instance, we specify a small part of a solution in advance so that only one optimal solution meets the specification. This paper formulates such a ``pre-assignment'' approach for the vertex cover problem as a typical combinatorial optimization problem and discusses its computational complexity. First, we show that the problem is ΣP2-complete in general, while the problem becomes NP-complete when an input graph is bipartite. We then present an O(2.1996^n)-time algorithm for general graphs and an O(1.9181^n)-time algorithm for bipartite graphs, where n is the number of vertices. The latter is based on an FPT algorithm with O*(3.6791^τ) time for vertex cover number τ. Furthermore, we show that the problem for trees can be solved in O(1.4143^n) time.

An approximation algorithm for the -prize-collecting multicut problem in trees with submodular penalties

AbstractLet $T=(V,E)$ be a tree in which each edge is assigned a cost; let $\mathcal{P}$ be a set of source–sink pairs of vertices in V in which each source–sink pair produces a profit. Given a lower bound K for the profit, the K-prize-collecting multicut problem in trees with submodular penalties is to determine a partial multicut $M\subseteq E$ such that the total profit of the disconnected pairs after removing M from T is at least K, and the total cost of edges in M plus the penalty of the set of still-connected pairs is minimized, where the penalty is determined by a nondecreasing submodular function. Based on the primal-dual scheme, we present a combinatorial polynomial-time algorithm by carefully increasing the penalty. In the theoretical analysis, we prove that the approximation factor of the proposed algorithm is $(\frac{8}{3}+\frac{4}{3}\kappa+\varepsilon)$ , where $\kappa$ is the total curvature of the submodular function and $\varepsilon$ is any fixed positive number. Experiments reveal that the objective value of the solutions generated by the proposed algorithm is less than 130% compared with that of the optimal value in most cases.

Algorithmic Aspects of Total Vertex-Edge Domination in Graphs

A vertex [Formula: see text] of a simple graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a total vertex-edge dominating set if every edge of [Formula: see text] is ve-dominated by a vertex of [Formula: see text] and the subgraph induced by [Formula: see text] has no isolated vertex. The total vertex-edge domination problem is to find a total vertex-edge dominating set of minimum cardinality. In this paper, we first show that the total vertex-edge domination problem is NP-complete for chordal graphs. Then we provide a linear-time algorithm for this problem in trees. Moreover, we show that the minimum total vertex-edge domination problem cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text]). Finally, we prove that the minimum total vertex-edge domination problem is APX-complete for bounded-degree graphs.

Approximation Algorithms for Partial Vertex Covers in Trees

This paper is concerned with designing algorithms for and analyzing the computational complexity of the partial vertex cover problem in trees. Graphs (and trees) are frequently used to model risk management in various systems. In particular, Caskurlu et al. in [4] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. In this paper, we focus on a special case of the partial vertex cover problem, when the input graph is a tree. We consider four possible versions of this setting, depending on whether or not, the vertices and edges are weighted. Two of these versions, where edges are assumed to be unweighted, are known to be polynomial-time solvable. However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, remained open. The main contribution of this paper is to resolve these questions by fully characterizing which variants of partial vertex cover remain NP-hard in trees, and which can be solved in polynomial time. In the paper, we propose two pseudo-polynomial DP-based algorithms for the most general case in which weights are present on both the edges and the vertices of the tree. One of these algorithms leads to a polynomial-time procedure, when weights are confined to the edges of the tree. The insights used in this algorithm are combined with additional scaling ideas to derive an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well.

A knapsack intersection hierarchy

We introduce a knapsack intersection hierarchy for strengthening linear programming relaxations of packing integer programs. In level t of the hierarchy, all valid cuts are added for the integer hull of the intersection of all t-row relaxations. This model captures the maximum possible strength of t-row cuts, an approach often used by solvers for small t. We investigate the integrality gap of the strengthened formulations on the all-or-nothing flow problem in trees (also called unsplittable flow on trees).

Primality, criticality and minimality problems in trees

In a graph [Formula: see text], a module is a vertex subset [Formula: see text] of [Formula: see text] such that every vertex outside [Formula: see text] is adjacent to all or none of [Formula: see text]. For example, [Formula: see text], [Formula: see text][Formula: see text] and [Formula: see text] are modules of [Formula: see text], called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex [Formula: see text] of a prime graph [Formula: see text] is critical if [Formula: see text] is decomposable. Moreover, a prime graph with [Formula: see text] noncritical vertices is called [Formula: see text]-critical graph. A prime graph [Formula: see text] is [Formula: see text]-minimal if there is some [Formula: see text]-vertex set [Formula: see text] of vertices such that there is no proper induced subgraph of [Formula: see text] containing [Formula: see text] is prime. From this perspective, Boudabbous proposes to find the [Formula: see text]-critical graphs and [Formula: see text]-minimal graphs for some integer [Formula: see text] even in a particular case of graphs. This research paper attempts to answer Boudabbous’s question. First, we describe the [Formula: see text]-critical tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-critical tree with [Formula: see text] vertices where [Formula: see text]. Second, we provide a complete characterization of the [Formula: see text]-minimal tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-minimal tree with [Formula: see text] vertices where [Formula: see text].

Lifetime prolongation of a wireless charging sensor network using a mobile robot via linear Diophantine fuzzy graph environment

Edge networking plays a major part in issues with computer networks and issues with the path. In this article, in linear Diophantine fuzzy (LDF) graphs, we present special forms of linear Diophantine fuzzy bridges, cut-vertices, cycles, trees, forests, and introduce some of their characteristics. Also, one of the most researched issues in linear Diophantine fuzzy sets (LDFS) and systems is the minimum spanning tree (MST) problem, where the arc costs have linear Diophantine fuzzy (LDF) values. In this work, we focus on an MST issue on a linear Diophantine fuzzy graph (LDFG), where each arc length is allocated a linear Diophantine fuzzy number (LDFG) rather than a real number. The LDFN can reflect the uncertainty in the LDFG’s arc costs. Two critical issues must be addressed in the MST problem with LDFG. One issue is determining how to compare the LDFNs, i.e., the cost of the edges. The other question is how to calculate the edge addition to determine the cost of the LDF-MST. To overcome these difficulties, the score function representation of LDFNs is utilized and Prim’s method is a well-known approach for solving the minimal spanning tree issue in which uncertainty is ignored, i.e., precise values of arc lengths are supplied. This technique works by providing more energy to nodes dependent on their position in the spanning tree. In addition, an illustrated example is provided to explain the suggested approach. By considering a mobile charger vehicle that travels across the sensor network on a regular basis, charging the batteries of each sensor node.

The weighted k-center problem in trees for fixed k

We present a linear time algorithm for the weighted k-center problem on trees for fixed k. This partially settles the long-standing question about the lower bound on the time complexity of the problem. The current time complexity of the best-known algorithm for the problem with k as part of the input is O(nlog⁡n) by Wang et al. (2018) [20]. Whether an O(n) time algorithm exists for arbitrary k is still open.

On star family packing of graphs

Let ℋ be a family of graphs. An ℋ-packing of a graph G is a set {G1, G2,…,Gk} of disjoint subgraphs of G such that each Gj is isomorphic to some element of ℋ. An ℋ-packing of a graph G that covers the maximum number of vertices of G is called a maximum ℋ-packing of G. The ℋ-packing problem seeks to find a maximum ℋ-packing of a graph. Let i be a positive integer. An i-star is a complete bipartite graph K1,i. This paper investigates the ℋ-packing problem with H being a family of stars. For an arbitrary family 𝒮 of stars, we design a linear-time algorithm for the 𝒮-packing problem in trees. Let t be a positive integer. An ℋ-packing is called a t+-star packing if ℋ consists of i-stars with i ≥ t. We show that the t+-star packing problem for t ≥ 2 is NP-hard in bipartite graphs. As a consequence, the 2+-star packing problem is NP-hard even in bipartite graphs with maximum degree at most 4. Let T and t be two positive integers with T > t. An ℋ-packing is called a T\t-star packing if ℋ={K1,1,K1,2,…,K1,T}\{K1,t}. For t ≥ 2, we present a t/t+1-approximation algorithm for the T\t-star packing problem that runs in 𝒪(mn1/2) time, where n is the number of vertices and m the number of edges of the input graph. We also design a 1/2-approximation algorithm for the 2+-star packing problem that runs in 𝒪(m) time, where m is the number of edges of the input graph. As a consequence, every connected graph with at least 3 vertices has a 2+-star packing that covers at least half of its vertices.

Algorithms for gerrymandering over graphs

We initiate the systematic algorithmic study for gerrymandering over graphs that was recently introduced by Cohen-Zemach, Lewenberg and Rosenschein. Namely, we study a strategic procedure for a political districting designer to draw electoral district boundaries so that a particular target candidate can win in an election. We focus on the existence of such a strategy under the plurality voting rule, and give interesting contrasts which classify easy and hard instances with respect to polynomial-time solvability. For example, we prove that the problem for trees is strongly NP-complete (thus unlikely to have a pseudo-polynomial-time algorithm), but has a pseudo-polynomial-time algorithm when the number of candidates is constant. Another example is to prove that the problem for complete graphs is NP-complete when the number of electoral districts is two, while is solvable in polynomial time when it is more than two.

Finding multi-objective supported efficient spanning trees

This article introduces a new algorithm for computing the set of supported non-dominated points in the objective space and all the corresponding efficient solutions in the decision space for the multi-objective spanning tree (MOST) problem. This algorithm is based on the connectedness property of the set of efficient supported solutions and uses a decomposition of the weight set in the weighting space defined for a parametric version of the MOST problem. This decomposition is performed through a space reduction approach until an indifference region for each supported non-dominated point is obtained. An adjacency relation defined in the decision space is used to compute all the supported efficient spanning trees associated to the same non-dominated supported point as well as to define the indifference region of the next points. An in-depth computational analysis of this approach for different types of networks with three objectives is also presented.

An $O(n\log n)$-Time Algorithm for the $k$-Center Problem in Trees

We consider a classical $k$-center problem in trees. Let $T$ be a tree of $n$ vertices such that every vertex has a nonnegative weight. The problem is to find $k$ centers on the edges of $T$ such t...

Complexity issues of perfect secure domination in graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

Об особенностях правового регулирования и алгоритме решения вопроса вырубки древесно-кустарниковой растительности в охранной зоне газопровода

We present a legal regulation analysis of procedure for cutting down emergency trees and trees and shrubs located in the protected zone of the gas pipeline, which poses a threat to the safety and functioning of gas networks. The purpose of study is to determine the subjects and the proce-dure for cutting down emergency trees in the security zone of gas pipeline, as well as to assess the regulatory regulation of procedure for cutting down emergency trees in the security zone of gas pipeline, taking into account the balance of interests of gas distribution organizations, local authorities and citizens – owners of land plots. The research methodology is based on a sys-tematic approach in the analysis of the legal regulation of the subject of re-search, formal legal, logical methods of cognition, which together ensure the reliability and validity of conclusions and proposals on the algorithm for solving the issue of cutting down trees and shrubs in the protective zone of the gas pipeline. We conclude that there is variability in the regulation of the elimination of emergency trees, taking into account such conditions as the location of the gas pipeline security zone (ownership of territory) and the owner of the land plot on which the trees are located. We state that the pres-ence of tree plantations in the security zone of gas distribution networks is not prohibited by law, but there is a procedure for the owner of land plots and gas networks to remove trees that potentially pose a threat to the safety of the gas pipeline within the security zone. The legislation unambiguously determines that the person who is obliged to carry out work on cutting down trees and sawing off branches is the owner of the land plot. All work on cutting trees must be carried out taking into account the interests of the gas distribution organization, the owner of the gas pipeline and a real threat to the safety of the gas pipeline. We formulate recommendations for effective building of interaction between subjects interested in resolving the problems of finding emergency trees in the protected zone of the gas pipeline.

ОПТИМИЗАЦИЯ КАБЕЛЬНОЙ СЕТИ СБОРА МОЩНОСТИ МОРСКИХ ВЕТРОЭЛЕКТРОСТАНЦИЙ С ПРИМЕНЕНИЕМ ПАРАМЕТРИЗОВАННОГО ЭВРИСТИЧЕСКОГО АЛГОРИТМА

The article discusses an approach to solving the problem of optimizing the routing of infield power cables layout to improve the efficiency and cost-effectiveness of offshore wind farms. Optimization seeks to reduce the total cost of the infield collection system while bearing in mind the constraints including use of sufficiently sized cables and the required absence of cable crossings in the circuit diagram. The problem is a degree-constrained capacitated minimum spanning tree (DCMST) problem with dependent node costs. Search for solu-tion is based on an integrated approach that uses a hybrid optimization algorithm, which combines a parameterized savings heuristic and particle swarm optimization to optimize the parameters of the primary algorithm, ultimately enabling better solutions. Several tests have been performed to compare the constructed circuit diagrams against solutions yielded by other algorithms; tests showed the proposed approach to significantly improve the efficiency of the constructed circuits as demonstrated in a series of tests and evaluated by comparison with other methods, as well as by comparing the efficiency and cost-effectiveness of the optimized routing against the actual layout of the Walney 1 offshore wind farm.

Fully Retroactive Minimum Spanning Tree Problem

Abstract This article describes an algorithm that solves a fully dynamic variant of the minimum spanning tree (MST) problem. The fully retroactive MST allows adding an edge to time $t$, or to obtain the current MST at time $t$. By using the square root technique and a data structure link-cut tree, it was possible to obtain an algorithm that runs each query in $O(\sqrt{m} \lg{|V(G)|})$ amortized, in which $|V(G)|$ is the number of nodes in graph $G$ and $m$ is the size of the timeline. We use a different approach to solve the MST problem instead of the standard algorithms, such as Prim or Kruskal, and this allows using the square root technique to improve the final complexity of the algorithm. Our empirical analysis shows that the proposed algorithm runs faster than re-executing the standard algorithms, and this difference only increases when the number of nodes in these graphs is larger.

An approximation algorithm for the submodular multicut problem in trees with linear penalties

In this paper, we consider the submodular multicut problem in trees with linear penalties (SMCLP(T) problem). In the SMCLP(T) problem, we are given a tree $$T=(V,E)$$ with a submodular function $$c(\cdot ): 2^{E}\rightarrow \mathbb {R}_{\ge 0}$$ , a set of k distinct pairs of vertices $$P=\{(s_1,t_1),(s_2,t_2),\ldots ,(s_k,t_k)\}$$ with non-negative penalty costs $$\pi _{j}$$ for the pairs $$(s_j,t_j)\in P$$ . The goal is to find a partial multicut $$M\subseteq E$$ such that the total cost, consisting of the submodular cost of M and the penalty cost of the pairs not cut by M, is minimized. Let $$P_j$$ be the unique path from $$s_j$$ to $$t_j$$ in the tree, where $$1\le j\le k$$ and let m be the maximal length of all $$P_j$$ . Our main work is to present an m-approximation algorithm for the SMCLP(T) problem via the primal-dual method.

The Multilocus Multispecies Coalescent: A Flexible New Model of Gene Family Evolution.

Incomplete lineage sorting (ILS), the interaction between coalescence and speciation, can generate incongruence between gene trees and species trees, as can gene duplication (D), transfer (T), and loss (L). These processes are usually modeled independently, but in reality, ILS can affect gene copy number polymorphism, that is, interfere with DTL. This has been previously recognized, but not treated in a satisfactory way, mainly because DTL events are naturally modeled forward-in-time, while ILS is naturally modeled backward-in-time with the coalescent. Here, we consider the joint action of ILS and DTL on the gene tree/species tree problem in all its complexity. In particular, we show that the interaction between ILS and duplications/transfers (without losses) can result in patterns usually interpreted as resulting from gene loss, and that the realized rate of D, T, and L becomes nonhomogeneous in time when ILS is taken into account. We introduce algorithmic solutions to these problems. Our new model, the multilocus multispecies coalescent, which also accounts for any level of linkage between loci, generalizes the multispecies coalescent (MSC) model and offers a versatile, powerful framework for proper simulation, and inference of gene family evolution. [Gene duplication; gene loss; horizontal gene transfer; incomplete lineage sorting; multispecies coalescent; hemiplasy; recombination.].

The Inverse Voronoi Problem in Graphs II: Trees

We consider the inverse Voronoi diagram problem in trees: given a tree T with positive edge-lengths and a collection $$\mathbb {U}$$ of subsets of vertices of V(T), decide whether $${\mathbb {U}}$$ is a Voronoi diagram in T with respect to the shortest-path metric. We show that the problem can be solved in $$O(N+n \log ^2 n)$$ time, where n is the number of vertices in T and $$N=n+\sum _{U\in {\mathbb {U}}}|U|$$ is the size of the description of the input. We also provide a lower bound of $$\Omega (n \log n)$$ time for trees with n vertices.