Showing papers on "Minimum weight published in 2023"
TL;DR: DeepOpt-MWDS as discussed by the authors is an efficient local search scheme that dynamically adjusts the number of added and removed vertices according to the information of the candidate solution and further develops three novel ideas to improve performance.
Abstract: The minimum weight dominating set (MWDS) problem is an important generalization of the minimum dominating set problem with various applications. In this work, we develop an efficient local search scheme that can dynamically adjust the number of added and removed vertices according to the information of the candidate solution. Based on this scheme, we further develop three novel ideas to improve performance, resulting in our so-called DeepOpt-MWDS algorithm. First, we use a new construction method with five reduction rules to significantly reduce massive graphs and construct an initial solution efficiently. Second, an improved configuration checking strategy called CC 2 V3+ is designed to reduce the cycling phenomenon in local search. Third, a general perturbation framework called deep optimization mechanism (DeepOpt) is proposed to help the algorithm avoid local optima and to converge to a new solution quickly. Extensive experiments based on eight popular benchmarks of different scales are carried out to evaluate the proposed algorithm. Compared to seven state-of-the-art heuristic algorithms, DeepOpt-MWDS performs better on random and classic benchmarks and obtains the best solutions on almost all massive graphs. We investigate three main algorithmic ingredients to understand their impacts on the performance of the proposed algorithm. Moreover, we adapt the proposed general framework DeepOpt to another NP-hard problem to verify its generality and achieve good performance.
6 citations
TL;DR: In this article , a novel approach based on chemical reaction optimization algorithm (CRO) has been proposed to solve the minimum weight dominating set (MWDS) problem, which is defined as the set of vertices that can cover all other vertices of the graph.
Abstract: Dominating set of a graph can be defined as the set of vertices that can cover all other vertices of the graph. The minimum weight dominating set (MWDS) is the minimum number of vertices in the dominating set with minimum total weight. In recent times, the chemical reaction optimization algorithm (CRO) has shown its supremacy in solving these types of problems. Therefore in this paper, a novel approach based on CRO has been proposed to solve the MWDS problem. The proposed method uses a repair-based technique to generate a molecule. To make the solution feasible by covering all vertices and to get better results, three supporting operators are implemented along with the CRO operators. Besides this, two repair operators are introduced. In the first repair operator, the searching procedure works based on the scaling properties of vertices, and the second one is a unique method for eliminating common neighbors of vertices of the dominating set. The performance of the proposed method is better than any other existing related algorithms. The performance is measured from different graphs of the benchmark datasets. It can be mentioned that the proposed method takes minimal running time to obtain the minimum weight compared to other benchmark methods.
1 citations
TL;DR: In this article , the real multilinear polynomial for the Boolean function which determines if a given bipartite graph G ⊆ K n , n has a perfect matching was given.
1 citations
12 Jun 2023-Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
TL;DR: In this paper , the authors presented the weight optimization of the main girder of a bridge crane with a non-symmetric box-like cross-section using water evaporation optimization (WEO).
Abstract: The proper choice of material and geometric properties of the girder of the bridge crane can significantly reduce its weight and production cost. This research presents the weight optimization of the main girder with a non-symmetric box-like cross-section. The strength analysis in characteristic points of the critical cross-section and the local stability of plates were conducted using Eurocodes. This study aimed to prove that a proper choice of geometry for the cross-section plates and their additional design elements can have a meaningful impact on the weight. The optimization procedure was done using Water Evaporation Optimization (WEO) algorithm, with the implementation of all necessary criteria and conditions which must be fulfilled. This study revealed that the application of the light-weight design philosophy to the structure of the main girder could significantly reduce its weight, which is verified in the existing examples of double-beam bridge cranes. Achieved savings in girder weight are between 24.43% and 34.73%, dependingly on the considered example. Also, the study showed the influence of the chosen material on the steel girder’s optimum weight and geometric parameters. Depending on the studied example of the bridge crane and selected material, the WEO algorithm gave the same solution through simulations. The value of the optimum weight had a slight deviation at the second decimal place, which is neglectable. The algorithm successfully avoided the trap of getting into the local minimum during the search. In the end, it can be stated that the application of the WEO algorithm was successful in the considered engineering problem since it was the optimization of a complex steel structure with 11 variables and more than 20 constraint functions.
TL;DR: In this article , the optimal weight for steel truss roof systems is determined while corrosion is considered, and the main program, which includes structural analysis (FEA) and optimization algorithms, is coded in MATLAB programming language.
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TL;DR: In this paper , the authors consider three basic optimization problems: minimum weight spanning trees, shortest paths, and minimum weight matchings, where the weights are random variables and the properties of the optimum value are also a random variable.
Abstract: There are many cases in which we put weights on the edges of a graph or digraph and ask for the minimum or maximum weight object. The optimization questions that arise from this are the backbone of Combinatorial optimization. When the weights are random variables, we can ask for properties of the optimum value, which will be also a random variable. In this chapter, we consider three of the most basic optimization problems: minimum weight spanning trees, shortest paths, and minimum weight matchings.
03 Apr 2023
TL;DR: In this article , the performance of Binary Particle Swarm Optimization (BPSO) algorithm was studied on unbraced and X-braced frames and the influence of the classification groups of elements.
Abstract: This research studies on the performance of Binary Particle Swarm Optimization (BPSO) algorithm. The first part is finding the best inertia weight of BPSO from various types of inertia weight. The second part is optimizing the cross-sectional area of steel structures and topology of bracing system under vertical and lateral load. The structures studied in the research include unbraced frames and X-braced frames. Moreover, the braced frame also investigates the influence of the classification groups of elements. The elements are classified into finer groups than the original group. The design of the structure follows the AISC code. From the investigation in the first part, a constant inertia weight of 0.98 is the best. In the second part, minimum weights of unbraced frames using BPSO are the lowest weight, except three-bays, twenty-four stories frame. For braced frames with original grouping, all examples get a lower weight than the unbraced frames. For studying the influence of group, the results of the two examples are contradictory. One bay, ten stories frame with new group has a minimum weight less than the original while three-bays but twenty-four stories frame is opposite.
TL;DR: The decision problem associated with the independent Roman dictatorship problem is NP-hard for bipartite graphs as discussed by the authors , and various upper bounds on the independent dictatorship number are established as well as exact values on it for some special graphs.
Abstract: An independent Roman dominating function (IRD-function) on a graph $G$ is a
function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the conditions that (i) every
vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which
$f(v)=2$, and (ii) the set of all vertices assigned non-zero
values under $f$ is independent. The weight of an IRD-function is
the sum of its function values over all vertices, and the independent Roman
domination number $i_{R}(G)$ of $G$ is the minimum weight of an
IRD-function on $G$. In this paper, we initiate the study of the independent
Roman bondage number $b_{iR}(G)$ of a graph $G$ having at least
one component of order at least three, defined as the smallest size of set of
edges $F\subseteq E(G)$ for which $i_{R}(G-F)>i_{R}(G)$. We begin by showing
that the decision problem associated with the independent Roman
bondage problem is NP-hard for bipartite graphs.
Then various upper bounds on $b_{iR}(G)$ are established as well
as exact values on it for some special graphs. In particular, for trees $T$
of order at least three, it is shown that $b_{iR}(T)\leq3,$
while for connected planar graphs the upper bounds are in terms of
the maximum degree with refinements depending on the girth of the graph.
04 May 2023
TL;DR: In this article , a closed-form expression for the enumeration of polar codewords with weight greater than the minimum weight was proposed, which is the only weight for which an explicit enumerator formula is available.
Abstract: The weight distribution of error correction codes is a critical determinant of their error-correcting performance, making enumeration of utmost importance. In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum distance $d$) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight $1.5\wm$. Our simulations demonstrate the potential for extending this method to higher weights.
22 Jan 2023
TL;DR: In this article , an auction algorithm using multiplicative instead of constant weight updates was proposed to compute a $(1-\eps)$-approximate maximum weight matching (MWM) in a bipartite graph with $n$ vertices and $m$ edges.
Abstract: $
ewcommand{\eps}{\varepsilon}$We present an auction algorithm using multiplicative instead of constant weight updates to compute a $(1-\eps)$-approximate maximum weight matching (MWM) in a bipartite graph with $n$ vertices and $m$ edges in time $O(m\eps^{-1}\log(\eps^{-1}))$, matching the running time of the linear-time approximation algorithm of Duan and Pettie [JACM '14]. Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a $(1-\eps)$-approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is $O(m\eps^{-1}\log(\eps^{-1}))$, where $m$ is the sum of the number of initially existing and inserted edges.
22 Mar 2023
TL;DR: In this paper , the authors introduced the problem of characterizing the code rate and designing constant-weight binary $B_2$-sequences, which are collections of binary strings of length n with the property that the real-valued sums of all distinct pairs of strings are distinct.
Abstract: Motivated by applications in polymer-based data storage we introduced the new problem of characterizing the code rate and designing constant-weight binary $B_2$-sequences. Binary $B_2$-sequences are collections of binary strings of length $n$ with the property that the real-valued sums of all distinct pairs of strings are distinct. In addition to this defining property, constant-weight binary $B_2$-sequences also satisfy the constraint that each string has a fixed, relatively small weight $\omega$ that scales linearly with $n$. The constant-weight constraint ensures low-cost synthesis and uniform processing of the data readout via tandem mass spectrometers. Our main results include upper bounds on the size of the codes formulated as entropy-optimization problems and constructive lower bounds based on Sidon sequences.
11 Jan 2023
TL;DR: In this article , the authors studied two examples of minimum weight random graphs with edge constraints and employed martingale difference techniques to describe the behaviour of the scaled minimum weight in terms of the edge constraint.
Abstract: In this paper, we study two examples of minimum weight random graphs with edge constraints. First we consider the complete graph on ${n}$ vertices equipped with uniformly heavy edge weights and use iteration methods to obtain deviation estimates for the minimum weight of subtrees with a given number of edges. Next we analyze edge constrained minimum weight paths in the integer lattice ${\mathbb{Z}^d}$ and employ martingale difference techniques to describe the behaviour of the scaled minimum weight in terms of the edge constraint.
TL;DR: In this article , a reformulated evolution strategy (ES) algorithm is proposed and employed to solve the resulting optimization problem effectively, a frame is first designed for the minimum weight considering sizing design variables only, where initial orientations of the column members are kept unchanged.
01 Jan 2023
TL;DR: In this paper , an auction algorithm using multiplicative instead of constant weight updates was proposed to compute a (1 − ε)-approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges.
Abstract: We present an auction algorithm using multiplicative instead of constant weight updates to compute a $$(1-\varepsilon )$$ -approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges in time $$O(m\varepsilon ^{-1}\log (\varepsilon ^{-1}))$$ , matching the running time of the linear-time approximation algorithm of Duan and Pettie [JACM ’14]. Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a $$(1-\varepsilon )$$ -approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is $$O(m\varepsilon ^{-1}\log (\varepsilon ^{-1}))$$ , where m is the sum of the number of initially existing and inserted edges.
22 Jan 2023
TL;DR: In this article , a new polynomial-time approximation algorithm for the minimum-weight CDS problem with approximation ratio 2H(delta-max+m-1) is presented.
Abstract: Consider a graph with nonnegative node weight. A vertex subset is called a CDS (connected dominating set) if every other node has at least one neighbor in the subset and the subset induces a connected subgraph. Furthermore, if every other node has at least $m$ neighbors in the subset, then the node subset is called a $(1,m)$CDS. The minimum-weight $(1,m)$CDS problem aims at finding a $(1,m)$CDS with minimum total node weight. In this paper, we present a new polynomial-time approximation algorithm for this problem with approximation ratio $2H(\delta_{\max}+m-1)$, where $\delta_{\max}$ is the maximum degree of the given graph and $H(\cdot)$ is the Harmonic function, i.e., $H(k)=\sum_{i=1}^k \frac{1}{i}$.
TL;DR: In this paper , it was shown that for any finite point sets A and B of points in the Euclidean plane, there exists a minimum (A, B )-network that is constructible by straightedge and compass.
Abstract: Abstract Given two finite sets A and B of points in the Euclidean plane, a minimum multi-source multi-sink Steiner network in the plane, or a minimum ( A , B )-network, is a directed graph embedded in the plane with a dipath from every node in A to every node in B such that the total length of all arcs in the network is minimised. Such a network may contain Steiner points—nodes appearing in the solution that are neither in A nor B . We show that for any finite point sets A , B in the plane, there exists a minimum ( A , B )-network that is constructible by straightedge and compass (this was claimed in a paper by Maxwell and Swanepoel, but their argument is incorrect). We use this property to formulate an algorithmic framework for exactly finding minimum ( A , B )-networks in the Euclidean plane. We also present several new structural and geometric properties of minimum ( A , B )-networks. In particular, we resolve a conjecture of Alfaro by proving that, for any terminal set A , adding an appropriate orientation to the edges of a minimum 2-edge-connected Steiner network on A yields a minimum ( A , A )-network.
12 Jan 2023
TL;DR: In this paper , a Roman dominating function is defined as a function satisfying the condition that every vertex u for which u is adjacent to exactly one vertex v for which ƒ(v) = 2 .
Abstract: Abstract Let G = (V, E) be a graph of order n . Let B(S) be the set of vertices in V\S that have a neighbor in the vertex set S . The differential of a vertex set S is defined as ∂(S) = |B(S)|−|S| and the maximum value of ∂(S) for any subset S of V is the differential of G . For S ⊆ V(G) , the set N p (S) is defined as the perfect neighborhood of S such that all vertices in V(G)\S have exactly one neighbor in S . The perfect differential of S is defined to be ∂ p (S) = |N p (S)|−|S| and the perfect differential of a graph is defined as ∂ p (G) = {∂ p (S)⊆V(G)} . A Roman dominating function of G is a function ƒ:V → {0,1,2} such that every vertex v for which ƒ(v) = 0 has a neighbor u with ƒ(u) = 2 . The weight of a Roman dominating function ƒ is w(ƒ) = ∑ v∈V ƒ(v) . The Roman domination number of a graph G , denoted by γ R (G) , is the minimum weight of all possible Roman dominating functions. A perfect Roman dominating function is defined as an Roman dominating function ƒ satisfying the condition that every vertex u for which ƒ(u) = 0 is adjacent to exactly one vertex v for which ƒ(v) = 2 . The perfect Roman domination number, denoted by γ P R (G) , is the minimum weight among all perfect Roman dominating functions on G , that is γ P R (G) = min{w(ƒ):ƒ is a perfect Roman dominating function on G} . This paper is devoted to the computation of differential, perfect differential and Roman domination, perfect Roman domination of probabilistic neural networks by the use of the proven Gallai-type results γ R (G) = n−∂(G), γ P R (G) + ∂ P (G) = n . Besides, existing Roman and perfect Roman graph classes of probabilistic neural networks are characterized. 2020 Mathematics Subject Classification: 05C69, 05C05, 05C82
TL;DR: In this article , the authors studied the problem of perfect double Italian dominating functions and proved the NP-completeness of the γdIp parameter for bipartite graphs.
TL;DR: In this paper , Gabow and Tarjan extended the depth-first search approach to find f-factors of general multigraphs and gave a blocking trail algorithm with running time O(m).
Abstract: Blocking flows were introduced by Dinic (Soviet Math Doklady 11: 1277–1280, 1970) to speed up the computation of maximum network flows. They have been used in algorithms for problems such as maximum cardinality matching of bipartite graphs Hopcroft and Karp (SIAM J Comput 2(4), 225–231, 1973) and general graphs Micali and Vazirani (in: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, 17–27, 1980), maximum weight matching of general graphs Gabow and Tarjan (J ACM 38(4), 815–853, 1991), and many others. The blocking algorithm of Gabow and Tarjan (1991) for matching is based on depth-first search. We extend the depth-first search approach to find f-factors of general multigraphs. Here f is an arbitrary integral-valued function on vertices, an f-matching is a subgraph where every vertex x has degree $$\le f(x)$$ , an f-factor has equality in every degree bound. A set of blocking trails for an f-matching M is a maximal collection $$\mathcal{A}$$ of edge-disjoint augmenting trails such that $$M\bigoplus _{A\in {\mathcal{A}}} A $$ is a valid f-matching. Blocking trails are needed in efficient algorithms for maximum cardinality f-matching Huang and Pettie (Algorithmica 84(7): 1952–1992, 2022), maximum weight f-factors/matchings by scaling Duan et al. (In: Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), Vol. 168 of LIPIcs, 41:1-41:17, 2020; Gabow (A weight-scaling algorithm for f-factors of multigraphs. arXiv:2010.01102 , 2020), and approximate maximum weight f-factors and f-edge covers Huang and Pettie (2022). Since these algorithms find many sets of blocking trails, the time to find blocking trails is a dominant factor in the running time. Our blocking trail algorithm runs in linear time O(m). In independent work and using a different approach, Huang and Pettie (2022) present a blocking trails algorithm using time $$O(m\alpha (m,n))$$ . As examples of the time bounds for the above applications, an approximate maximum weight f-factor is found in time $$O(m\,\alpha (m,n))$$ using Huang and Pettie (2022), and our algorithm eliminates the factor $$\alpha (m,n)$$ . Similarly a maximum weight f-factor is found in time $$O(\sqrt{\Phi \log \Phi }\, m\,\alpha (m,n)\, \log (\Phi W))\,$$ using Huang and Pettie (2022) , ( $$\Phi =\sum _{v\in V} f(v)$$ , W the maximum edge weight) and our algorithm eliminates the $$\alpha (m,n)$$ factor, making the time within a factor $$\sqrt{\log {\Phi }}$$ of the bound for bipartite multigraphs. The technical difficulty for this work stems from the fact that a fixed vertex can occur many times in a given search. This does not occur in ordinary matching or in algorithms for maximum cardinality or maximum weight f-matching. These multiple occurrences can create a new variant of blossom, the “skew blossom”. Also they can make blossoms become “incomplete”, i.e., partially processed yet still relevant in future searches.