Exact Algorithms for the Minimum Load Spanning Tree Problem

2021 ◽  
Author(s):  
Xiaojun Zhu ◽  
Shaojie Tang
Keyword(s):  
Spanning Tree ◽  
Exact Algorithm ◽  
Maximum Load ◽  
Exact Algorithms ◽  
Mixed Integer ◽  
Bounded Degree ◽  
Linear Programming Formulation ◽  
Previous Algorithm ◽  
Objective Value ◽  
Polynomial Factor

In a minimum load spanning tree (MLST) problem, we are given an undirected graph and nondecreasing load functions for nodes defined on nodes’ degrees in a spanning tree, and the objective is to find a spanning tree that minimizes the maximum load among all nodes. We propose the first [Formula: see text] time exact algorithm for the MLST problem, where [Formula: see text] is the number of nodes and [Formula: see text] ignores polynomial factor. The algorithm is obtained by repeatedly querying whether a candidate objective value is feasible, where each query can be formulated as a bounded degree spanning tree problem (BDST). We propose a novel solution to BDST by extending an inclusion-exclusion based algorithm. To further enhance the time efficiency of the previous algorithm, we then propose a faster algorithm by generalizing the concept of branching walks. In addition, for the purpose of comparison, we give the first mixed integer linear programming formulation for MLST. In numerical analysis, we consider various load functions on a randomly generated network. The results verify the effectiveness of the proposed algorithms. Summary of Contribution: Minimum load spanning tree (MLST) plays an important role in various applications such as wireless sensor networks (WSNs). In many applications of WSNs, we often need to collect data from all sensors to some specified sink. In this paper, we propose the first exact algorithms for the MLST problem. Besides having theoretical guarantees, our algorithms have extraordinarily good performance in practice. We believe that our results make significant contributions to the field of graph theory, internet of things, and WSNs.

scholarly journals Secure the IoT Networks as Epidemic Containment Game

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 156
Author(s):  
Juntao Zhu ◽  
Hong Ding ◽  
Yuchen Tao ◽  
Zhen Wang ◽  
Lanping Yu
Keyword(s):  
Heuristic Algorithms ◽  
Heuristic Method ◽  
Exact Algorithms ◽  
Epidemic Threshold ◽  
Solution Quality ◽  
Sis Model ◽  
Linear Programming Formulation ◽  
Iot Devices ◽  
General Graphs ◽  
The Internet Of Things

The spread of a computer virus among the Internet of Things (IoT) devices can be modeled as an Epidemic Containment (EC) game, where each owner decides the strategy, e.g., installing anti-virus software, to maximize his utility against the susceptible-infected-susceptible (SIS) model of the epidemics on graphs. The EC game’s canonical solution concepts are the Minimum/Maximum Nash Equilibria (MinNE/MaxNE). However, computing the exact MinNE/MaxNE is NP-hard, and only several heuristic algorithms are proposed to approximate the MinNE/MaxNE. To calculate the exact MinNE/MaxNE, we provide a thorough analysis of some special graphs and propose scalable and exact algorithms for general graphs. Especially, our contributions are four-fold. First, we analytically give the MinNE/MaxNE for EC on special graphs based on spectral radius. Second, we provide an integer linear programming formulation (ILP) to determine MinNE/MaxNE for the general graphs with the small epidemic threshold. Third, we propose a branch-and-bound (BnB) framework to compute the exact MinNE/MaxNE in the general graphs with several heuristic methods to branch the variables. Fourth, we adopt NetShiled (NetS) method to approximate the MinNE to improve the scalability. Extensive experiments demonstrate that our BnB algorithm can outperform the naive enumeration method in scalability, and the NetS can improve the scalability significantly and outperform the previous heuristic method in solution quality.

A mixed integer linear formulation for the minimum label spanning tree problem

2009 ◽  
Vol 36 (11) ◽  
pp. 3082-3085 ◽  
Author(s):  
M.Eugénia Captivo ◽  
João C.N. Clímaco ◽  
Marta M.B. Pascoal
Keyword(s):  
Spanning Tree ◽  
Linear Formulation ◽  
Mixed Integer ◽  
Mixed Integer Linear Formulation

scholarly journals A Synchronous Optimization Model for Multiship Shuttle Tanker Fleet Design and Scheduling Considering Hard Time Window Constraint

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Zhenfeng Jiang ◽  
Dongxu Chen ◽  
Zhongzhen Yang
Keyword(s):  
Time Window ◽  
Programming Model ◽  
Transportation Cost ◽  
Exact Algorithm ◽  
Mixed Integer ◽  
Routing Problem ◽  
Hard Time ◽  
Proposed Model ◽  
Total Transportation Cost ◽  
Time Window Constraints

A Synchronous Optimization for Multiship Shuttle Tanker Fleet Design and Scheduling is solved in the context of development of floating production storage and offloading device (FPSO). In this paper, the shuttle tanker fleet scheduling problem is considered as a vehicle routing problem with hard time window constraints. A mixed integer programming model aiming at minimizing total transportation cost is proposed to model this problem. To solve this model, we propose an exact algorithm based on the column generation and perform numerical experiments. The experiment results show that the proposed model and algorithm can effectively solve the problem.

scholarly journals Computing Simplicial Depth by Using Importance Sampling Algorithm and Its Application

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Fanyu Meng ◽  
Wei Shao ◽  
Yuxia Su
Keyword(s):  
Importance Sampling ◽  
Testing Machine ◽  
Exact Algorithm ◽  
Real Data ◽  
Exact Algorithms ◽  
Approximate Algorithm ◽  
Sampling Algorithm ◽  
Simplicial Depth ◽  
Low Efficiency ◽  
Np Problem

Simplicial depth (SD) plays an important role in discriminant analysis, hypothesis testing, machine learning, and engineering computations. However, the computation of simplicial depth is hugely challenging because the exact algorithm is an NP problem with dimension d and sample size n as input arguments. The approximate algorithm for simplicial depth computation has extremely low efficiency, especially in high-dimensional cases. In this study, we design an importance sampling algorithm for the computation of simplicial depth. As an advanced Monte Carlo method, the proposed algorithm outperforms other approximate and exact algorithms in accuracy and efficiency, as shown by simulated and real data experiments. Furthermore, we illustrate the robustness of simplicial depth in regression analysis through a concrete physical data experiment.

scholarly journals Algorithms for the power-p Steiner tree problem in the Euclidean plane

2018 ◽  
Vol 25 (4) ◽  
pp. 28
Author(s):  
Christina Burt ◽  
Alysson Costa ◽  
Charl Ras
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Valid Inequalities ◽  
Minimum Cost ◽  
Performance Ratio ◽  
Mixed Integer ◽  
Steiner Tree Problem ◽  
Steiner Points ◽  
Tree Construction ◽  
The Cost

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case.

scholarly journals On the Location of Fog Nodes in Fog-Cloud Infrastructures

Sensors ◽  
2019 ◽  
Vol 19 (11) ◽  
pp. 2445 ◽  
Author(s):  
Rodrigo A. C. da Silva ◽  
Nelson L. S. da Fonseca
Keyword(s):  
Fog Computing ◽  
Real Data ◽  
Mixed Integer ◽  
End User ◽  
Computing Paradigm ◽  
Linear Programming Formulation ◽  
Cloud Infrastructures ◽  
New Applications ◽  
Integer Linear Programming Formulation

In the fog computing paradigm, fog nodes are placed on the network edge to meet end-user demands with low latency, providing the possibility of new applications. Although the role of the cloud remains unchanged, a new network infrastructure for fog nodes must be created. The design of such an infrastructure must consider user mobility, which causes variations in workload demand over time in different regions. Properly deciding on the location of fog nodes is important to reduce the costs associated with their deployment and maintenance. To meet these demands, this paper discusses the problem of locating fog nodes and proposes a solution which considers time-varying demands, with two classes of workload in terms of latency. The solution was modeled as a mixed-integer linear programming formulation with multiple criteria. An evaluation with real data showed that an improvement in end-user service can be obtained in conjunction with the minimization of the costs by deploying fewer servers in the infrastructure. Furthermore, results show that costs can be further reduced if a limited blocking of requests is tolerated.

scholarly journals Bi-objective load balancing multiple allocation hub location: a compromise programming approach

2020 ◽  
Vol 296 (1-2) ◽  
pp. 363-406 ◽  
Author(s):  
Rahimeh Neamatian Monemi ◽  
Shahin Gelareh ◽  
Anass Nagih ◽  
Dylan Jones
Keyword(s):  
Spatial Distribution ◽  
Transportation Cost ◽  
Compromise Programming ◽  
Mixed Integer ◽  
Programming Approach ◽  
Hub Location ◽  
Hub And Spoke ◽  
Classical Models ◽  
Linear Programming Formulation ◽  
Multiple Allocation

AbstractIn this paper we address unbalanced spatial distribution of hub-level flows in an optimal hub-and-spoke network structure of median-type models. Our study is based on a rather general variant of the multiple allocation hub location problems with fixed setup costs for hub nodes and hub edges in both capacitated and uncapacitated variants wherein the number of hub nodes traversed along origin-destination pairs is not constrained to one or two as in the classical models.. From the perspective of an infrastructure owner, we want to make sure that there exists a choice of design for the hub-level sub-network (hubs and hub edges) that considers both objectives of minimizing cost of transportation and balancing spatial distribution of flow across the hub-level network. We propose a bi-objective (transportation cost and hub-level flow variance) mixed integer non-linear programming formulation and handle the bi-objective model via a compromise programming framework. We exploit the structure of the problem and propose a second-order conic reformulation of the model along with a very efficient matheuristics algorithm for larger size instances.

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