scholarly journals Algorithms for Euclidean Degree Bounded Spanning Tree Problems

2019 ◽  
Vol 29 (02) ◽  
pp. 121-160 ◽  
Author(s):  
Patrick J. Andersen ◽  
Charl J. Ras
Keyword(s):  
Approximation Algorithms ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Euclidean Plane ◽  
Minimum Spanning Tree ◽  
Computational Experiments ◽  
Optimal Solutions ◽  
New Type ◽  
Disjoint Sets ◽  
Set Of Points

Given a set of points in the Euclidean plane, the Euclidean [Formula: see text]-minimum spanning tree ([Formula: see text]-MST) problem is the problem of finding a spanning tree with maximum degree no more than [Formula: see text] for the set of points such the sum of the total length of its edges is minimum. Similarly, the Euclidean [Formula: see text]-minimum bottleneck spanning tree ([Formula: see text]-MBST) problem, is the problem of finding a degree-bounded spanning tree for a set of points in the plane such that the length of the longest edge is minimum. When [Formula: see text], these two problems may yield disjoint sets of optimal solutions for the same set of points. In this paper, we perform computational experiments to compare the accuracies of a variety of heuristic and approximation algorithms for both these problems. We develop heuristics for these problems and compare them with existing algorithms. We also describe a new type of edge swap algorithm for these problems that outperforms all the algorithms we tested.

ON THE STEINER RATIO IN $\mathcal{R}_{n}$

2011 ◽  
Vol 03 (04) ◽  
pp. 473-489
Author(s):  
HAI DU ◽  
WEILI WU ◽  
ZAIXIN LU ◽  
YINFENG XU
Keyword(s):  
Euclidean Space ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Minimum Length ◽  
Steiner Ratio ◽  
Line Segments ◽  
Finite Set ◽  
Steiner Minimum Tree ◽  
Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

Integer Programming Formulations for Minimum Spanning Tree Interdiction

2021 ◽  
Author(s):  
Ningji Wei ◽  
Jose L. Walteros ◽  
Foad Mahdavi Pajouh
Keyword(s):  
Integer Programming ◽  
Convex Hull ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Solution Space ◽  
Integer Program ◽  
Computational Experiments ◽  
Network Interdiction ◽  
Computational Aspects ◽  
Feasible Solutions

We consider a two-player interdiction problem staged over a graph where the attacker’s objective is to minimize the cost of removing edges from the graph so that the defender’s objective, that is, the weight of a minimum spanning tree in the residual graph, is increased up to a predefined level r. Standard approaches for graph interdiction frame this type of problems as bilevel formulations, which are commonly solved by replacing the inner problem by its dual to produce a single-level reformulation. In this paper, we study an alternative integer program derived directly from the attacker’s solution space and show that this formulation yields a stronger linear relaxation than the bilevel counterpart. Furthermore, we analyze the convex hull of the feasible solutions of the problem and identify several families of facet-defining inequalities that can be used to strengthen this integer program. We then proceed by introducing a different formulation defined by a set of so-called supervalid inequalities that may exclude feasible solutions, albeit solutions whose objective value is not better than that of an edge cut of minimum cost. We discuss several computational aspects required for an efficient implementation of the proposed approaches. Finally, we perform an extensive set of computational experiments to test the quality of these formulations, analyzing and comparing the benefits of each model, as well as identifying further enhancements. Summary of Contribution: Network interdiction has received significant attention over the last couple of decades, with a notable peak of interest in recent years. This paper provides an interesting balance between the theoretical and computational aspects of solving a challenging network interdiction problem via integer programming. We present several technical developments, including a detailed study of the problem's solution space, multiple formulations, and a polyhedral analysis of the convex hull of feasible solutions. We then analyze the results of an extensive set of computational experiments that were used to validate the effectiveness of the different methods we developed in this paper.

CLUSTERING WITH MINIMUM SPANNING TREES

2011 ◽  
Vol 20 (01) ◽  
pp. 139-177 ◽  
Author(s):  
YAN ZHOU ◽  
OLEKSANDR GRYGORASH ◽  
THOMAS F. HAIN
Keyword(s):  
Spanning Tree ◽  
Clustering Algorithm ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Clustering Algorithms ◽  
Predefined Criterion ◽  
Point Set ◽  
Color Clustering ◽  
Representative Points ◽  
Set Of Points

We propose two Euclidean minimum spanning tree based clustering algorithms — one a k-constrained, and the other an unconstrained algorithm. Our k-constrained clustering algorithm produces a k-partition of a set of points for any given k. The algorithm constructs a minimum spanning tree of a set of representative points and removes edges that satisfy a predefined criterion. The process is repeated until k clusters are produced. Our unconstrained clustering algorithm partitions a point set into a group of clusters by maximally reducing the overall standard deviation of the edges in the Euclidean minimum spanning tree constructed from a given point set, without prescribing the number of clusters. We present our experimental results comparing our proposed algorithms with k-means, X-means, CURE, Chameleon, and the Expectation-Maximization (EM) algorithm on both artificial data and benchmark data from the UCI repository. We also apply our algorithms to image color clustering and compare them with the standard minimum spanning tree clustering algorithm as well as CURE, Chameleon, and X-means.

scholarly journals CONSTRUCTING DEGREE-3 SPANNERS WITH OTHER SPARSENESS PROPERTIES

1996 ◽  
Vol 07 (02) ◽  
pp. 121-135 ◽  
Author(s):  
GAUTAM DAS ◽  
PAUL J. HEFFERNAN
Keyword(s):  
Euclidean Space ◽  
Shortest Path ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Euclidean Distance ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Edge Weight

Let V be any set of n points in k-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for all u and υ in V, the length of the shortest path from u to υ in the spanner is at most t times the Euclidean distance between u and υ. We show that for any δ>1, there exists a t-spanner (where t is a constant that depends only on δ and k) with the following properties: its maximum degree is 3, it has at most n·δ edges, its total edge weight is at most O(1) times the weight of the minimum spanning tree of V, and it can be constructed in O(n log n) time. The constants implicit in the O-notation depend on δ and k.

scholarly journals Novel Degree Constrained Minimum Spanning Tree Algorithm Based on an Improved Multicolony Ant Algorithm

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Xuemei Sun ◽  
Cheng Chang ◽  
Hua Su ◽  
Chuitian Rong
Keyword(s):  
Experimental Data ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Optimal Algorithm ◽  
Minimum Weight ◽  
Ant Algorithm ◽  
Random Perturbations ◽  
Optimal Solutions ◽  
Ant System ◽  
Multiple Groups

Degree constrained minimum spanning tree (DCMST) refers to constructing a spanning tree of minimum weight in a complete graph with weights on edges while the degree of each node in the spanning tree is no more thand(d≥ 2). The paper proposes an improved multicolony ant algorithm for degree constrained minimum spanning tree searching which enables independent search for optimal solutions among various colonies and achieving information exchanges between different colonies by information entropy. Local optimal algorithm is introduced to improve constructed spanning tree. Meanwhile, algorithm strategies in dynamic ant, random perturbations ant colony, and max-min ant system are adapted in this paper to optimize the proposed algorithm. Finally, multiple groups of experimental data show the superiority of the improved algorithm in solving the problems of degree constrained minimum spanning tree.

scholarly journals PROXIMITY DRAWINGS OF HIGH-DEGREE TREES

2013 ◽  
Vol 23 (03) ◽  
pp. 213-230
Author(s):  
FERRAN HURTADO ◽  
GIUSEPPE LIOTTA ◽  
DAVID R. WOOD
Keyword(s):  
Spanning Tree ◽  
Maximum Degree ◽  
Minimum Spanning Tree ◽  
Bounded Degree ◽  
Natural Properties ◽  
Vertex Set ◽  
High Degree ◽  
The Given

A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the partition or covering satisfies some natural properties, then there is a drawing of the entire tree such that each of the given subtrees is drawn as a minimum spanning tree of its vertex set.

GA-BASED ALTERNATIVE APPROACHES FOR THE DEGREE-CONSTRAINED SPANNING TREE PROBLEM

2009 ◽  
Vol 23 (01) ◽  
pp. 129-144
Author(s):  
GENGUI ZHOU ◽  
ZHENYU CAO ◽  
ZHIQING MENG ◽  
JIAN CAO
Keyword(s):  
Genetic Algorithm ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Practical Importance ◽  
Numerical Results ◽  
Tree Structure ◽  
Np Hard ◽  
Optimal Solutions ◽  
Alternative Approaches ◽  
Prüfer Number

The degree-constrained minimum spanning tree (dc-MST) problem is of high practical importance. Up to now there are few effective algorithms to solve this problem because of its NP-hard complexity. More recently, a genetic algorithm (GA) approach for this problem was tried by using Prüfer number to encode a spanning tree. The Prüfer number is a skillful encoding for tree but not efficient enough to deal with the dc-MST problem. In this paper, a new tree-based encoding is developed directly based on the tree structure. We denote it as tree-based permutation encoding and apply it to the dc-MST problem by using the GA approach. Compared with the numerical results and CPU runtimes between two encodings, the new tree-based permutation is effective to deal with the dc-MST problem and even more efficient than the Prüfer number to evolve to the optimal or near-optimal solutions.

scholarly journals Parallel strategies for a multi-criteria GRASP algorithm

Production ◽  
2007 ◽  
Vol 17 (1) ◽  
pp. 84-93 ◽  
Author(s):  
Dalessandro Soares Vianna ◽  
José Elias Claudio Arroyo ◽  
Pedro Sampaio Vieira ◽  
Thiago Ribeiro de Azeredo
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Efficient Solutions ◽  
Search Problem ◽  
Pareto Optimal ◽  
Complete Graphs ◽  
Pareto Optimal Solutions ◽  
Adaptive Search ◽  
Optimal Solutions

This paper proposes different strategies of parallelizing a multi-criteria GRASP (Greedy Randomized Adaptive Search Problem) algorithm. The parallel GRASP algorithm is applied to the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem, a vector of costs is defined for each edge of the graph and the goal is to find all the efficient or Pareto optimal spanning trees (Pareto-optimal solutions). Each process finds a subset of efficient solutions. These subsets are joined using different strategies to obtain the final set of efficient solutions. The multi-criteria GRASP algorithm with the different parallel strategies are tested on complete graphs with n = 20, 30 and 50 nodes and r = 2 and 3 criteria. The computational results show that the proposed parallel algorithms reduce the execution time and the results obtained by the sequential version were improved.

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