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.

scholarly journals Quantifying geographic range measures and their utility as extinction risk proxies

2017 ◽  
Author(s):  
James Boyle
Keyword(s):  
Convex Hull ◽  
Sample Size ◽  
Cell Count ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Extinction Risk ◽  
Geographic Range ◽  
Western Hemisphere ◽  
True Value ◽  
Latitudinal Range

Geographic range is used as a correlate of extinction risk for extant and extinct organisms across the fields of conservation and paleobiology. However, the exact method used to measure geographic range, the biases, and the limitations of each are rarely discussed explicitly despite their potential to impact conclusions. Here I examine and quantify properties of five commonly used measures of geographic range (convex hull area, maximum pairwise great circle distance, latitudinal range, longitudinal range, and cell count) along with a rarely used measure (minimum spanning tree distance) in the context of three datasets. A simulated dataset of two shapes with known areal limits, a paleontological occurrence dataset of pre-Cenozoic brachiopod genera from the Paleobiology Database (PBDB), and 50000 occurrence records of birds species in the western hemisphere from the eBird database. Simulated distributions indicate that when a distribution is simple (i.e. a rectangle) all measures are similarly accurate and precise at varying sample sizes and all measures converge toward the true value with increasing sample size. However, given a more complex distribution (i.e. horseshoe shape) the convex hull area becomes increasingly inaccurate as sample size increases even as it becomes more precise. Within the PBDB dataset the minimum spanning tree was heavily favored by Akaike Information Criterion as the most effective predictor of extinction risk. Analysis of the eBird data set indicated differences between IUCN Red List Categories were significant for all geographic range measures. Overall, these results suggest that for the purpose of categorical comparisons, such as those between victims and survivors of an extinction event, all six geographic range measures are interchangeable. However, the magnitude of the relationship between geographic range and extinction risk is strongly dependent on the choice of measure. Simple linear measures, such as latitudinal range, were relatively poor predictors while minimum spanning tree and cell count measures were strong predictors, especially after sampling was accounted for. The minimum spanning tree measure was found to perform at the same level or better than most other measures with the main drawback being that it is computationally expensive.

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.

scholarly journals Analysis and optimisation of looped water distribution networks

2000 ◽  
Vol 41 (4) ◽  
pp. 508-526 ◽  
Author(s):  
Brian Young
Keyword(s):  
Water Distribution ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Solution Space ◽  
Distribution Networks ◽  
Global Optimum ◽  
Water Distribution Networks ◽  
Pipe Flows ◽  
Non Linear ◽  
Feasible Solutions

AbstractA three stage procedure for the analysis and least-cost design of looped water distribution networks is considered in this paper. The first stage detects spanning trees and identifies the true global optimum for the system. The second stage determines hydraulically feasible pipe flows for the network by the numerical solution of a set of non-linear simultaneous equations and shows that these solutions are contained within closed convex polygonal regions in the solution space bounded by singularities resulting from zero flows in individual pipes. Ideal pipe diameters, consistent with the pipe flows and the constant velocity constraint adopted to prevent the system degenerating into a branched network, are selected and costed. It is found that the most favourable optimum is in the vicinity of a vertex in the solution space corresponding to the minimum spanning tree. In the third stage, commercial pipes are specified and the design finalised. Upper bound formulae for the number of spanning trees and hydraulically feasible solutions in a network have also been proposed. The treatment of large networks by a heuristic procedure is described which is shown to result in significant economies compared with designs obtained by non-linear programming.

scholarly journals Quantifying geographic range measures and their utility as extinction risk proxies

2017 ◽  
Author(s):  
James Boyle
Keyword(s):  
Convex Hull ◽  
Sample Size ◽  
Cell Count ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Extinction Risk ◽  
Geographic Range ◽  
Western Hemisphere ◽  
True Value ◽  
Latitudinal Range

Geographic range is used as a correlate of extinction risk for extant and extinct organisms across the fields of conservation and paleobiology. However, the exact method used to measure geographic range, the biases, and the limitations of each are rarely discussed explicitly despite their potential to impact conclusions. Here I examine and quantify properties of five commonly used measures of geographic range (convex hull area, maximum pairwise great circle distance, latitudinal range, longitudinal range, and cell count) along with a rarely used measure (minimum spanning tree distance) in the context of three datasets. A simulated dataset of two shapes with known areal limits, a paleontological occurrence dataset of pre-Cenozoic brachiopod genera from the Paleobiology Database (PBDB), and 50000 occurrence records of birds species in the western hemisphere from the eBird database. Simulated distributions indicate that when a distribution is simple (i.e. a rectangle) all measures are similarly accurate and precise at varying sample sizes and all measures converge toward the true value with increasing sample size. However, given a more complex distribution (i.e. horseshoe shape) the convex hull area becomes increasingly inaccurate as sample size increases even as it becomes more precise. Within the PBDB dataset the minimum spanning tree was heavily favored by Akaike Information Criterion as the most effective predictor of extinction risk. Analysis of the eBird data set indicated differences between IUCN Red List Categories were significant for all geographic range measures. Overall, these results suggest that for the purpose of categorical comparisons, such as those between victims and survivors of an extinction event, all six geographic range measures are interchangeable. However, the magnitude of the relationship between geographic range and extinction risk is strongly dependent on the choice of measure. Simple linear measures, such as latitudinal range, were relatively poor predictors while minimum spanning tree and cell count measures were strong predictors, especially after sampling was accounted for. The minimum spanning tree measure was found to perform at the same level or better than most other measures with the main drawback being that it is computationally expensive.

scholarly journals Towards Merging Binary Integer Programming Techniques with Genetic Algorithms

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Reza Zamani
Keyword(s):  
Genetic Algorithm ◽  
Integer Programming ◽  
Optimization Problems ◽  
Mathematical Formulation ◽  
Integer Program ◽  
Binary Integer Programming ◽  
Programming Technique ◽  
Combinatorial Optimization Problems ◽  
Programming Techniques ◽  
Feasible Solutions

This paper presents a framework based on merging a binary integer programming technique with a genetic algorithm. The framework uses both lower and upper bounds to make the employed mathematical formulation of a problem as tight as possible. For problems whose optimal solutions cannot be obtained, precision is traded with speed through substituting the integrality constrains in a binary integer program with a penalty. In this way, instead of constraining a variable u with binary restriction, u is considered as real number between 0 and 1, with the penalty of Mu(1-u), in which M is a large number. Values not near to the boundary extremes of 0 and 1 make the component of Mu(1-u) large and are expected to be avoided implicitly. The nonbinary values are then converted to priorities, and a genetic algorithm can use these priorities to fill its initial pool for producing feasible solutions. The presented framework can be applied to many combinatorial optimization problems. Here, a procedure based on this framework has been applied to a scheduling problem, and the results of computational experiments have been discussed, emphasizing the knowledge generated and inefficiencies to be circumvented with this framework in future.

Sign in / Sign up

Export Citation Format

Share Document