Inverse Spanning Tree Problem Research Articles

A Chance-Constrained Programming Model for Inverse Spanning Tree Problem with Uncertain Edge Weights

The inverse spanning tree problem is to make a given spanning tree be a minimum spanning tree on a connected graph via a minimum perturbation on its edge weights. In this paper, a chance-constrained programming model is proposed to handle the inverse spanning tree problem where the edge weights are assumed to be uncertain variables. It is shown that such an uncertain minimum spanning tree can be characterized by some constraints on the paths of the graph. Consequently, the proposed model can be reformulated into a deterministic programming model. Furthermore, when the edge weights are linear uncertain variables, the corresponding model reduces to a linear programming problem and can be solved efficiently.

The Inverse Spanning Tree of a Fuzzy Graph Based on Credibility Measure

An inverse spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree. In this paper, based on the notion of fuzzy α-minimum spanning tree, the inverse spanning tree problem with fuzzy edge weights is discussed and formulated as a fuzzy programming model with some chance constraints. It shows that when all the edge weights are assumed to be independent fuzzy variables with regular credibility distributions, the proposed model can be reformulated into a traditional nonlinear programming according to the equivalent condition of fuzzy α-minimum spanning tree characterized by a set of constraints on non-tree edges and their tree paths. Moreover, if all the fuzzy weights are triangular fuzzy numbers, a linear programming problem can be obtained and solved efficiently with the help of some well developed software packages.

Solving the Convex Cost Integer Dual Network Flow Problem

In this paper1, we consider an integer convex optimization problem where the objective function is the sum of separable convex functions (that is, of the form ∑(i,j)ϵQF̄ij(wij) + ∑iϵPB̄i(μi)), the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the form μi− μj≤ wij, (i,j)∈Q), with lower and upper bounds on variables. Letn= |P|,m= |Q| , andUbe the largest magnitude in the lower and upper bounds of variables. We call this problemthe convex cost integer dual network flow problem. In this paper, we describe several applications of the convex cost integer dual network flow problem arising in a dial-a-ride transit problem, inverse spanning tree problem, project management, and regression analysis. We develop network flow-based algorithms to solve the convex cost integer dual network flow problem. We show that using the Lagrangian relaxation technique, the convex cost integer dual network flow problem can be transformed into a convex cost primal network flow problem where each cost function is a piecewise linear convex function with integer slopes. Its special structure allows the convex cost primal network flow problem to be solved inO(nmlog(n 2/m)log(nU)time. This time bound is the same time needed to solve the minimum cost flow problem using the cost-scaling algorithm, and is also is best available time bound to solve the convex cost integer dual network flow problem.

Robust partial inverse network flow problems

In this paper, a new model for inverse network flow problems, robust partial inverse problem is presented. For a given partial solution, the robust partial inverse problem is to modify the coefficients optimally such that all full solutions containing the partial solution become optimal under new coefficients. It has been shown that the robust partial inverse spanning tree problem can be formulated as a combinatorial linear program, while the robust partial inverse minimum cut problem and the robust partial inverse assignment problem can be solved by combinatorial strongly polynomial algorithms.

A Faster Algorithm for the Inverse Spanning Tree Problem

In this paper, we consider the inverse spanning tree problem. Given an undirected graph G0=(N0,A0) with n nodes, m arcs, an arc cost vector c, and a spanning tree T0, the inverse spanning tree problem is to perturb the arc cost vector c to a vector d so that T0 is a minimum spanning tree with respect to the cost vector d and the cost of perturbation given by |d−c|=∑(i,j)∈A|dij−cij| is minimum. We show that the dual of the inverse spanning tree problem is a bipartite node weighted matching problem on a specially structured graph (which we call the path graph) that contains m nodes and as many as (m−n+1)(n−1)=O(nm) arcs. We first transform the bipartite node weighted matching problem into a specially structured minimum cost flow problem and use its special structure to develop an O(n3) algorithm. We next use its special structure more effectively and develop an O(n2logn) time algorithm. This improves the previous O(n3) time algorithm due to Sokkalingam et al. (1999, Oper. Res.47, 291–298).

Solving Inverse Spanning Tree Problems Through Network Flow Techniques

Given a solution x* and an a priori estimated cost vector c, the inverse optimization problem is to identify another cost vector d so that x* is optimal with respect to the cost vector d and its deviation from c is minimum. In this paper, we consider the inverse spanning tree problem on an undirected graph G = (N, A) with n nodes and m arcs, and where the deviation between c and d is defined by the rectilinear distance between the two vectors, that is, L1 norm. We show that the inverse spanning tree problem can be formulated as the dual of an assignment problem on a bipartite network G0 = (N0, A0) with N0 = N1 ∪ N2 and A0 ⊆ N1 × N2. The bipartite network satisfies the property that |N1| = (n − 1), |N2| = (m − n + 1), and |A0| = O(nm). In general, |N1| ≪ |N2|. Using this special structure of the assignment problem, we develop a specific implementation of the successive shortest path algorithm that runs in O(n3) time. We also consider the weighted version of the inverse spanning tree problem in which the objective function is to minimize the sum of the weighted deviations of arcs. The weighted inverse spanning tree can be formulated as the dual of the transportation problem. Using a cost scaling algorithm, this transportation problem can be solved in O(n2m log(nC)) time, where C denotes the largest arc cost in the data. Finally, we consider a minimax version of the inverse spanning tree problem and show that it can be solved in O(n2) time.