Solving inverse spanning tree problems through network flow techniques
成果类型：
Article
署名作者：
Sokkalingam, PT; Ahuja, RK; Orlin, JB
署名单位：
Cisco Systems Inc; Cisco India; State University System of Florida; University of Florida; Massachusetts Institute of Technology (MIT)
刊物名称：
OPERATIONS RESEARCH
ISSN/ISSBN：
0030-364X
DOI：
10.1287/opre.47.2.291
发表日期：
1999
页码：
291-298
关键词：
摘要：
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, L-1 norm. We show that the inverse spanning tree problem can be formulated as the dual of an assignment problem on a bipartite network G(0) = (N-0, A(0)) with N-0 = N-1 boolean OR N-2 and A(0) subset of or equal to N-1 x N-2. The bipartite network satisfies the property that \N-1\ = (n - 1), \N-2\ = (m - n + 1), and \A(0)\ = O(nm). In general, \N-1\ much less than \N-2\. Using this special structure of the assignment problem, we develop a specific implementation of the successive shortest path algorithm that runs in O(n(3)) 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(n(2) mlog(nC)) time, where C denotes the largest are 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(n(2)) time.