Inverse max \(+\) sum spanning tree problem under Hamming distance by modifying the sum-cost vector

  • Xiucui Guan
  • Xinyan He
  • Panos M. Pardalos
  • Binwu Zhang
Article

Abstract

The inverse max \(+\) sum spanning tree (MSST) problem is considered by modifying the sum-cost vector under the Hamming Distance. On an undirected network G(VEwc), a weight w(e) and a cost c(e) are prescribed for each edge \(e\in E\). The MSST problem is to find a spanning tree \(T^*\) which makes the combined weight \(\max _{e\in T}w(e)+\sum _{e\in T}c(e)\) as small as possible. It can be solved in \(O(m\log n)\) time, where \(m:=|E|\) and \(n:=|V|\). Whereas, in an inverse MSST problem, a given spanning tree \(T_0\) of G is not an optimal MSST. The sum-cost vector c is to be modified to \(\bar{c}\) so that \(T_0\) becomes an optimal MSST of the new network \(G(V,E,w,\bar{c})\) and the cost \(\Vert \bar{c}-c\Vert \) can be minimized under Hamming Distance. First, we present a mathematical model for the inverse MSST problem and a method to check the feasibility. Then, under the weighted bottleneck-type Hamming distance, we design a binary search algorithm whose time complexity is \(O(m log^2 n)\). Next, under the unit sum-type Hamming distance, which is also called \(l_0\) norm, we show that the inverse MSST problem (denoted by IMSST\(_0\)) is \(NP-\)hard. Assuming \({\textit{NP}} \nsubseteq {\textit{DTIME}}(m^{{\textit{poly}} \log m})\), the problem IMSST\(_0\) is not approximable within a factor of \(2^{\log ^{1-\varepsilon } m}\), for any \(\varepsilon >0\). Finally, We consider the augmented problem of IMSST\(_0\) (denoted by AIMSST\(_0\)), whose objective function is to multiply the \(l_0\) norm \(\Vert \beta \Vert _0\) by a sufficiently large number M plus the \(l_1\) norm \(\Vert \beta \Vert _1\). We show that the augmented problem and the \(l_1\) norm problem have the same Lagrange dual problems. Therefore, the \(l_1\) norm problem is the best convex relaxation (in terms of Lagrangian duality) of the augmented problem AIMSST\(_0\), which has the same optimal solution as that of the inverse problem IMSST\(_0\).

Keywords

Max \(+\) sum spanning tree problem Inverse optimization problem Hamming distance \(l_0\) norm Approximability 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Xiucui Guan
    • 1
  • Xinyan He
    • 2
  • Panos M. Pardalos
    • 3
    • 5
  • Binwu Zhang
    • 4
  1. 1.Department of MathematicsSoutheast UniversityNanjingChina
  2. 2.Zhenjiang High SchoolZhenjiangChina
  3. 3.Department of Industrial and Systems Engineering, Center for Applied OptimizationUniversity of FloridaGainesvilleUSA
  4. 4.Department of Mathematics and Physics, Changzhou CampusHohai UniversityChangzhouChina
  5. 5.LATNAHigher School of EconomicsMoscowRussia

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