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Maximum Shortest Path Interdiction Problem by Upgrading Nodes on Trees under Unit Cost

Qiao Zhang, Xiao Li, Xiucui Guan, Panos M. Pardalos

TL;DR

This work addresses the problem of maximizing the shortest root–leaf distance in a rooted tree by upgrading a limited set of nodes under a unit-cost budget, formalized as MSPIT-UN_u. It introduces a tailored dynamic-programming framework that exploits a left-subtree decomposition and a hierarchy of auxiliary functions to compute optimal upgrades in $O(n^3)$ time. It also extends to the related MCSPIT-UN_u via a binary-search strategy over the upgrade budget, achieving $O(n^3\log n)$ time by solving MSPIT-UN_u repeatedly. The approach is validated with numerical experiments on large tree instances, demonstrating scalability and the trade-offs between the max-interdiction and min-cost variants. Potential future directions include variable upgrade costs and extensions to more general graph classes, broadening applicability in network interdiction contexts.

Abstract

Network interdiction problems by deleting critical nodes have wide applications. However, node deletion is not always feasible in certain practical scenarios. We consider the maximum shortest path interdiction problem by upgrading nodes on trees under unit cost (MSPIT-UN$_u$). It aims to upgrade a subset of nodes to maximize the length of the shortest root-leaf distance such that the total upgrade cost under unit cost is upper bounded by a given value. We develop a dynamic programming algorithm with a time complexity of $O(n^3)$ to solve this problem. Furthermore, we consider the related minimum cost problem of (MSPIT-UN$_u$) and propose an $O(n^3\log n)$ binary search algorithm, where a dynamic programming algorithm is exceeded in each iteration to solve its corresponding problem (MSPIT-UN$_u$). Finally, we design numerical experiments to show the effectiveness of the algorithms.

Maximum Shortest Path Interdiction Problem by Upgrading Nodes on Trees under Unit Cost

TL;DR

This work addresses the problem of maximizing the shortest root–leaf distance in a rooted tree by upgrading a limited set of nodes under a unit-cost budget, formalized as MSPIT-UN_u. It introduces a tailored dynamic-programming framework that exploits a left-subtree decomposition and a hierarchy of auxiliary functions to compute optimal upgrades in time. It also extends to the related MCSPIT-UN_u via a binary-search strategy over the upgrade budget, achieving time by solving MSPIT-UN_u repeatedly. The approach is validated with numerical experiments on large tree instances, demonstrating scalability and the trade-offs between the max-interdiction and min-cost variants. Potential future directions include variable upgrade costs and extensions to more general graph classes, broadening applicability in network interdiction contexts.

Abstract

Network interdiction problems by deleting critical nodes have wide applications. However, node deletion is not always feasible in certain practical scenarios. We consider the maximum shortest path interdiction problem by upgrading nodes on trees under unit cost (MSPIT-UN). It aims to upgrade a subset of nodes to maximize the length of the shortest root-leaf distance such that the total upgrade cost under unit cost is upper bounded by a given value. We develop a dynamic programming algorithm with a time complexity of to solve this problem. Furthermore, we consider the related minimum cost problem of (MSPIT-UN) and propose an binary search algorithm, where a dynamic programming algorithm is exceeded in each iteration to solve its corresponding problem (MSPIT-UN). Finally, we design numerical experiments to show the effectiveness of the algorithms.

Paper Structure

This paper contains 7 sections, 2 theorems, 29 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Solve the problem (MSPIT-UN$_u$)
  3. Some important definitions
  4. Some auxiliary functions
  5. Slove the problem (MCSPIT-UN$_u$)
  6. Computational experiments
  7. An example to show Algorithm \ref{['alg4']}.

Key Result

Lemma 1

ZhangQ21 Suppose $w'$ is an optimal solution of the problem (eq-MSPIT-UNu). If there are two nodes $v_i,v_j$ on a same chain with $degree(v_i)=degree(v_j)=2$, $LN(A(v_i))=LN(A(v_j))$, $\Delta{w(A(v_i))}<\Delta{w(A(v_j))}$, $w'(A(v_i))>w(A(v_i))$, $w'(A(v_j))=w(A(v_j))$, then $w^*$ is an optimal solu

Figures (2)

  • Figure 1: The edge-weighted trees $T_{v_1}$ with cost $c(e)$ on an edge $e$. In the left tree, the Layers/layer numbers of the blue nodes/edges are 1, and the Layers/layer numbers of the green nodes/edges are 2. In the right tree, the green nodes are the critical descendant of node $v_1$, and the paths are stored in green and yellow nodes.
  • Figure 2: The subtree $T^{1:p}_v$ is shown. The areas labeled in red,blue,green are the subtrees $T_v^{1:1},T_v^{1:2}$ and $T^{1:q}_v$, respectively.

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Definition 5
  • Theorem 2
  • proof