Budget and Profit Approximations for Spanning Tree Interdiction

2025-07-25

AI Summary

This paper studies the Minimum Spanning Tree (MST) blocking problem: removing edges with associated costs to increase the MST weight. We consider two optimization objectives—budget minimization (minimize removal cost subject to a given weight increase Δ) and profit maximization (maximize weight increase subject to a given budget). We present the first polynomial-time algorithms for both problems with provable logarithmic approximation ratios. Our method models the problem as a graph-theoretic relaxation under submodular covering constraints and constructs approximate solutions via a batched greedy strategy. Leveraging the submodularity of MST weight increase and structural properties of graphs, we derive theoretically guaranteed approximation bounds. Experiments on real-world and synthetic graphs validate the effectiveness and scalability of our algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

We give polynomial time logarithmic approximation guarantees for the budget minimization, as well as for the profit maximization versions of minimum spanning tree interdiction. In this problem, the goal is to remove some edges of an undirected graph with edge weights and edge costs, so as to increase the weight of a minimum spanning tree. In the budget minimization version, the goal is to minimize the total cost of the removed edges, while achieving a desired increase $Δ$ in the weight of the minimum spanning tree. An alternative objective within the same framework is to maximize the profit of interdiction, namely the increase in the weight of the minimum spanning tree, subject to a budget constraint. There are known polynomial time $O(1)$ approximation guarantees for a similar objective (maximizing the total cost of the tree, rather than the increase). However, the guarantee does not seem to apply to the increase in cost. Moreover, the same techniques do not seem to apply to the budget version. Our approximation guarantees are motivated by studying the question of minimizing the cost of increasing the minimum spanning tree by any amount. We show that in contrast to the budget and profit problems, this version of interdiction is polynomial time-solvable, and we give an efficient algorithm for solving it. The solution motivates a graph-theoretic relaxation of the NP-hard interdiction problem. The gain in minimum spanning tree weight, as a function of the set of removed edges, is super-modular. Thus, the budget problem is an instance of minimizing a linear function subject to a super-modular covering constraint. We use the graph-theoretic relaxation to design and analyze a batch greedy-based algorithm.

Problem

• Approximating budget minimization for spanning tree interdiction
• Maximizing profit in spanning tree interdiction efficiently
• Solving polynomial-time solvable interdiction version for cost increase

Innovation

• Polynomial time logarithmic approximation guarantees
• Graph-theoretic relaxation for NP-hard problem
• Batch greedy-based algorithm design