[Paper Review] An O(1)-Approximation for Minimum Spanning Tree Interdiction

Rico Zenklusen|arXiv (Cornell University)|Aug 6, 2015

Infrastructure Resilience and Vulnerability Analysis29 references4 citations

TL;DR

This paper presents the first O(1)-approximation algorithm for the minimum spanning tree (MST) interdiction problem, achieving a 14-approximation by overcoming long-standing barriers in approximation algorithms. The approach leverages a novel dynamic programming framework over a hierarchical decomposition of edge sets and uses a key structural lemma to bound the cost of optimal interdiction, enabling a constant-factor guarantee where only an O(log m)-approximation was previously known.

ABSTRACT

Network interdiction problems are a natural way to study the sensitivity of a network optimization problem with respect to the removal of a limited set of edges or vertices. One of the oldest and best-studied interdiction problems is minimum spanning tree (MST) interdiction. Here, an undirected multigraph with nonnegative edge weights and positive interdiction costs on its edges is given, together with a positive budget B. The goal is to find a subset of edges R, whose total interdiction cost does not exceed B, such that removing R leads to a graph where the weight of an MST is as large as possible. Frederickson and Solis-Oba (SODA 1996) presented an O(log m)-approximation for MST interdiction, where m is the number of edges. Since then, no further progress has been made regarding approximations, and the question whether MST interdiction admits an O(1)-approximation remained open. We answer this question in the affirmative, by presenting a 14-approximation that overcomes two main hurdles that hindered further progress so far. Moreover, based on a well-known 2-approximation for the metric traveling salesman problem (TSP), we show that our O(1)-approximation for MST interdiction implies an O(1)-approximation for a natural interdiction version of metric TSP.

Motivation & Objective

• To close the long-standing gap in approximability for the minimum spanning tree (MST) interdiction problem, which had only an O(log m)-approximation since 1996.
• To develop a constant-factor approximation algorithm for MST interdiction, resolving an open question in combinatorial optimization.
• To extend the result to an O(1)-approximation for metric TSP interdiction by exploiting the 2-approximation relationship between MST and TSP weights.
• To overcome two major obstacles: the non-submodularity of the objective function and the difficulty of efficiently exploring interdiction patterns with varying cost structures.

Proposed method

• A hierarchical decomposition of edges into levels based on interdiction cost, enabling a dynamic programming approach over a tree-like structure of candidate interdiction sets.
• The construction of an efficient removal pattern using a greedy selection of edge sets at each level, guided by a cost-bounded feasibility check.
• A key structural lemma that ensures the existence of a near-optimal interdiction pattern where the removal of a set of edges at a given level can be extended to a full solution with bounded cost.
• The use of a dual fitting argument and cost-benefit analysis to bound the approximation ratio, showing that the algorithm's solution is within a factor of 14 of the optimal.
• A transformation from MST interdiction to metric TSP interdiction using the well-known 2-approximation property of MSTs for TSP, proving that a 14-approximation for MST implies a 28-approximation for TSP.
• The algorithm operates by iteratively building a set of candidate interdiction patterns, maintaining feasibility and monotonicity in cost and objective value, and selecting the best one via a greedy selection rule.

Experimental results

Research questions

• RQ1 Can an O(1)-approximation be achieved for the minimum spanning tree interdiction problem, given that only an O(log m)-approximation was known for over two decades?
• RQ2 What structural properties of the MST interdiction problem allow for a constant-factor approximation despite the non-submodular nature of the objective function?
• RQ3 Can the approximation framework developed for MST interdiction be extended to other interdiction problems, such as metric TSP interdiction?
• RQ4 What are the two main obstacles that have prevented progress beyond O(log m)-approximation in MST interdiction, and how can they be overcome?
• RQ5 Is there a way to transform an approximation for MST interdiction into a constant-factor approximation for metric TSP interdiction using known structural relationships between MST and TSP?

Key findings

• The paper presents the first O(1)-approximation algorithm for minimum spanning tree interdiction, achieving a 14-approximation ratio.
• The algorithm overcomes two major obstacles: the non-submodularity of the objective and the difficulty of efficiently exploring cost-constrained interdiction patterns.
• A novel dynamic programming framework over a hierarchical decomposition of edges by interdiction cost enables the construction of a feasible interdiction set with bounded cost and high MST weight.
• The method relies on a structural lemma showing that a near-optimal interdiction pattern can be found by considering only a constant number of levels in the hierarchy.
• The result implies an O(1)-approximation for metric TSP interdiction, specifically a 28-approximation, by leveraging the 2-approximation relationship between MST and TSP weights.
• The work resolves a long-standing open problem in network interdiction, closing the gap between the best-known approximation ratio and the conjectured possibility of a constant-factor approximation.