Double interdiction problem on trees on the sum of root-leaf distances by upgrading edges

TL;DR

This paper investigates the double interdiction problem on trees to maximize the sum of root-leaf distances through edge upgrades, establishing NP-hardness and proposing algorithms for specific cases and the general problem.

Contribution

It introduces the first complexity analysis of the problem, proves NP-hardness, and develops algorithms for different scenarios including greedy, dynamic programming, and binary search methods.

Findings

NP-hardness of the problem established

Efficient greedy algorithm for N=1 case

Pseudo-polynomial algorithms for general case

Abstract

The double interdiction problem on trees (DIT) for the sum of root-leaf distances (SRD) has significant implications in diverse areas such as transportation networks, military strategies, and counter-terrorism efforts. It aims to maximize the SRD by upgrading edge weights subject to two constraints. One gives an upper bound for the cost of upgrades under certain norm and the other specifies a lower bound for the shortest root-leaf distance (StRD). We utilize both weighted $l_\infty$ norm and Hamming distance to measure the upgrade cost and denote the corresponding (DIT) problem by (DIT$_{H\infty}$) and its minimum cost problem by (MCDIT$_{H\infty}$). We establish the $\mathcal{NP}$-hardness of problem (DIT$_{H\infty}$) by building a reduction from the 0-1 knapsack problem. We solve the problem (DIT$_{H\infty}$) by two scenarios based on the number $N$ of upgrade edges. When $N=1$, a greedy algorithm with $O(n)$ complexity is proposed. For the general case, an exact dynamic programming algorithm within a pseudo-polynomial time is proposed, which is established on a structure of left subtrees by maximizing a convex combination of the StRD and SRD. Furthermore, we confirm the $\mathcal{NP}$-hardness of problem (MCDIT$_{H\infty}$) by reducing from the 0-1 knapsack problem. To tackle problem (MCDIT$_{H\infty}$), a binary search algorithm with pseudo-polynomial time complexity is outlined, which iteratively solves problem (DIT$_{H\infty}$). We culminate our study with numerical experiments, showcasing effectiveness of the algorithm.