Improved Algorithms for MST and Metric-TSP Interdiction

We consider the {\em MST-interdiction} problem: given a multigraph $G = (V, E)$, edge weights ${we\geq 0}{e \in E}$, interdiction costs ${ce\geq 0}{e \in E}$, and an interdiction budget $B\geq 0$, the goal is to remove a set $R\subseteq E$ of edges of total interdiction cost at most $B$ so as to maximize the $w$-weight of an MST of $G-R:=(V,E\setminus R)$. Our main result is a $4$-approximation algorithm for this problem. This improves upon the previous-best $14$-approximation~\cite{Zenklusen15}. Notably, our analysis is also significantly simpler and cleaner than the one in~\cite{Zenklusen15}. Whereas~\cite{Zenklusen15} uses a greedy algorithm with an involved analysis to extract a good interdiction set from an over-budget set, we utilize a generalization of knapsack called the {\em tree knapsack problem} that nicely captures the key combinatorial aspects of this "extraction problem." We prove a simple, yet strong, LP-relative approximation bound for tree knapsack, which leads to our improved guarantees for MST interdiction. Our algorithm and analysis are nearly tight, as we show that one cannot achieve an approximation ratio better than 3 relative to the upper bound used in our analysis (and the one in~\cite{Zenklusen15}). Our guarantee for MST-interdiction yields an $8$-approximation for {\em metric-TSP interdiction} (improving over the $28$-approximation in~\cite{Zenklusen15}). We also show that the {\em maximum-spanning-tree interdiction} problem is at least as hard to approximate as the minimization version of densest-$k$-subgraph.