Reoptimization of Path Vertex Cover Problem

Most optimization problems are notoriously hard. Considerable efforts must be spent in obtaining an optimal solution to certain instances that we encounter in the real world scenarios. Often it turns out that input instances get modified locally in some small ways due to changes in the application world. The natural question here is, given an optimal solution for an old instance $I_O$, can we construct an optimal solution for the new instance $I_N$, where $I_N$ is the instance $I_O$ with some local modifications. Reoptimization of NP-hard optimization problem precisely addresses this concern. It turns out that for some reoptimization versions of the NP-hard problems, we may only hope to obtain an approximate solution to a new instance. In this paper, we specifically address the reoptimization of path vertex cover problem. The objective in $k$-$path$ vertex cover problem is to compute a minimum subset $S$ of the vertices in a graph $G$ such that after removal of $S$ from $G$ there is no path with $k$ vertices in the graph. We show that when a constant number of vertices are inserted, reoptimizing unweighted $k$-$path$ vertex cover problem admits a PTAS. For weighted $3$-$path$ vertex cover problem, we show that when a constant number of vertices are inserted, the reoptimization algorithm achieves an approximation factor of $1.5$, hence an improvement from known $2$-approximation algorithm for the optimization version. We provide reoptimization algorithm for weighted $k$-$path$ vertex cover problem $(k \geq 4)$ on bounded degree graphs, which is also an NP-hard problem. Given a $\rho$-approximation algorithm for $k$-$path$ vertex cover problem on bounded degree graphs, we show that it can be reoptimized within an approximation factor of $(2-\frac{1}{\rho})$ under constant number of vertex insertions.