Weighted Partition Vertex and Edge Cover

We study generalizations of the classical Vertex Cover and Edge Cover problems that incorporate group-wise coverage constraints. Our first focus is the \emph{Weighted Prize-Collecting Partition Vertex Cover} (WP-PVC) problem: given a graph with weights on both vertices and edges, and a partition of the edge set into $\omega$ groups, the goal is to select a minimum-weight subset of vertices such that, in each group, the total weight (profit) of covered edges meets a specified threshold. This formulation generalizes classical vertex cover, partial vertex cover and partition vertex cover. We present two algorithms for WP-PVC. The first is a simple 2-approximation that solves $n^{\omega}$ LP's, improving over prior work by Bandyapadhyay et al.\ by removing an enumerative step and the extra $\epsilon$-factor in approximation, while also extending to the weighted setting. The second is a bi-criteria algorithm that applies when $\omega$ is large, approximately meeting profit targets with a bounded LP-relative cost. We also study a natural generalization of the edge cover problem, the \emph{Weighted Partition Edge Cover} (W-PEC) problem, where each edge has an associated weights, and the vertex set is partitioned into groups. For each group, the goal is to cover at least a specified number of vertices using incident edges, while minimizing the total weight of the selected edges. We present the first exact polynomial-time algorithm for the weighted case, improving runtime from $O(\omega n^3)$ to $O(mn+n^2 \log n)$ and simplifying the algorithmic structure over prior unweighted approaches. We also show that the prize-collecting variant of the W-PEC problem is NP-Complete via a reduction from the knapsack problem.