Capacitated Partition Vertex Cover and Partition Edge Cover

Our first focus is the Capacitated Partition Vertex Cover (C-PVC) problem in hypergraphs. In C-PVC, we are given a hypergraph with capacities on its vertices and a partition of the hyperedge set into $ω$ distinct groups. The objective is to select a minimum size subset of vertices that satisfies two main conditions: (1) in each group, the total number of covered hyperedges meets a specified threshold, and (2) the number of hyperedges assigned to any vertex respects its capacity constraint. A covered hyperedge is required to be assigned to a selected vertex that belongs to the hyperedge. This formulation generalizes classical Vertex Cover, Partial Vertex Cover, and Partition Vertex Cover. We investigate two primary variants: soft capacitated (multiple copies of a vertex are allowed) and hard capacitated (each vertex can be chosen at most once). Let $f$ denote the rank of the hypergraph. Our main contributions are: $(i)$ an $(f+1)$-approximation algorithm for the weighted soft-capacitated C-PVC problem, which runs in $n^{O(ω)}$ time, and $(ii)$ an $(f+ε)$-approximation algorithm for the unweighted hard-capacitated C-PVC problem, which runs in $n^{O(ω/ε)}$ time. 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 weight, 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 $O(ωn^3)$ to $O(mn+n^2 \log n)$ and simplifying the algorithmic structure over prior unweighted approaches.

💡 Research Summary

This paper introduces and studies two novel combinatorial optimization problems that blend vertex/edge covering with group‑wise constraints and capacity limits. The first problem, Capacitated Partition Vertex Cover (C‑PVC), is defined on a hypergraph H = (V, E) where each vertex v has a weight w_v and a capacity k_v, and the hyperedges are partitioned into ω groups E₁,…,E_ω. For each group g a threshold ρ_g specifies the minimum number of hyperedges that must be covered. A feasible solution consists of an integer copy count x_v (soft capacities allow any non‑negative integer, hard capacities restrict x_v ∈ {0,1}) and an assignment φ that maps each covered hyperedge to a selected vertex belonging to that hyperedge, respecting the per‑vertex capacity (the number of assigned edges ≤ x_v·k_v). The objective is to minimize Σ_v w_v·x_v.

The second problem, Weighted Partition Edge Cover (W‑PEC), is defined on a simple graph G = (V, E) with edge weights p(e). The vertex set is partitioned into ω groups V₁,…,V_ω, and each group g requires at least ρ_g vertices to be incident to selected edges. The goal is to pick a minimum‑weight edge set satisfying all group requirements.

Main Results

1. Soft‑capacity C‑PVC: An (f + 1)‑approximation algorithm (where f is the hypergraph rank) running in n^{O(ω)} time. The algorithm first guesses the ω most expensive vertices of an optimal solution, solves a strengthened LP (LP‑M) that forces those vertices to be selected, and then performs a “responsibility assignment” that concentrates each hyperedge’s fractional coverage onto a single endpoint. This step inflates the cost by at most a factor f. The remaining fractional part is handled by a sparse LP (LP‑S) whose variables are limited to the ω groups; solving LP‑S yields a fractional vector α that is rounded without further loss. The total cost is bounded by (f + 1)·OPT.

2. Hard‑capacity C‑PVC: An (f + ε)‑approximation algorithm for any fixed ε > 0, running in n^{O(ω/ε)} time. The method adapts iterative rounding: at each iteration, variables with value at least 1 − ε are fixed to 1, and the LP is re‑solved. A new counting argument shows that after fixing, the number of remaining fractional variables is O(ω/ε), independent of f. The small residual instance is enumerated exhaustively, yielding the claimed approximation ratio.

3. Weighted Partition Edge Cover: An exact polynomial‑time algorithm running in O(mn + n² log n) time, improving upon the previous O(ω n³) algorithm for the unweighted case. The authors construct a min‑cost flow network: a source connects to every edge node with capacity 1, each edge node connects to its incident vertices, each vertex connects to the sink, and for each group g a dummy demand node D_g receives flow equal to ρ_g. Edge‑to‑vertex arcs have cost equal to the edge weight. Computing a minimum‑cost flow that satisfies all group demands directly yields an optimal edge set. The flow can be found with standard algorithms in O(mn) time; the additional O(n² log n) term comes from sorting vertices within each group. The approach naturally extends to arbitrary non‑negative edge weights, something the prior tropical‑matching based method could not handle.

Technical Contributions

• The paper unifies several classic covering problems (Vertex Cover, Partial Vertex Cover, Partition Vertex Cover, Capacitated Vertex Cover) into a single framework, showing that the added dimensions of capacity and partition constraints can be tackled simultaneously.
• The “responsibility assignment” technique generalizes the rounding step used in earlier capacitated vertex‑cover algorithms, but must handle hyperedges of size up to f, leading to the f‑factor loss.
• The sparse LP (LP‑S) isolates the effect of the guessed expensive vertices and reduces the fractional rounding problem to a small linear program whose size depends only on ω.
• The iterative‑rounding analysis for hard capacities introduces a novel counting argument that incorporates the ω partition constraints, thereby bounding the number of remaining fractional variables by a function of ω/ε rather than f.
• For W‑PEC, the flow‑based formulation eliminates the need for tropical matching and budgeted matching reductions, yielding a simpler and faster algorithm that also works for weighted instances.

Implications and Future Directions
The algorithms are particularly relevant for applications where resources (servers, routers, etc.) have limited capacity and service level agreements are expressed per‑region or per‑customer group. The parameterized dependence on ω suggests that the methods are practical when the number of groups is modest, which aligns with many real‑world scenarios (e.g., geographic zones, service tiers). Open questions include designing fixed‑parameter tractable (FPT) algorithms when ω is large, extending the techniques to online or stochastic settings, and exploring tighter approximation ratios for the hard‑capacity case that might eliminate the additive ε term. Overall, the paper makes substantial progress in bridging capacity‑aware covering with group‑wise requirements, offering both theoretical insights and algorithmic tools for a broad class of network design problems.