On the Complexity of Edge Packing and Vertex Packing

This paper studies the computational complexity of the Edge Packing problem and the Vertex Packing problem. The edge packing problem (denoted by $\bar{EDS}$) and the vertex packing problem (denoted by $\bar{DS} $) are linear programming duals of the edge dominating set problem and the dominating set problem respectively. It is shown that these two problems are equivalent to the set packing problem with respect to hardness of approximation and parametric complexity. It follows that $\bar{EDS}$ and $\bar{DS}$ cannot be approximated asymptotically within a factor of $O(N^{1/2-\epsilon})$ for any $\epsilon>0$ unless $NP=ZPP$ where, $N$ is the number of vertices in the given graph. This is in contrast with the fact that the edge dominating set problem is 2-approximable where as the dominating set problem is known to have an $O(\log$ $|V|)$ approximation algorithm. It also follows from our proof that $\bar{EDS}$ and $\bar{DS}$ are $W[1]$-complete.

💡 Research Summary

The paper investigates two graph optimization problems that have received relatively little attention compared to their well‑studied duals: the Edge Packing problem (denoted (\bar{EDS})) and the Vertex Packing problem (denoted (\bar{DS})). Both problems arise as linear‑programming (LP) duals of the classic Edge Dominating Set (EDS) and Dominating Set (DS) problems. In (\bar{EDS}) one seeks a maximum set of edges such that no two chosen edges share an endpoint, while every unchosen edge must be adjacent to at least one chosen edge. In (\bar{DS}) one seeks a maximum independent set of vertices with the property that every vertex not in the set has a neighbor inside the set. The authors first formalize these definitions and then demonstrate a tight structural equivalence between (\bar{EDS}), (\bar{DS}) and the well‑known Set Packing problem.

The reduction proceeds by interpreting each edge (or vertex) of the input graph as an element of a family of sets, and encoding adjacency as a conflict relation. Consequently, a feasible packing in the graph corresponds exactly to a collection of pairwise non‑conflicting sets, i.e., a set packing. This transformation is polynomial‑time and size‑preserving, which allows the authors to transfer known hardness results for Set Packing directly to the two packing problems.

From the approximation‑complexity perspective, Set Packing is known to be hard to approximate within a factor of (O(N^{1/2-\epsilon})) for any (\epsilon>0) unless (NP = ZPP). By the aforementioned equivalence, the same lower bound applies to (\bar{EDS}) and (\bar{DS}). This result is striking because the primal problems behave very differently: EDS admits a constant‑factor (2‑approximation) algorithm, and DS has a logarithmic‑factor approximation algorithm ((O(\log|V|))). The paper argues that the additional “non‑adjacency” constraint inherent in the packing versions dramatically reduces the flexibility of approximation algorithms, making the dual problems substantially harder.

On the parameterized‑complexity side, the authors give a parameter‑preserving reduction from the (k)-Clique problem, which is (W