On the Complexity of Edge Packing and Vertex Packing

TL;DR

This paper analyzes the computational complexity of edge and vertex packing problems, showing they are as hard to approximate as set packing and are W[1]-complete, contrasting with related problems that have better approximation algorithms.

Contribution

It establishes the hardness and W[1]-completeness of the edge and vertex packing problems, revealing their equivalence to set packing in complexity.

Findings

Edge and vertex packing are as hard as set packing in approximation.

These problems are W[1]-complete.

They cannot be approximated within certain bounds unless NP=ZPP.

Abstract

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.