Recognizing When Heuristics Can Approximate Minimum Vertex Covers Is Complete for Parallel Access to NP

TL;DR

This paper investigates the complexity of recognizing graphs where certain heuristics approximate minimum vertex covers well, showing these recognition problems are complete for parallel NP access, and proving NP-hardness for specific graph classes.

Contribution

It establishes the parallel NP-completeness of recognizing graphs suitable for heuristic approximation and proves NP-hardness for graphs where heuristics find optimal solutions.

Findings

Recognition problems are complete for parallel access to NP.

Recognizing graphs where heuristics approximate within a factor is NP-complete.

Finding optimal solutions with heuristics remains NP-hard for certain graph classes.

Abstract

For both the edge deletion heuristic and the maximum-degree greedy heuristic, we study the problem of recognizing those graphs for which that heuristic can approximate the size of a minimum vertex cover within a constant factor of r, where r is a fixed rational number. Our main results are that these problems are complete for the class of problems solvable via parallel access to NP. To achieve these main results, we also show that the restriction of the vertex cover problem to those graphs for which either of these heuristics can find an optimal solution remains NP-hard.