5 research outputs found
Recognizing When Heuristics Can Approximate Minimum Vertex Covers Is Complete for Parallel Access to NP
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.Comment: 16 pages, 2 figure
The Complexity of Computing Minimal Unidirectional Covering Sets
Given a binary dominance relation on a set of alternatives, a common thread
in the social sciences is to identify subsets of alternatives that satisfy
certain notions of stability. Examples can be found in areas as diverse as
voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08]
proved that it is NP-hard to decide whether an alternative is contained in some
inclusion-minimal upward or downward covering set. For both problems, we raise
this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and
provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other
natural problems regarding minimal or minimum-size covering sets are hard or
complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of
our results is that neither minimal upward nor minimal downward covering sets
(even when guaranteed to exist) can be computed in polynomial time unless P=NP.
This sharply contrasts with Brandt and Fischer's result that minimal
bidirectional covering sets (i.e., sets that are both minimal upward and
minimal downward covering sets) are polynomial-time computable.Comment: 27 pages, 7 figure