Complexity of the Exact Domatic Number Problem and of the Exact Conveyor Flow Shop Problem

TL;DR

This paper establishes the computational complexity of exactly determining the domatic number and related problems, showing they are complete for specific levels of the boolean hierarchy over NP, thus highlighting their inherent difficulty.

Contribution

It proves the exact domatic number problem and related problems are complete for the boolean hierarchy levels over NP, extending understanding of their computational hardness.

Findings

Exact domatic number problem is complete for the 2k-th level of the boolean hierarchy.

Deciding if the domatic number equals a specific value is DP-complete for k=1.

Reductions apply Wagner's conditions to establish hardness levels.

Abstract

We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is {\em exactly} one of k given values is complete for the 2k-th level of the boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not the domatic number of a given graph equals exactly a given integer. Note that DP is the second level of the boolean hierarchy over NP. We obtain similar results for the exact versions of generalized dominating set problems and of the conveyor flow shop problem. Our reductions apply Wagner's conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP.