Complexity of Stability

TL;DR

This paper investigates the computational complexity of determining graph stability for key parameters like chromatic number, showing that this problem is complete for the complexity class ^p, indicating high computational difficulty.

Contribution

It introduces the study of graph stability for central parameters and proves the ^p-completeness of deciding stability, a novel complexity result in this area.

Findings

Stability decision problems are ^p-complete.

Graph stability for key parameters is computationally hard.

The results connect graph stability to the second level of the polynomial hierarchy.

Abstract

Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph (i.e., the smallest number of colors needed to color all vertices such that no two adjacent vertices are of the same color) can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable for them in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs for such parameters in terms of their computational complexity. We show that, for various central graph parameters, the problem of determining whether or not a given graph is stable is complete for \Theta_2^p, a well-known complexity class in the second level of the polynomial hierarchy, which is also known as "parallel access to NP."