Abstract
Graph parameters such as the clique number and the chromatic 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 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 in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs in terms of their computational complexity. We show for various central graph parameters that deciding the stability of a given graph is complete for Θ2p, a well-known complexity class in the second level of the polynomial hierarchy.
Highlights
In this first section, we motivate our research topic, introduce the necessary notions and notation, and provide an overview of both the related work and our contribution
We initiate a systematic study of stability of graphs in terms of their computational complexity and present some tools to stabilize specific parts of a graph
The class 2, whose name is due to Wagner [41], belongs to the second level of the polynomial hierarchy; it can be defined as 2 = PNP[O(log n)], which is the class of problems that can be solved in polynomial time by an algorithm with access to an oracle that decides arbitrary instances for an NP-complete problem—with one instance per call and each such query taking constant time—restricted to a logarithmic number of queries. (Without the last restriction, we would get the p p p class 2 = PNP .) Results due to Hemachandra [23, Theorem 4.10] usefully characterize 2 as Ptt, the class of languages that are polynomial-time truth-table reducible to NP
Summary
Stated, a graph is stable with respect to some graph parameter (such as the chromatic number) if some type of small perturbation of the graph (a local modification such as adding an edge or deleting a vertex) does not change the parameter. Other graph parameters we consider are the clique number, the independence number, and the vertex cover number. This notion of stability formalizes the robustness of graphs for these parameters, which is important in many applications. In various applied areas of computer science, graph coloring has been used for register allocation in compiler optimization [7], pattern matching and pattern mining [39], and scheduling tasks [31]. To ensure that these applications of graph parameters are robust, graphs need to be stable for them with respect to certain operations. We initiate a systematic study of stability of graphs in terms of their computational complexity and present some tools to stabilize specific parts of a graph
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