Complexity of stability

Stability of Special Graph Classes

“…Our work is based on the assumption that most of the real-world applications of stability of graph parameters do not use arbitrarily complex graphs but may often be restricted to certain special graph classes. Consequently, our studies show that-despite the completeness results by Frei et al [6]-there are tractable solutions to these problems when one limits the scope of the problem to a special graph class. We study seven different classes of special graphs: empty graphs consisting of only isolated vertices and no edges (I), complete graphs that have all possible edges (K), paths (P), trees (T ), forests (F ), bipartite graphs (B), and co-graphs (C).…”

“…Frei et al [6] comprehensively studied the problem of how stable certain central graph parameters are when a given graph is slightly modified, i.e., under operations such as adding or deleting either edges or vertices. Given a graph parameter ξ (like, e.g., the independence number or the chromatic number), they formally introduced the problems ξ-Stability, ξ-VertexStability, ξ-Unfrozenness, and ξ-VertexUnfrozenness and showed that they are, typically, Θ P 2 -complete, that is, they are complete for the complexity class known as "parallel access to NP," which was introduced by Papadimitriou and Zachos [18] and intensely studied by, e.g., Wagner [21,22], Hemaspaandra et al [8,10], and Rothe et al [20]; see the survey by Hemaspaandra et al [9].…”

“…Given a graph parameter ξ (like, e.g., the independence number or the chromatic number), they formally introduced the problems ξ-Stability, ξ-VertexStability, ξ-Unfrozenness, and ξ-VertexUnfrozenness and showed that they are, typically, Θ P 2 -complete, that is, they are complete for the complexity class known as "parallel access to NP," which was introduced by Papadimitriou and Zachos [18] and intensely studied by, e.g., Wagner [21,22], Hemaspaandra et al [8,10], and Rothe et al [20]; see the survey by Hemaspaandra et al [9]. 1 Furthermore, Frei et al [6] proved that some more specific versions of these problems, namely k-χ-Stability and k-χ-VertexStability, are NP-complete for k = 3 and DP-complete for k ≥ 4, respectively, where DP was introduced by Papadimitriou and Yannakakis [17] as the class of problems that can be written as the difference of NP problems.…”

“…Alternatively, Θ P 2 = P NP[O(log n)] can be viewed as the class of problems solvable in polynomial time via adaptively accessing its NP oracle (i.e., computing the next query depending on the answer to the previous query) logarithmically often (in the input size). Overall, the results of Frei et al [6] indicate that these problems are rather intractable and there exist no efficient algorithms solving them exactly. Considering the vast number of real-world applications for these problems mentioned by Frei et al [6]-e.g., the design of infrastructure, coloring algorithms for biological networks [15,13] or for complex information, social, and economic networks [12], etc.-these results are rather disappointing and unsatisfying.…”

“…Overall, the results of Frei et al [6] indicate that these problems are rather intractable and there exist no efficient algorithms solving them exactly. Considering the vast number of real-world applications for these problems mentioned by Frei et al [6]-e.g., the design of infrastructure, coloring algorithms for biological networks [15,13] or for complex information, social, and economic networks [12], etc.-these results are rather disappointing and unsatisfying.…”