On the Threshold of Intractability
We study the computational complexity of the graph modification problems Open image in new window and Open image in new window , adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are Open image in new window -hard, resolving a conjecture by Natanzon, Shamir, and Sharan (2001). On the positive side, we show that these problems admit quadratic vertex kernels. Furthermore, we give a subexponential time parameterized algorithm solving Open image in new window in Open image in new window time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to Open image in new window , as well as the completion and deletion variants of both problems.
KeywordsChordal Graph Split Graph Discrete Apply Mathematic Chain Graph Threshold Graph
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