On the threshold of intractability

The computational complexity of the graph modification problems THRESHOLD EDITING and CHAIN EDITING has been an important open question in computational graph theory for more than 15 years. These problems consist of adding and deleting the fewest possible edges to transform a graph into a threshold or chain graph. We show that both problems are NP-hard, resolving a conjecture by Natanzon, Shamir, and Sharan from 2001. On the positive side, we show both problems admit quadratic vertex kernels and give subexponential time parameterized algorithms solving both problems in 2(O(root klogk)) + n(O(1)) time. Few natural problems are known to be in this complexity class. These results also extend to the completion and deletion variants of both problems, and threshold/chain graphs are the only known graph classes for which all three versions-completion, deletion, and editing-are both NP-complete and solvable in subexponential parameterized time. (C) 2021 Elsevier Inc. All rights reserved.