On the Threshold of Intractability

TL;DR

This paper proves that Threshold and Chain Editing problems are NP-complete, but also provides a quadratic kernel and a subexponential parameterized algorithm, advancing understanding of their computational complexity.

Contribution

The paper establishes NP-completeness for Threshold and Chain Editing, and introduces a quadratic kernel and a subexponential algorithm for these problems.

Findings

Both problems are NP-complete.

Existence of a quadratic vertex kernel.

Subexponential time parameterized algorithm.

Abstract

We study the computational complexity of the graph modification problems Threshold Editing and Chain Editing, 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 NP-complete, resolving a conjecture by Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128, 2001). On the positive side, we show the problem admits a quadratic vertex kernel. Furthermore, we give a subexponential time parameterized algorithm solving Threshold Editing in $2^{O(\surd k \log k)} + \text{poly}(n)$ 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 (ISAAC, 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 the related problem of Chain Editing, as well as the completion and deletion variants of both problems.