Being even slightly shallow makes life hard

TL;DR

This paper investigates the computational difficulty of finding dense substructures in graphs, showing that even slight relaxations of the problem are NP-hard and unlikely to be efficiently solvable with current algorithms.

Contribution

It proves NP-hardness and tight complexity bounds for detecting dense shallow topological minors and subdivisions, even in sparse graphs and for small relaxations.

Findings

NP-hardness for r=1 in sparse graphs

No subexponential algorithms parameterized by treewidth for r≥2

Complexity bounds hold unless ETH fails

Abstract

We study the computational complexity of identifying dense substructures, namely $r/2$-shallow topological minors and $r$-subdivisions. Of particular interest is the case when $r=1$, when these substructures correspond to very localized relaxations of subgraphs. Since Densest Subgraph can be solved in polynomial time, we ask whether these slight relaxations also admit efficient algorithms. In the following, we provide a negative answer: Dense $r/2$-Shallow Topological Minor and Dense $r$-Subdivsion are already NP-hard for $r = 1$ in very sparse graphs. Further, they do not admit algorithms with running time $2^{o(\mathbf{tw}^2)} n^{O(1)}$ when parameterized by the treewidth of the input graph for $r \geq 2$ unless ETH fails.