# Complexity Dichotomies for the Minimum F-Overlay Problem

## TL;DR

This paper establishes a clear dichotomy between polynomial-time solvable and NP-hard cases for the Minimum F-Overlay problem, which generalizes network design and biological assembly problems, based on properties of the family F.

## Contribution

It provides a comprehensive complexity classification (dichotomy) for the Minimum F-Overlay problem and analyzes its parameterized complexity with respect to different families F.

## Key findings

- Easy cases are polynomial; complex cases are NP-complete.
- Characterization of F leading to W[1]-hard, W[2]-hard, or FPT parameterized problems.
- FPT/W[1]-hard dichotomy for a relaxed version of the problem.

## Abstract

For a (possibly infinite) fixed family of graphs F, we say that a graph G overlays F on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of F as a spanning subgraph.While it is easy to see that the complete graph on |V(H)| overlays F on a hypergraph H whenever the problem admits a solution, the Minimum F-Overlay problem asks for such a graph with the minimum number of edges.This problem allows to generalize some natural problems which may arise in practice. For instance, if the family F contains all connected graphs, then Minimum F-Overlay corresponds to the Minimum Connectivity Inference problem (also known as Subset Interconnection Design problem) introduced for the low-resolution reconstruction of macro-molecular assembly in structural biology, or for the design of networks.Our main contribution is a strong dichotomy result regarding the polynomial vs. NP-hard status with respect to the considered family F. Roughly speaking, we show that the easy cases one can think of (e.g. when edgeless graphs of the right sizes are in F, or if F contains only cliques) are the only families giving rise to a polynomial problem: all others are NP-complete.We then investigate the parameterized complexity of the problem and give similar sufficient conditions on F that give rise to W[1]-hard, W[2]-hard or FPT problems when the parameter is the size of the solution.This yields an FPT/W[1]-hard dichotomy for a relaxed problem, where every hyperedge of H must contain some member of F as a (non necessarily spanning) subgraph.