Automated Search Tree Generation ( 2004 ; Gramm , Guo , Hüffner , Niedermeier ) Falk

This problem is concerned with the automated development and analysis of search tree algorithms. Search tree algorithms are a popular way to find optimal solutions to NP-complete problems. The idea is to recursively solve several smaller instances in such a way that at least one branch is a yes-instance iff the original instance is. Typically, this is done by trying all possibilities to contribute to a solution certificate for a small part of the input, yielding a small local modification of the instance in each branch. For example, consider the NP-complete Cluster Editing problem: can a graph be transformed by adding or deleting up to k edges into a cluster graph, that is, a disjoint union of cliques? To give a search tree algorithm for Cluster Editing, one can use the fact that cluster graphs are exactly the graphs that do not contain a P3 (a path of 3 vertices) as induced subgraph. One can thus solve Cluster Editing by finding a P3 and splitting into 3 branches: delete the first edge, delete the second edge, or add the missing edge. By the characterization, whenever no P3 is found, one already has a cluster graph. The original instance has a solution with k modifications iff at least one of the branches has a solution with k − 1 modifications.