Final Report Summary - PARAMTIGHT (Parameterized complexity and the search for tight complexity results)
The goal of the project is to use the framework of parameterized complexity for obtaining tight understanding of the algorithmic complexity of various combinatorial problems. By designing new algorithms and providing complexity-theoretic lower bounds, we determined the best possible running time that can be achieved for various concrete problems and discovered the maximum generality in which a problem can be solved efficiently. The project has achieved substantial progress in understanding hard combinatorial problems on planar networks by designing significantly improved algorithms, together with matching complexity lower bounds proving their optimality. Another highlight of the project is obtaining tight characterization results for the fundamental problems of finding and counting small subgraph patterns in large graphs, discovering all the tractable special cases of the problems, thereby determining the maximum generality in which these problems can be solved efficiently.