From the point of view of algorithm design and complexity theory, it is of primary importance to understand the computational complexity of a given task. For example, computing shortest path between two given points can be done efficiently (as witnessed by navigation apps such as Google Maps), while finding a shortest sightseeing tour through a few dozen of points of interest (in unspecified order) can be computationally challenging as the problem is NP-hard.
The project focuses on the interplay of graph theory and algorithm design: which graph-theoretic developments can help in designing efficient algorithms and in which scenarios this is possible.
One of the main directions in the grant is a systematic study of complexity of basic computational problems, such as Maximum Independent Set, in various graph classes, that is, when the input of the problem is restricted to have some particular properties. Surprisingly, there is a wide range of graph classes for which the complexity status of these basic problems is open. The graph classes in question largely coincide with the graph classes on the front of many questions in structural and extremal graph theory, such as the Erdos-Hajnal conjecture or the theory of chi-boundedness. This suggested that these two areas may benefit from an exchange of ideas and techniques, and this is exactly what was witnessed in the project.
In other main direction of the grant, we predicted applicability of the recent toolbox for decomposing graphs into smaller, simpler pieces, for the Graph Isomorphism problem in sparse graphs. This classic problem, asking whether two given graphs are isomorphic, is not known to be efficiently (i.e. polynomial-time) solvable in theory, but is efficiently solvable in practice by heuristic methods.
A third main direction of the grant considered algorithms for graph separation problems in directed graphs. As an example, consider the st-Cut problem: given a directed graph with a designated source and sink vertices, delete as few arcs as possible so that no path from the source to the sink remains in the graph. This classic problem is known to be efficiently solvable even in the weighted setting (i.e. arcs have weights and we want to delete arcs of minimum total weight), but becomes hard if one restricts both the cardinality and total weight of the cut (so-called bicriteria setting). This problem, and many other related problems, lacked fixed-pameter algorithm: an algorithm that is efficient if the cardinality of the cut in question is small. We conjectured that a algorithmic or graph theory technique is missing that will provide a uniform reason for tractability.
In all three aforementioned main directions, we have obtained major progress during the grant, fulfilling all or almost all objectives. Furthermore, in the first mentioned direction we have went far beyond the original plan of work.
