Limits of Structural Tractability

The combination of methods from logic and graph theory has been extremely successful in the design of algorithms, in complexity theory, and other areas of theoretical computer science.

A success story exemplifying the power of this approach is the recent development in the algorithmic structure theory of sparse graphs.

In this line of research, structural results stemming from Robertson and Seymour’s graph minor theory, and the more recent sparsity theory of Nešetřil and Ossona de Mendez, were com- bined with logical methods in order to obtain a systematic understanding of tractability.

An example result in this area states that every graph property definable in first order logic can be decided in linear time, for all planar graphs.

Culminating a long line of research, Grohe, Kreutzer, and Siebertz gener- alized this result to all nowhere dense graph classes.

Those are very general classes of sparse graphs, which include the class of planar graphs, classes of bounded maximum degree, or classes excluding a fixed minor. Moreover, this result completely delimits the tractability frontier for sparse graph classes. However, many classes are tractable, but not sparse.

The recent twin-width theory, drawing on deep connections between logic and enumerative combinatorics, achieves an analogue of the result of Grohe et al. for all ordered graphs. Thus, algorithmic tractability is now understood in two contexts: of sparse graphs, and of ordered graphs. This project sets out to characterize all tractable graph classes.

This requires developing a systematic understanding of the logical structure underlying algorithmic tractability.

The tools I intend to apply and develop originate from graph structure theory, and from stability theory, one of the most successful areas in logic recently.

The expected results will be of foundational nature, and of interest primarily to theoretical computer scientists, graph theorists, and logicians.