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Logical Methods in the Complexity Analysis of Graph Algorithms

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A classical observation in complexity theory is that many common algorithmic problems on graphs are hard to solve. Consequently, much work has gone into studying restricted classes of admissible inputs on which tractability results can be retained. A particular rich source of structural properties which guarantee the existence of efficient algorithms for many problems on graphs come from structural graph theory, especially algorithmic graph minor theory. It has been found that most generally hard problems become tractable on graph classes of bounded tree-width and many remain tractable on planar graphs or graph classes excluding a fixed minor.

Besides many specific results giving algorithms for individual problems, of particular interest are results that establish tractability of a large class of problems on specific classes of instances. These results come in various flavours. In this tutorial we will focus on results that take a descriptive approach, i.e. results that use a logic to describe algorithmic problems and then establish general tractability results for all problems definable in that logic on specific classes of inputs. These results are usually referred to as algorithmic meta-theorems.

In some sense the first theorem of this form is Courcelle’s famous result that all monadic second-order definable problems are solvable in linear time on all classes of structures of bounded tree-width. Subsequently many further theorems of this form and many tools for obtaining such results have been developed.

In this tutorial we will present the main methods and results obtained in this area. In the first part, the focus will be on logical tools that can be used to obtain algorithmic meta-theorems. In the second part of the tutorial we will focus on methods to establish corresponding lower bounds, i.e. intractability results for particular logics on classes of graphs that do not have any of the nice properties that make algorithmic meta-theorems possible. In particular, we will present a reasonably tight lower bound for Courcelle’s theorem mentioned above.

This talk is part of the Isaac Newton Institute Seminar Series series.

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