Contemporary Mathematics Methods for Algorithmic Meta Theorems Martin Grohe and Stephan Kreutzer Abstract. Algorithmic meta-theorems state that certain families of algorithmic problems, usually defined in terms of logic, can be solved efficiently. This is a survey of algorithmic meta-theorems, highlighting the general methods avail- able to prove such theorems rather than specific results. 1. Introduction Faced with the seeming intractability of many common algorithmic problems, much work has been devoted to studying restricted classes of admissible inputs on which tractability results can be retained. A particularly rich source of structural properties which guarantee the existence of efficient algorithms for many problems on graphs comes from structural graph theory, especially 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 par- ticular interest are results that establish tractability of a large class of problems on specific classes of instances. These results come in various flavours. Here we are mostly interested in results that take a descriptive approach, i.e. results that use a logic to describe algorithmic problems and then provide general tractability results for all problems definable in that logic on specific classes of inputs. Results of this form are usually referred to as algorithmic meta-theorems. The first explicit algorithmic meta-theorem was proved by Courcelle [3] establishing tractability of decision problems definable in monadic second-order logic (even with quantification over edge sets) on graph classes of bounded tree-width, followed by similar results for monadic second-order logic with only quantification over vertex sets on graph classes of bounded clique-width [4], for first-order logic on graph classes of bounded degree [45], on planar graphs and more generally graph classes of bounded local tree-width [23], on graph classes excluding a fixed minor [20], on graph classes locally excluding a minor [6] and graph classes of bounded local expansion [13]. The natural counterpart to any algorithmic meta-theorem establishing tract- ability for all problems definable in a given logic L on specific classes of structures are corresponding lower bounds, i.e. results establishing intractability results for L 1991 Mathematics Subject Classification. Primary 68Q19; Secondary 68Q25. c 0000 (copyright holder) 1