Complexity in databases, games, and logics

Databases are ubiquitous in our society. This dissertation presents a complexity analysis of two important problems related to databases. First, we show the complexity of establishing strong k-consistency, which is a common heuristic for database query evaluation. Second, we classify the complexity of the existence-of-solutions problem for data exchange. The first part of this dissertation presents a computational complexity analysis of the problem of determining the winner of the existential k-pebble game. Existential k-pebbles were originally introduced to aid in the analysis of the expressive power of Datalog and related infinitary logics with finitely many variables. It has since been observed that strong k-consistency can be established for an instance of constraint satisfaction if and only if the Duplicator has a winning strategy for the existential k-pebble game on structures related to the given instance of constraint satisfaction. The main result is that determining the winner of the existential k-pebble game is EXPTIME-complete when k is part of the input. This implies that establishing strong k-consistency is inherently exponential. We then use this result to prove complexity results for a collection of related games. The second part of this dissertation addresses the combined complexity of the existence-of-solutions problem for data exchange. Data exchange is the problem of transforming data from a source schema to a target schema so that all constraints of a schema mapping are satisfied. The existence-of-solutions problem is the question: given a source instance, is there a target instance that satisfies the constraints of the schema mapping? Earlier work showed that for schema mappings specified by embedded implicational dependencies, this problem is solvable in polynomial time, if the schema mapping is fixed and the constraints of the schema mapping satisfy a certain structural condition, called weak acyclicity. We show that if the schema mapping is also part of the input, then the complexity of the existence-of-solutions problem rises to 2EXPTIME. We also show that there are restricted classes of inputs to the existence-of-solutions problem for which the problem can be EXPTIME-complete or CoNP-complete.