Exact and approximation algorithms for graph problems with some biological applications

In this thesis we study several combinatorial problems in algorithmic graph theory and computational biology, and different algorithmical approaches for solving them.
In particular, we focus on graph algorithms, seeking for the most part polynomial or sub-exponential exact solutions, but in some cases also approximate solutions.
In the first part we study two problems on phylogenetic trees, the problem of finding a consensus tree for a set of phylogenetic trees, and the problem of combining a set of not necessarily consistent three-taxon trees into a phylogenetic tree that represents the input trees as closely as possible.
The second part is devoted to the problem of the embedding of graphs into integer lattices. In particular, we extend the problem to integer lattices of higher dimensions. We present a subexponential-time method for optimal embedding of an arbitrary graph into a d-dimensional integer lattice. In particular, it yields a subexponential-time algorithm for optimal protein folding in the HP-model.
Next, we study the problem of efficiently finding shortest cycles, and present approximation algorithms for both the weighted and unweighted variants of this problem.
Finally, we consider the approximability of the maximum and minimum edge clique partition problems, giving upper and lower bounds, and present approximation algorithms for both variants.