Making de Bruijn Graphs Eulerian

International audienceA directed multigraph is called Eulerian if it has a circuit which uses each edge exactly once. Euler's theorem tells us that a weakly connected directed multigraph is Eulerian if and only if every node is balanced. Given a collection S of strings over an alphabet Σ, the de Bruijn graph (dBG) of order k of S is a directed multigraph G S,k (V, E), where V is the set of length-(k − 1) substrings of the strings in S, and G S,k contains an edge (u, v) with multiplicity mu,v, if and only if the string u[0] • v is equal to the string u • v[k − 2] and this string occurs exactly mu,v times in total in strings in S. Let G Σ,k (V Σ,k , E Σ,k) be the complete dBG of Σ k. The Eulerian Extension (EE) problem on G S,k asks to extend G S,k with a set B of nodes from V Σ,k and a smallest multiset A of edges from E Σ,k to make it Eulerian. Note that extending dBGs is algorithmically much more challenging than extending general directed multigraphs because some edges in dBGs are by definition forbidden. Extending dBGs lies at the heart of sequence assembly [Medvedev et al., WABI 2007], one of the most important tasks in bioinformatics. The novelty of our work with respect to existing works is that we allow not only to duplicate existing edges of G S,k but to also add novel edges and nodes, in an effort to (i) connect multiple components and (ii) reduce the total EE cost. It is easy to show that EE on G S,k is NP-hard via a reduction from shortest common superstring. We further show that EE remains NP-hard, even when we are not allowed to add new nodes, via a highly non-trivial reduction from 3-SAT. We thus investigate the following two problems underlying EE in dBGs