Structure of conflict graphs in constrained alignment problems and algorithms

We consider the constrained graph alignment problem which has applications in biological network analysis. Given two input graphs G(1) = (V-1, E-1), G(2) = (V-2, E-2), two vertices u(1), v(1) of G(1) paired respectively to two vertices u(2), v(2 )of G(2) induce an edge conservation if u(1), v(1) and u(2), v(2) are adjacent in their respective graphs. The goal is to provide a one-to-one mapping between some vertices of the input graphs in order to maximize edge conservation. However the allowed mappings are restricted since each vertex from V-1 (resp. V-2) is allowed to be mapped to at most m(1) (resp. m(2)) specified vertices in V-2 (resp. V-1). Most of the results in this paper deal with the case m(2) = 1 which attracted most attention in the related literature. We formulate the problem as a maximum independent set problem in a related conflict graph and investigate structural properties of this graph in terms of forbidden subgraphs. We are interested, in particular, in excluding certain wheels, fans, cliques or claws (all terms are defined in the paper), which in turn corresponds to excluding certain cycles, paths, cliques or independent sets in the neighborhood of each vertex. Then, we investigate algorithmic consequences of some of these properties, which illustrates the potential of this approach and raises new horizons for further works. In particular this approach allows us to reinterpret a known polynomial case in terms of conflict graph and to improve known approximation and fixed-parameter tractability results through efficiently solving the maximum independent set problem in conflict graphs. Some of our new approximation results involve approximation ratios that are functions of the optimal value, in particular its square root; this kind of results cannot be achieved for maximum independent set in general graphs.