Consensus Genetic Maps as Median Orders from Inconsistent Sources

A genetic map is an ordering of geneticmarkers calculated from a population of known lineage.While traditionally a map has been generated from a singlepopulation for each species, recently researchers have createdmaps from multiple populations. In the face of thesenew data, we address the need to find a consensus map — a map that combines the information from multiple partialand possibly inconsistent input maps. We model eachinput map as a partial order and formulate the consensusproblem as finding a median partial order. Finding themedian of multiple total orders (preferences or rankings)is a well studied problem in social choice. We choose tofind the median using the weighted symmetric differencedistance, a more general version of both the symmetricdifference distance and the Kemeny distance. Finding amedian order using this distance is NP-hard. We showthat for our chosen weight assignment, a median ordersatisfies the positive responsiveness, extended Condorcet,and unanimity criteria. Our solution involves finding themaximum acyclic subgraph of a weighted directed graph.We present a method that dynamically switches betweenan exact branch and bound algorithm and a heuristicalgorithm, and show that for real data from closely relatedorganisms, an exact median can often be found.We presentexperimental results using seven populations of the cropplant Zea mays.