Fast Local Search Algorithm for Weighted Feedback Arc Set in Tournaments

We present a fast local search algorithm that finds an improved solution (if there is any) in the k-exchange neighborhood of the given solution to an instance of WEIGHTED FEEDBACK ARC SET IN TOURNAMENTS. More precisely, given an arc weighted tournament T on n vertices and a feedback arc set F (a set of arcs whose deletion from T turns it into a directed acyclic graph), our algorithm decides in time O(2n log n) if there is a feedback arc set of smaller weight and that differs from F in at most k arcs. To our knowledge this is the first algorithm searching the k-exchange neighborhood of an NP-complete problem that runs in (parameterized) subexponential time. Using this local search algorithm for WEIGHTED FEEDBACK ARC SET IN TOURNAMENTS, we obtain subexponential time algorithms for a local search variant of KEMENY RANKING – a problem in social choice theory and of ONE-SIDED CROSS MINIMIZATION – a problem in graph drawing.