Kernelization, Permutation CSPs Parameterized above Average

Let r be an integer and let V be a set of n variables. An ordering is a bijection from V to f1; 2; : : : ; ng; a constraint is an ordered r-tuple .v1; v2; : : : ; vr/ of distinct variables of V; satisfies .v1; v2; : : : ; vr/ if .v1/ < .v2/ < < .vr/: An instance of MAX-r-LIN-ORDERING consists of a multiset C of constraints, and the objective is to find an ordering that satisfies the maximum number of constraints. Note that MAX-2-LIN ORDERING is equivalent to the problem of finding a maximum weight acyclic subgraph in an integer-weighted directed graph. Since the FEEDBACK ARC SET problem is NP-hard, MAX-2-LIN ORDERING is NP-hard, and thus MAXr-LIN-ORDERING is NP-hard for each r 2. Let be an ordering chosen randomly and uniformly from all orderings and let c 2 C be a constraint. Then the probability that satisfies c is 1=rS. Thus the expected number of constraints in C satisfied by equals jCj=rS. This is a lower bound on the maximum number of constraints satisfied by an ordering, and, in fact, it is a tight lower bound. This allows us to consider the following parameterized problem (AA stands for Above Average).