The linear ordering problem consists in finding a linear order $<$ on a finite set $A$ so as to minimize the sum of costs associated with pairs of elements $a, b$ for which $a < b$. The problem is NP-hard and APX-hard. We introduce algorithms for solving the problem *partially* by deciding efficiently for some pairs $(a,b)$ whether $a < b$ is in an optimal solution. To do so, we construct maps from the feasible set of orders to itself and establish efficiently testable conditions on the cost function of the problem for which these maps are improving. We examine the effectiveness and efficiency of these conditions and algorithms empirically, on two data sets.

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.