We define a graph structure over predicates that represents entailment relations as directed edges, and use a global transitivity constraint on the graph to learn the optimal set of edges, by formulating the optimization problem as an Integer Linear Program .

In this paper we tackle a similar problem of learning a transitive relation, but we use linear programming .

A Linear Program (LP) is an optimization problem, where a linear function is minimized (or maximized) under linear constraints.

variables are integers, the problem is termed an Integer Linear Program (ILP).

The optimization problem is formulated as an Integer Linear Program (ILP) and solved with an ILP solver.

Since we seek a global solution under transitivity and other constraints, linear programming is a natural choice, enabling the use of state of the art optimization packages.

We describe two formulations of integer linear programs that learn the edges: one maximizing a global score function, and another maximizing a global probability function.

Since the variables are binary, both formulations are integer linear programs with O(|V|2) variables and O(|V|3) transitivity constraints that can be solved using standard ILP packages.

Using an integer linear programming formulation, the model learns to select and combine phrases subject to length, coverage and grammar constraints.

Grammaticality, length and coverage requirements are encoded as constraints in an integer linear program .

We encode these constraints through the use of integer linear programming (ILP), a well-studied optimization framework that is able to search the entire solution space efficiently.