Target Cuts from Relaxed Decision Diagrams

Abstract

The most common approach to generate cuts in integer programming is to derive them from the linear programming relaxation. We study an alternative approach that extracts cuts from discrete relaxations known as relaxed decision diagrams. Through a connection between decision diagrams and polarity, the algorithm generates cuts that are facet-defining for the convex hull of a decision diagram relaxation. As proof of concept, we provide computational evidence that this algorithm generates strong cuts for the maximum independent set problem and the minimum set covering problem.

Improving the filtering of Branch-And-Bound MDD solver (extended)

“…Being able to derive good lower and upper bounds for some optimization problem P is useful when the goal is to use these bounds to strengthen algorithms [13,32,33]. But it is not the only way these approximations can be used.…”

“…For instance, by using Lagrangian relaxation [23] or by solving a MIP [6] to derive with very tight bounds. But the other direction is also under active investigation: for example, [32,33] use DD to derive tight bounds which are used to replace LP relaxation in a cutting planes solver. Very recently, a third hybridization approach has been proposed by Gonzàlez et al [18].…”

A Combinatorial Cut-and-Lift Procedure with an Application to 0-1 Second-Order Conic Programming

“…We show that the set of cuts derived from this model defines the convex hull of the solutions encoded by the BDD, i.e., X. Moreover, in contrast to recent cutting-plane algorithms based on BDDs [28,51], our CGLP does not require any additional information about X, such as interior points or normalization constraints. Finally, for practical purposes, we build on this model to present a weaker but computationally faster alternative that solves a combinatorial max-flow/min-cut problem over the BDD to generate valid inequalities.…”

“…Behle (2007) [13] formalized this procedure and proposed a branch-and-cut algorithm that employs BDDs to generate exclusion and implication cuts. The author also introduced the network flow model employed by most BDD cutting-plane procedures [28,40,51].…”

“…Tjandraatmadja and van Hoeve (2019) [51] recently demonstrated how to generate target cuts from polar sets, using relaxed BDDs to obtain more computationally tractable procedures. Davarnia and van Hoeve (2020) [28] proposed an iterative method to generate outer-approximations for non-linear inequalities.…”