Integer Programming, Constraint Programming, and Hybrid Decomposition Approaches to Discretizable Distance Geometry Problems

Abstract

Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in [Formula: see text] such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called discretization assumptions reduce the search space of the realization to a finite discrete one, which can be explored via the branch-and-prune (BP) algorithm. Given a discretization vertex order in G, the BP algorithm constructs a binary tree where the nodes at a layer provide all possible coordinates of the vertex corresponding to that layer. The focus of this paper is on finding optimal BP trees for a class of discretizable DGPs. More specifically, we aim to find a discretization vertex order in G that yields a BP tree with the least number of branches. We propose an integer programming formulation and three constraint programming formulations that all significantly outperform the state-of-the-art cutting-plane algorithm for this problem. Moreover, motivated by the difficulty in solving instances with a large and low-density input graph, we develop two hybrid decomposition algorithms, strengthened by a set of valid inequalities, which further improve the solvability of the problem. Summary of Contribution: We present a new model to solve a combinatorial optimization problem on graphs, MIN DOUBLE, which comes from the highly active area of distance geometry and has applications in a wide variety of fields. We use integer programming (IP) and present the first constraint programming (CP) models and hybrid decomposition methods, implemented as a branch-and-cut procedure, for MIN DOUBLE. Through an extensive computational study, we show that our approaches advance the state of the art for MIN DOUBLE. We accomplish this by not only combining generic techniques from IP and CP but also exploring the structure of the problem in developing valid inequalities and variable fixing rules. Our methods significantly improve the solvability of MIN DOUBLE, which we believe can also provide insights for tackling other problem classes and applications.

The Referenced Vertex Ordering Problem: Theory, Applications, and Solution Methods

“…The authors of [22] also established that, even for L = 1, the greedy search does not approximate the optimal value of this problem within a constant factor. Moreover, recent works dedicated to the exact solution of special cases of min revorder have also reported that several methods based on integer programming (IP), constraint programming and decomposition techniques were unable to deal with instances containing as few as 60 vertices within a reasonable computation time [21,16].…”

“…The benefit is evident in this context, since the implication is that there is no branching at layer v of the search tree. As a consequence, a particular case of min revorder where U = L + 1 has been considered recently in [16] and [21] with the aim of selecting the most promising discretization order. This optimization problem has been called min double, because two positions may have to be enumerated for partially-referenced vertices.…”

“…In the following, recall that the literature about the search of optimal discretization orders can be easily extended to min revorder. In this section, we focus on the IP formulations that were proposed for min double [21,16].…”

“…The authors then show that both formulations are outperformed by a constraint generation procedure where cycle constraints are iteratively added to the model. In [16], Bodur and MacNeil present a third compact formulation, vertexrank, where binary variables indicate whether a vertex v is at rank r in the order for all v ∈ V and all r ∈ [ [1, n]]. They also develop a decomposition scheme, witness, where they introduce the notion of witness, which can be defined as a necessary reference.…”

“…The decomposition yields a constraint generation algorithm that is also based on the separation of cycle constraints. One important difference between the models developed in [21] and in [16] is that the former enumerate all the feasible initial cliques and include one extra binary variable per clique, whereas the latter add constraints ensuring that the first vertices form a clique.…”