LEARNING TO GENERATE COLUMNS WITH APPLICATION TO VERTEX COLORING

Abstract

We present a new column generation approach based on Machine Learning (ML) for solving combinatorial optimization problems. The aim of our method is to generate high-quality columns that belong to an optimal integer solution, in contrast to the traditional approach that aims at solving linear programming relaxations. To achieve this aim, we design novel features to characterize a column, and develop an effective ML model to predict whether a column belongs to an optimal integer solution. We then use the ML model as a filter to select high-quality columns generated from a sampling method and use the selected columns to construct an integer solution. Our method is computationally fast compared to the traditional methods that generate columns by repeatedly solving a pricing problem. We demonstrate the efficacy of our method on the vertex coloring problem, by empirically showing that the columns selected by our ML model are significantly better, in terms of the integer solution that can be constructed from them, than those selected randomly or based only on their reduced cost. Further, we show that the columns generated by our method can be used as a warm start to boost the performance of a column generation-based heuristic.

1. INTRODUCTION

Machine Learning (ML) has been increasingly used to tackle combinatorial optimization problems (Bengio et al., 2020) , such as learning to branch in Mixed Integer Program (MIP) solvers (Khalil et al., 2016; Balcan et al., 2018; Gupta et al., 2020) , learning heuristic search algorithms (Dai et al., 2017; Li et al., 2018) , and learning to prune the search space of optimization problems (Lauri & Dutta, 2019; Sun et al., 2021b; Hao et al., 2020) . Given a large amount of historical data to learn from, ML techniques can often outperform random approaches and hand-crafted methods typically used in the existing exact and heuristic algorithms. Predicting optimal solutions for combinatorial optimization problems via ML has attracted much attention recently. A series of studies (Li et al., 2018; Lauri & Dutta, 2019; Sun et al., 2021b; Grassia et al., 2019; Fischetti & Fraccaro, 2019; Lauri et al., 2020; Sun et al., 2021a; 2022; Ding et al., 2020; Abbasi et al., 2020; Zhang et al., 2020) have demonstrated that predicting optimal solution values for individual variables can achieve a reasonable accuracy. The predicted solution can be used in various ways (e.g., to prune the search space of a problem (Lauri & Dutta, 2019; Sun et al., 2021b; Hao et al., 2020) or warm-start a search method (Li et al., 2018; Zhang et al., 2020; Sun et al., 2022) ) to facilitate the solving of combinatorial optimization problems. However, for a symmetric optimization problem, predicting optimal values for individual decision variables does not provide much benefit for solving the problem. For example, in the vertex coloring problem (VCP) (See Section 2 for a formal problem definition), a random permutation of the colors in an optimal solution results in an alternative optimal solution, and thus predicting the optimal colors for individual vertices is not very useful. On the other hand, predicting a complete optimal solution for a problem directly is too difficult. This is partially due to the NP-hardness of a problem and the difficulty in designing a generic representation for solutions of different sizes. In this paper, we take an intermediate step by developing an effective ML model to predict columns (or fragments (Alyasiry et al., 2019) ) that belong to an optimal solution of a combinatorial optimization problem. To illustrate our method, we use the VCP as an example, in which a column is a Maximal Independent Set (MIS) (Tarjan & Trojanowski, 1977) , whose vertices can share the same color in a feasible solution. The aim of our ML model is to predict which MISs belong to an optimal solution for a given problem instance. To train our ML model, we construct a training set using solved problem instances with known optimal solutions, where each training instance corresponds to a column in a training graph. Three categories of features are designed to represent a column, including 1) problem-specific features computed from the graph data, 2) statistical measures computed from sample solutions, and 3) linear program (LP) features (e.g., reduced cost) computed from the LP relaxation of the MIP model. A training instance is labeled as positive if the corresponding column belongs to an optimal solution; otherwise it is labeled as negative. This is then a standard binary classification problem and any existing classification algorithm can be used for this task. We use the trained ML model to evaluate the quality of columns, and combine it with a sampling method to generate high-quality columns for unseen problem instances. Specifically, our method starts by randomly generating a subset of columns. It then computes the features for each column in the current subset and uses the trained ML model to evaluate the quality of the columns. The lowquality columns predicted by ML are replaced by the new ones generated by a sampling method. This process is repeated for multiple iterations with low-quality columns filtered out and high-quality columns remaining. A subproblem formed by the selected columns is then solved to generate an integer solution. We call this method Machine Learning-based Column Generation (MLCG). Our MLCG method is a significant departure from the traditional CG methods (Lübbecke & Desrosiers, 2005; Mehrotra & Trick, 1996) . Firstly, the traditional methods typically generate columns to solve the LP relaxation of the MIP, while our MLCG method aims at generating columns that are included in a high-quality integer solution. Secondly, the traditional methods select columns only based on their reduced cost, while our method learns a more robust criterion via ML based on a set of features for selecting columns. Thirdly, the traditional methods typically generate columns by repeatedly solving a pricing problem, while our method samples columns on the fly and uses the trained ML model to filter out low-quality columns. To demonstrate the effectiveness of our MLCG method, we evaluate it on the VCP, though the same idea is generally applicable to other combinatorial optimization problems. We empirically show that our ML model can achieve a high accuracy in predicting which columns are part of an optimal solution on the problem instances considered. The columns selected by our ML model are significantly better, in terms of the integer solution that can be constructed from them, than those selected randomly or based purely on their reduced cost. Furthermore, we use the subset of columns generated by our method to warm-start a CG-based heuristic, the Restricted Master Heuristic (RMH) (Taillard, 1999; Bianchessi et al., 2014) , and the results show that our method combined with RMH significantly outperforms RMH alone in terms of both solution quality and run time.

2. BACKGROUND AND RELATED WORK

Vertex Coloring Problem Formulation. Given an undirected graph G(V, E), where V is the set of vertices and E is the set of edges, the objective of VCP is to assign a color to each vertex, such that the adjacent vertices have different colors and the total number of colors used is minimized. Since adjacent vertices cannot share the same color by the problem definition, the vertices that are of the same color in any feasible solution must form an Independent Set. Therefore, the VCP is equivalent to a set partitioning problem which aims to select the minimum number of Independent Sets from a graph such that each vertex is covered exactly once. This is also equivalent to a set covering problem, the objective of which is to minimize the number of Maximal Independent Sets (MISs) selected such that each vertex in the graph is covered at least once (Mehrotra & Trick, 1996) . Let S denote a MIS, S denote the set of all MISs of a graph G, and S v denote the set of MISs that contain vertex v ∈ V . We use a binary variable x S to denote whether a MIS S is selected. The set covering formulation of VCP is defined in (1)-(3) (Mehrotra & Trick, 1996) . A variable x S corresponds to a MIS in the graph and also a column of the constraint matrix of the MIP. As the number of MISs in a graph is potentially exponential in |V |, the 



S∈Svx S ≥ 1, v ∈ V ; (2) x S ∈ {0, 1}, S ∈ S. (3)number of columns of the MIP can be very large. It can be shown that the LP relaxation of (1)-(3)

