Sector duration optimization (SDO) is a problem arising in treatment planning for stereotactic radiosurgery on Gamma Knife. Given a set of isocenter locations, SDO aims to select collimator size configurations and irradiation times thereof such that target tissues receive prescribed doses in a reasonable amount of treatment time and healthy tissues nearby are spared. We present a multiobjective linear programming model for SDO to generate a diverse collection of solutions so that clinicians can select the most appropriate treatment. We develop a generic two-phase solution strategy based on the ε-constraint method for solving multiobjective optimization models, 2phasε, which aims to systematically increase the number of high-quality solutions obtained, instead of conducting a traditional uniform search. To improve solution quality further and to accelerate the procedure, we incorporate some general and problem-specific enhancements. Moreover, we propose an alternative version of 2phasε, which makes use of machine learning tools to reduce the computational effort. In our computational study on eight previously treated real test cases, a significant portion of 2phasε solutions outperformed clinical results and those from a single-objective model from the literature. In addition to significant benefits of the algorithmic enhancements, our experiments illustrate the usefulness of machine learning strategies to reduce the overall run times nearly by half while maintaining or besting the clinical practice. History: Accepted by Paul Brooks, Area Editor for Applications in Biology, Medicine, and Healthcare. Funding: This work was supported in part by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant RGPIN-2019-05588]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [ https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1252 ] or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at [ http://dx.doi.org/10.5281/zenodo.7048848 ].

Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph such that no two adjacent vertices receive the same color. The selective graph coloring problem is a generalization of the standard graph coloring problem; given a graph with a partition of its vertex set into clusters, the objective is to choose exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimum. This study focuses on a decomposition based exact solution framework for selective coloring in some perfect graph families: in particular, permutation, generalized split, and chordal graphs where the selective coloring problem is known to be NP‐hard. Our method combines integer programming techniques and combinatorial algorithms for the graph classes of interest. We test our method on graphs with different sizes and densities, present computational results and compare them with solving an integer programming formulation of the problem by CPLEX, and a state‐of‐the art algorithm from the literature. Our computational experiments indicate that our decomposition approach significantly improves solution performance in low‐density graphs, and regardless of edge‐density in the class of chordal graphs.