Prof. Claudia Archetti
Title: Branch-and-cut algorithms for colorful components problems
Joint work with Carmine Sorgente and Martina Cerulli.
We tackle three optimization problems in which a colored graph, where each node is assigned a color, must be partitioned into colorful connected components. A component is defined as colorful if each color appears at most once. The problems differ in the objective function, which determines which partition is the best one. These problems have applications in community detection, cybersecurity, and bioinformatics. We present integer non-linear formulations, which are then linearized using standard techniques. To solve these formulations, we develop exact branch-and-cut algorithms, embedding various improving techniques, such as valid inequalities, bounds limiting the number of variables, and warm-start and preprocessing techniques. Extensive computational tests on benchmark instances demonstrate the effectiveness of the proposed procedures. The branch-and-cut algorithms can solve reasonably sized instances efficiently. To the best of our knowledge, we are the first to propose an exact algorithm for solving these problems.
Prof. Chinh Hoang
Title: Coloring graphs with forbidden induced subgraphs
The Coloring Problem is to decide if a given graph can be colored with
at most k colors, given an integer k, such that no two adjacent adjacent
vertices receive the same color. Given a set L of graphs, a graph G is
L-free if G does not contain any graph in L as an induced subgraph. The
complexity of the Coloring Problem on L-free graphs is known whenever
L contains a single graph. There has been keen interest in coloring graphs
whose forbidden list L contains basic graphs such as induced paths, in-
duced cycles and their complements. In this talk, I will provide a survey
of recent progress on this topic.