Maximum Independent Set And Maximum Induced Matching Problems For Competitive Programming

Competitive programming is a growing interest among students, with some students training for years to be competitive in national and international competitions. Competitive programming problems continue to become more complex; yet they are always solvable with skills learned in an undergraduate algorithms class. This makes competitive programming a great way for undergraduates to develop their coding skills and learn complex algorithms. However, there are very few competitive programming problems on particular graph classes, despite the fact that the field of graph theory is rich with complexity and algorithms results for over eighty years. This may be because of the overwhelming amount of graph classes and terminology that students need to be familiar with to understand even the simplest results in graph theory, sometimes overlooking the connection between graph theory and the study of algorithms. Some of these algorithms, like that of computing a maximum independent set (MIS) or a maximum induced matching (MIM) on special graph classes, only require techniques learned in an undergraduate algorithms course. However, in the literature, they are hidden behind results for generalized classes, often using terminology and notation far beyond what undergraduate students are exposed to. Some of these graph theoretic results are either so old that the original papers are hard to find or they are held behind payment gateways from publishers. Therefore, there needs to be substantive work done to improve the expositions of old (and some new) graph theory algorithms that can be solved using topics learned in an undergraduate course. This will allow students in algorithm classes to be exposed to topics in graph theory, while fundamental problems on graphs can be used as excellent motivating examples for topics in algorithms.