Some Problems in Graph Theory and Scheduling

Abstract

In this dissertation, we present three results related to combinatorial algorithms in graph theory and scheduling, both of which are important subjects in the area of discrete mathematics and theoretical computer science. In graph theory, a graph is a set of vertices and edges, where each edge is a pair of vertices. A coloring of a graph is a function that assigns each vertex a color such that no two adjacent vertices share the same color. The first two results are related to coloring graphs belonging to specific classes. In scheduling problems, we are interested in how to efficiently schedule a set of jobs on machines. The last result is related to a scheduling problem in an environment where there is uncertainty on the number of machines. The first result of this thesis is a polynomial time algorithm that determines if an input graph containing no induced seven-vertex path is 3-colorable. This affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002. Our algorithm also solves the list-coloring version of the 3-coloring problem, where every vertex is assigned a list of colors that is a subset of {1, 2, 3}, and gives an explicit coloring if one exists. This is joint work with Flavia Bonomo, Maria Chundnovsky, Peter Maceli, Oliver Schaudt, and Maya Stein. A graph is H-free if it has no induced subgraph isomorphic to H. In the second part of this thesis, we characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable H-free graphs. This solves a problem posed by Golovach et al. We also characterize all graphs H for which there are only finitely many H-free minimal obstructions for list 3-colorability. This is joint work with Maria Chudnovsky, Jan Goedgebeur and Oliver Schaudt. The last result of this thesis deals with a scheduling problem addressing the uncertainty regarding the machines. We study a scheduling environment in which jobs first need to be grouped into some sets before the number of machines is known, and then the sets need to be scheduled on machines without being separated. In order to evaluate algorithms in such an environment, we introduce the idea of an alpha-robust algorithm, one which is guaranteed to return a schedule on any number m of machines that is within an alpha factor of the optimal schedule on m machines, where the optimum is not subject to the restriction that the sets cannot be separated. Under such environment, we give a (5/3+epsilon)-robust algorithm for scheduling on parallel machines to minimize makespan, and show a lower bound of 4/3. For the special case when the jobs are infinitesimal, we give a 1.233-robust algorithm with an asymptotic lower bound of 1.207. This is joint work with Clifford Stein