Computational Complexity in Graph Theory

Title: Computational Complexity in Graph Theory Author: Eva Jeĺınková Department: Department of Applied Mathematics Supervisor: Prof. RNDr. Jan Kratochv́ıl, CSc. , Department of Applied Mathematics Abstract: We address problems from graph theory, especially from the computational complexity point of view. In the first part of the thesis we address the computational complexity of problems related to Seidel’s switching of graphs. We prove that the problem to decide if a given graph can be switched to contain at most a given number of edges is NP-complete, even for graphs with bounded density. We thus partially answer a question of Matoušek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014]. We also describe infinitely many graphs H such that it is NP-complete to decide if a given graph is switching-equivalent to a graph that does not contain H as an induced subgraph. We thus close an open problem of Kratochv́ıl, Nešetřil and Zýka [Annals of Discrete Math. 51, 1992]. In the second part of the thesis we address the topic of matchings under preferences. We focus on the housing market problem, in particular, on the model with duplicate houses. We present a 2-approximation algorithm for the maximum number of satisfied agents when the preference lists of agents are trichotomic. On the other hand, we prove that the number is NP-hard to approximate within a factor smaller than 21/19, and if the preferences are not required to be trichotomic, then the number is NP-hard to approximate within a factor smaller than 1.2. In addition to these results, we show an asymptotically optimal improvement of the equilibrium algorithm in the setting where agents’ preferences are strict linear orderings of house types. We address problems from graph theory, especially from the computational complexity point of view. In the first part of the thesis we address the computational complexity of problems related to Seidel’s switching of graphs. We prove that the problem to decide if a given graph can be switched to contain at most a given number of edges is NP-complete, even for graphs with bounded density. We thus partially answer a question of Matoušek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014]. We also describe infinitely many graphs H such that it is NP-complete to decide if a given graph is switching-equivalent to a graph that does not contain H as an induced subgraph. We thus close an open problem of Kratochv́ıl, Nešetřil and Zýka [Annals of Discrete Math. 51, 1992]. In the second part of the thesis we address the topic of matchings under preferences. We focus on the housing market problem, in particular, on the model with duplicate houses. We present a 2-approximation algorithm for the maximum number of satisfied agents when the preference lists of agents are trichotomic. On the other hand, we prove that the number is NP-hard to approximate within a factor smaller than 21/19, and if the preferences are not required to be trichotomic, then the number is NP-hard to approximate within a factor smaller than 1.2. In addition to these results, we show an asymptotically optimal improvement of the equilibrium algorithm in the setting where agents’ preferences are strict linear orderings of house types.