Combinatorial Games Theory

Title: Combinatorial Games Theory Author: Tomáš Valla Department / Institute: IUUK MFF UK Supervisor: Prof. RNDr. Jaroslav Nešetřil, DrSc., IUUK MFF UK Abstract: In this thesis we study the complexity that appears when we consider the competitive version of a certain environment or process, using mainly the tools of al- gorithmic game theory, complexity theory, and others. For example, in the Internet environment, one cannot apply any classical graph algorithm on the graph of connected computers, because it usually requires existence of a central authority, that manipu- lates with the graph. We describe a local and distributed game, that in a competitive environment without a central authority simulates the computation of the weighted vertex cover, together with generalisation to hitting set and submodular weight func- tion. We prove that this game always has a Nash equilibrium and each equilibrium yields the same approximation of optimal cover, that is achieved by the best known ap- proximation algorithms. More precisely, the Price of Anarchy of our game is the same as the best known approximation ratio for this problem. All previous results in this field do not have the Price of Anarchy bounded by a constant. Moreover, we include the results in two more fields, related to the complexity of competitive..