An algorithmic approach for multi-color Ramsey graphs

The classical Ramsey number R(r1,r2,...,rm) is defined to be the smallest integer n such that no matter how the edges of Kn are colored with the m colors, 1, 2, 3, . . . ,m, there exists some color i such that there is a complete subgraph of size ri, all of whose edges are of color i. The problem of determining Ramsey numbers is known to be very difficult and is usually split into two problems, finding upper and lower bounds. Lower bounds can be obtained by the construction of a, so called, Ramsey graph. There are many different methods to construct Ramsey graphs that establish lower bounds. In this thesis mathematical and computational methods are exploited to construct Ramsey graphs. It was shown that the problem of checking that a graph coloring gives a Ramsey graph is NP-complete. Hence it is almost impossible to find a polynomial time algorithm to construct Ramsey graphs by searching and checking. Consequently, a method such as backtracking with good pruning techniques should be used. Algebraic methods were developed to enable such a backtrack search to be feasible when symmetry is assumed. With the algorithm developed in this thesis, two new lower bounds were established: R(3,3,5)≥45 and R(3,4,4)≥55. Other best known lower bounds were matched, such as R(3,3,4)≥30. The Ramsey graphs giving these lower bounds were analyzed and their full symmetry groups were determined. In particular it was shown that there are unique cyclic graphs up to isomorphism giving R(3,3,4)≥30 and R(3,4,4)≥55, and 13 non-isomorphic cyclic graphs giving R(3,3,5)≥45

A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.

Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph

Sequential correlated equilibrium in stopping games

In many situations, such as trade in stock exchanges, agents have many instances to act even though the duration of interactions take a relatively short time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential constant-expectation normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium

Perfect correlated equilibria in stopping games

In many situations, such as trade in stock exchanges, agents have many instances to act even though the duration of interactions take a relatively short time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential constant-expectation normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium

Sequential correlated equilibrium in stopping games

In many situations, such as trade in stock exchanges, agents have many instances to act even though the duration of interactions take a relatively short time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential constant-expectation normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium

逆数学と可算代数系

Tohoku University山崎武課

Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view