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計算量クラスTFNPに対する不動点定理およびアルゴリズム的ゲーム理論からのアプローチ

石塚, 天 ISHIZUKA, Takashi イシヅカ, タカシ 九州大学

2022.09.22

概要

While NP-hardness has played a crucial role in guaranteeing the intractability of many fascinating problems, there still are left many natural problems that are seemingly not effortless to solve and are not known to be NP-hard. The complexity class TFNP captures the complexity of such problems. This class is made up of total search problems, and the correctness of solutions is polynomial-time verifiable. The TFNP classes, like PLS (Polynomial Local Search) and PPAD (Polynomial Parity Argument for Digraphs), constitute an essential and fascinating complexity-theoretical research field. This thesis provides new results for search problems belonging to TFNP classes.

In the first part, we consider the complexity of search problems on an exponentially-large graph whose vertices have positive values. We show the robustness of the definition of EOPL by END OF POTENTIAL LINE. Furthermore, we provide new PPA ∩ PLS-complete problems.

The rest of this thesis focuses on the computational aspects of some search problems from the viewpoints of Fixed Point Theory and Algorithmic Game Theory. In the second part, we consider the complexity of finding a fixed point whose existence is guaranteed by Caritsti’s fixed point theorem. We prove the PLS completeness of this problem. Thus, Caristi’s fixed point theorem characteries the complexity class PLS. Furthermore, we discuss the complexity of computing a fixed point whose ex- istence is guaranteed by Brøndsted’s fixed point theorem.

The final part deal with the complexity of equilibrium computation. First, we consider the hardness of distinguishing the existence of uniform Nash equilibria on a two-player strategic-form game. Second, we consider the complexity of finding a pure Nash equilibrium on a discrete preference game and a network coordination game; it is well known that these graphical games always have a pure Nash equilibrium. Finally, we study the complexity of finding a stable fractional matching on a hypergraphic preference system, a generalization of a roommate matching model. We provide a tractable-intractable boundary for this problem.

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