Online computation is a concept to model uncertainty where not all information on a problem instance is known in advance. An online algorithm receives requests which reveal the instance piecewise and has to respond with irrevocable decisions. Often, an adversary is assumed that constructs the instance knowing the deterministic behavior of the algorithm. Thus, the adversary is able to tailor the input to any online algorithm. From a game theoretical point of view, the adversary and the online algorithm are players in an asymmetric two-player game. To overcome this asymmetry, the online algorithm is equipped with an isomorphic copy of the graph, which is referred to as unlabeled map. By applying the game theoretical perspective on online graph problems, where the solution is a subset of the vertices, we analyze the complexity of these online vertex subset games. For this, we introduce a framework for reducing online vertex subset games from TQBF. This framework is based on gadget reductions from 3-SATISFIABILITY to the corresponding offline problem. We further identify a set of rules for extending the 3-SATISFIABILITY-reduction and provide schemes for additional gadgets which assure that these rules are fulfilled. By extending the gadget reduction of the vertex subset problem with these additional gadgets, we obtain a reduction for the corresponding online vertex subset game. At last, we provide example reductions for online vertex subset games based on VERTEX COVER, INDEPENDENT SET, and DOMINATING SET, proving that they are PSPACE-complete. Thus, this paper establishes that the online version with a map of NP-complete vertex subset problems form a large class of PSPACE-complete problems.

翻译：在线计算是一种用于建模不确定性的概念，即问题实例的所有信息并非事先已知。在线算法会接收逐步揭示实例的请求，并需作出不可撤销的决策。通常，假设存在一个知晓算法确定性行为的对手来构造实例，因此对手能够针对任何在线算法定制输入。从博弈论的角度看，对手与在线算法构成了一个非对称双人博弈中的两个玩家。为克服这种非对称性，在线算法配备了一个图的同构副本，称为无标记地图。通过将博弈论视角应用于解为顶点子集的在线图问题，我们分析了这类在线顶点子集博弈的复杂性。为此，我们引入了一个将在线顶点子集博弈从TQBF归约的框架。该框架基于从3-SATISFIABILITY到相应离线问题的子句归约。我们进一步识别了一组扩展3-SATISFIABILITY归约的规则，并提供了确保这些规则得以满足的附加子句方案。通过用这些附加子句扩展顶点子集问题的子句归约，我们得到了对应在线顶点子集博弈的归约。最后，我们基于VERTEX COVER、INDEPENDENT SET和DOMINATING SET提供了在线顶点子集博弈的示例归约，证明了它们是PSPACE完全的。因此，本文确立了带有地图的NP完全顶点子集问题的在线版本构成了一个大型PSPACE完全问题类。