The Complexity of Online Graph Games

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. From a game theoretical point of view, the adversary and the online algorithm are players in a two-player game. By applying this view on combinatorial graph problems, especially on problems where the solution is a subset of the vertices, we analyze their complexity. For this, we introduce a framework based on gadget reductions from 3-Satisfiability and extend it to an online setting where the graph is a priori known by a map. This is done by identifying a set of rules for the reductions and providing schemes for gadgets. The extension of the framework to the online setting enable reductions from TQBF. We provide example reductions to the well-known problems Vertex Cover, Independent Set and Dominating Set and prove that they are PSPACE-complete. Thus, this paper establishes that the online version with a map of NP-complete graph problems form a large class of PSPACE-complete problems

Gadgets, approximation, and linear programming (Extended Abstract)

We present a linear-programming based method for finding "gadgets", i.e., combinatorial structures reducing constraints of one optimization problems to constraints of another. A key step in this method is a simple observation which limits the search space to a nite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by [1] on how to prove the optimality of gadgets -- we show how LP duality gives such proofs. The new gadgets improve hardness results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 60=61 and 44=45 respectively is NP-hard (improving upon the previous hardness of 71=72 for both problems [1]). We also use the gadgets to obtain an improved approximation algorithm for MAX 3SAT which guarantees an approximation ratio of :801. This improves upon the previous best bound (implicit in [6, 3]) of :7704