Jochen Messner

A kolmogorov complexity proof of the lovász local lemma for satisfiability

Recently, Moser and Tardos [MT10] came up with a constructive proof of the Lovasz Local Lemma. In this paper, we give another constructive proof of the lemma, based on Kolmogorov complexity. Actually, we even improve the Local Lemma slightly.

On the Structure of the Simulation Order of Proof Systems

We examine the degree structure of the simulation relation on the proof systems for a set L. As observed, this partial order forms a distributive lattice. A greatest element exists iff L has an optimal proof system. In case L is infinite there is no least element, and the class of proof systems for L is not presentable. As we further show the simulation order is dense. In fact any partial order can be embedded into the interval determined by two proof systems f and g such that f simulates g but g does not simulate f. Finally we obtain that for any non-optimal proof system h an infinite set of proof systems that are pairwise incomparable with respect simulation and that are also incomparable to h.

Planarizing Gadgets for Perfect Matching Do Not Exist

To reduce a graph problem to its planar version, a standard technique is to replace crossings in a drawing of the input graph by planarizing gadgets. We show unconditionally that such a reduction is not possible for the perfect matching problem and also extend this to some other problems related to perfect matching. We further show that there is no planarizing gadget for the Hamiltonian cycle problem.

Exact Perfect Matching in Complete Graphs

A red-blue graph is a graph where every edge is colored either red or blue. The exact perfect matching problem asks for a perfect matching in a red-blue graph that has exactly a given number of red edges. We show that for complete and bipartite complete graphs, the exact perfect matching problem is logspace equivalent to the perfect matching problem. Hence, an efficient parallel algorithm for perfect matching would carry over to the exact perfect matching problem for this class of graphs. We also report some progress in extending the result to arbitrary graphs.

Game Values and Computational Complexity: An Analysis via Black-White Combinatorial Games

A black-white combinatorial game is a two-person game in which the pieces are colored either black or white. The players alternate moving or taking elements of a specific color designated to them before the game begins. A player loses the game if there is no legal move available for his color on his turn.

Planarizing gadgets for perfect matching do not exist

To reduce a graph problem to its planar version, a standard technique is to replace crossings in a drawing of the input graph by planarizing gadgets. We show unconditionally that such a reduction is not possible for the perfect matching problem and also extend this to some other problems related to perfect matching. We further show that there is no planarizing gadget for the Hamiltonian cycle problem.