Philipp Zumstein

Polychromatic colorings of plane graphs

We show that the vertices of any plane graph in which every face is of size at least g can be colored by (3g Àý 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete even for graphs in which all faces are of size 3 or 4 only. If all faces are of size 3 this can be decided in polynomial time.

How many conflicts does it need to be unsatisfiable

A pair of clauses in a CNF formula constitutes a conflict if there is a variable that occurs positively in one clause and negatively in the other. Clearly, a CNF formula has to have conflicts in order to be unsatisfiable--in fact, there have to be many conflicts, and it is the goal of this paper to quantify how many. An unsatisfiable k-CNF has at least 2k clauses; a lower bound of 2k for the number of conflicts follows easily. We improve on this trivial bound by showing that an unsatisfiable k-CNF formula requires Ω(2.32k) conflicts. On the other hand there exist unsatisfiable k-CNF formulas with O(4k log3 k/k) conflicts. This improves the simple bound O(4k) arising from the unsatisfiable k-CNF formula with the minimum number of clauses.

Satisfiability with exponential families

Fix a set S ⊆ {0, 1}* of exponential size, e.g. |S ∩ {0, 1}n| ∈ Ω(αn), α > 1. The S-SAT problem asks whether a propositional formula F over variables v1, . . . , vn has a satisfying assignment (v1, . . . , vn) ∈ {0, 1}n ∩ S. Our interest is in determining the complexity of S-SAT. We prove that S-SAT is NP-complete for all context-free sets S. Furthermore, we show that if S-SAT is in P for some exponential S, then SAT and all problems in NP have polynomial circuits. This strongly indicates that satisfiability with exponential families is a hard problem. However, we also give an example of an exponential set S for which the S-SAT problem is not NP-hard, provided P ≠ NP.

Minimum and maximum against k lies

A neat 1972 result of Pohl asserts that ⌈3n/2 ⌉−2 comparisons are sufficient, and also necessary in the worst case, for finding both the minimum and the maximum of an n-element totally ordered set. The set is accessed via an oracle for pairwise comparisons. More recently, the problem has been studied in the context of the Renyi–Ulam liar games, where the oracle may give up to k false answers. For large k, an upper bound due to Aigner shows that

Linked Open Citation Database: Enabling Libraries to Contribute to an Open and Interconnected Citation Graph

(k+{\mathcal O}(\sqrt{k}))n

Semiautomatische Katalogisierung und Normdatenverknüpfung mit Zotero im Index Theologicus

comparisons suffice. We improve on this by providing an algorithm with at most

Verbesserung der OCR in digitalen Sammlungen von Bibliotheken

(k+1+C)n+{\mathcal O}(k^3)

Literaturauswahl und -bestellung leichter gemacht!

comparisons for some constant C. The known lower bounds are of the form (k+1+ck)n−D, for some constant D, where c0=0.5,

Extremal colorings and extremal satisfiability

c_1=\frac{23}{32}= 0.71875

On the minimum degree of minimal Ramsey graphs

, and