# 9th workshop of the Center for Foundations of Modern Computer Science

# 9. workshop Centra základů moderní informatiky

The nineth workshop of the Center for Foundations of Modern Computer Science took place on Tuesday August 2, 2022, in Prague as a part of MCW XXVII.

9:00 | Martin Balko: Bounding and computing obstacle numbers of graphs |

9:25 | Jakub Bulín: Coloring oriented trees |

9:50 | Pavel Hubáček: On the Complexity of Inefficient Proofs of Existence in Combinatorics |

## Abstracts

### Martin Balko: Bounding and computing obstacle numbers of graphs

An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each n-vertex graph is O(nlog n) and that there are n-vertex graphs whose obstacle number is Ω(n/(loglog n)^2). We improve this lower bound to Ω(n/loglog n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. Our bounds are asymptotically tight in several instances.

This is a joint work with Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, and Alexander Wolff.

### Jakub Bulín: Coloring oriented trees

We study the computational and descriptive complexity of the H-coloring problem in the case where H is an orientation of a tree. We present the smallest NP-hard oriented tree, and we discuss mathematical as well as experimental results towards resolving three open problems: conjectured classification of trees solvable in L, in NL, and a conjecture that tractable trees lack the 'ability to count'.

Based on joint work with Manuel Bodirsky, Florian Starke, and Michael Wernthaler (TU Dresden).

### Pavel Hubáček: On the Complexity of Inefficient Proofs of Existence in Combinatorics

Many famous existential theorems lack efficient constructive proofs, giving rise to important computational problems without an obvious efficient algorithm. In my talk, I will introduce computational problems motivated by classical combinatorial theorems such as, for example, the Erdős-Ko-Rado theorem and show that they characterize the complexity classes PPP (Papadimitriou, J. Comput. Syst. Sci. 1994) and PWPP (Jeřábek, J. Comput. Syst. Sci. 2016).

Based on joint work with Romain Bourneuf, Lukáš Folwarczný, Alon Rosen, and Nikolaj I. Schwartzbach