Viktor Kiss

Chip-firing games on Eulerian digraphs and NP-hardness of computing the rank of a divisor on a graph

Abstract Baker and Norine introduced a graph-theoretic analogue of the Riemann–Roch theory. A central notion in this theory is the rank of a divisor. In this paper we prove that computing the rank of a divisor on a graph is NP -hard, even for simple graphs. The determination of the rank of a divisor can be translated to a question about a chip-firing game on the same underlying graph. We prove the NP -hardness of this question by relating chip-firing on directed and undirected graphs.

Ranks on the Baire class functions

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class 1 functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class functions, and generalize most of the results from the Baire class 1 case. As an application, we solve a problem concerning the so called solvability cardinals of systems of dierence equations, arising from the theory of geometric equidecomposability. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in !1.

On the complexity of the chip-firing reachability problem

In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in strongly polynomial time, even if the digraph has multiple edges. We also show a special case when the reachability problem can be decided in polynomial time for general digraphs: if the target distribution is recurrent restricted to each strongly connected component. As a further positive result, we show that the chip-firing reachability problem is in co-NP for general digraphs. We also show that the chip-firing halting problem is in co-NP for Eulerian digraphs.

Classification of bounded Baire class

Kechris and Louveau showed that each real-valued bounded Baire class 1 function defined on a compact metric space can be written as an alternating sum of a decreasing countable transfinite sequence of upper semi-continuous functions. Moreover, the length of the shortest such sequence is essentially the same as the value of certain natural ranks they defined on the Baire class 1 functions. They also introduced the notion of pseudouniform convergence to generate some classes of bounded Baire class 1 functions from others. The main aim of this paper is to generalize their results to Baire class

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functions

functions. For our proofs to go through, it was essential to first obtain similar results for Baire class 1 functions defined on not necessary compact Polish spaces. Using these new classifications of bounded Baire class

Unions of Regular Polygons with Large Perimeter-to-Area Ratio

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The structure of random automorphisms

functions, one can define natural ranks on these classes. We show that these ranks essentially coincide with those defined by Elekes et. al.

The structure of random homeomorphisms.

Keleti [Acta Univ Carol Math Phys 39(1–2):111–118, 1998] asked whether the ratio of the perimeter to the area of a finite union of unit squares is always at most 4. In this paper we present an example where the ratio is greater than 4. We also answer the analogous question for regular triangles negatively and list a number of open problems.