arXiv subjects

Rudolf Berghammer

Publications and source records attributed to Rudolf Berghammer.

5 recordsLinked to original sources

arXiv2026

Counting Specific Classes of Relations Regarding Fixed Points and Reflexive Points

Given a finite and non-empty set $X$ and randomly selected specific functions and relations on $X$, we investigate the existence and non-existence of fixed points and reflexive points, respectively. First, we consider the class of functions, weaken it to the classes of partial functions, total relations and general relations and also strengthen it to the class of permutations. Then we investigate the class of involutions and the subclass of proper involutions. Finally, we treat idempotent functions, partial idempotent functions and related concepts. We count relations, calculate corresponding probabilities and also calculate the limiting values of the latter in case that the cardinality of $X$ tends to infinity. All these results have been motivated and also supported by numerous experiments performed with the RelView tool.

cs.DM↗

arXiv2020

Algorithmic Counting of Zero-Dimensional Finite Topological Spaces With Respect to the Covering Dimension

Taking the covering dimension dim as notion for the dimension of a topological space, we first specify thenumber zdim_{T_0}(n) of zero-dimensional T_0-spaces on {1,...,n}$ and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1,\ldots,n} by means oftwo mappings po and P that yieldthe number po(n) of partial orders on {1,...,n} and the set P(n) of partitions of {1,...,n}, respectively. Algorithms for both mappings exist. Assuming one for po to be at hand, we use our specification of zdim_{T_0}(n) and modify one for P in such a way that it computes zdim_{T_0}(n) instead of P(n). The specification of zdim(n) then allows to compute this number from zdim_{T_0}(1) to zdim_{T_0}(n) and the Stirling numbers of the second kind S(n,1) to S(n,n). The resulting algorithms have been implemented in C and we also present results of practical experiments with them. To considerably reduce the running times for computing zdim_{T_0}(n), we also describe a backtracking approach and its parallel implementation in C using the OpenMP library.

cs.DM↗

arXiv2018

Relational Characterisations of Paths

Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in many areas of mathematics and computing, researchers usually fall back to point-wise reasoning when it comes to arguments about paths in a graph. We present a purely algebraic way to specify different kinds of paths in relation algebras. We study the relationship between paths with a designated root vertex and paths without such a vertex. Since we stay in first-order logic this development helps with mechanising proofs.To demonstrate the applicability of the algebraic framework we verify the correctness of three basic graph algorithms. All results of this paper are formally verified using the interactive proof assistant Isabelle/HOL.

cs.LO↗

arXiv2015

Solving Hard Control Problems in Voting Systems via Integer Programming

Voting problems are central in the area of social choice. In this article, we investigate various voting systems and types of control of elections. We present integer linear programming (ILP) formulations for a wide range of NP-hard control problems. Our ILP formulations are flexible in the sense that they can work with an arbitrary number of candidates and voters. Using the off-the-shelf solver Cplex, we show that our approaches can manipulate elections with a large number of voters and candidates efficiently.

cs.GT↗

arXiv2013

Relation-algebraic and Tool-supported Control of Condorcet Voting

We present a relation-algebraic model of Condorcet voting and, based on it, relation-algebraic solutions of the constructive control problem via the removal of voters. We consider two winning conditions, viz. to be a Condorcet winner and to be in the (Gilles resp. upward) uncovered set. For the first condition the control problem is known to be NP-hard; for the second condition the NP-hardness of the control problem is shown in the paper. All relation-algebraic specifications we will develop in the paper immediately can be translated into the programming language of the BDD-based computer system RelView. Our approach is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other voting rules and control problems.

cs.GT↗