arXiv subjects

Jens Maßberg

Publications and source records attributed to Jens Maßberg.

6 recordsLinked to original sources

arXiv2015

The Depth-Restricted Rectilinear Steiner Arborescence Problem is NP-complete

In the rectilinear Steiner arborescence problem the task is to build a shortest rectilinear Steiner tree connecting a given root and a set of terminals which are placed in the plane such that all root-terminal-paths are shortest paths. This problem is known to be NP-hard. In this paper we consider a more restricted version of this problem. In our case we have a depth restrictions $d(t)\in\mathbb{N}$ for every terminal $t$. We are looking for a shortest binary rectilinear Steiner arborescence such that each terminal $t$ is at depth $d(t)$, that is, there are exactly $d(t)$ Steiner points on the unique root-$t$-path is exactly $d(t)$. We prove that even this restricted version is NP-hard.

cs.CC↗

arXiv2015

The Union-Closed Sets Conjecture for Small Families

We prove that the union-closed sets conjecture is true for separating union-closed families $\mathcal{A}$ with $|\mathcal{A}| \leq 2\left(m+\frac{m}{\log_2(m)-\log_2\log_2(m)}\right)$ where $m$ denotes the number of elements in $\mathcal{A}$.

math.CO↗

arXiv2015

Solitaire Chess is NP-complete

"Solitaire Chess" is a logic puzzle published by Thinkfun, that can be seen as a single person version of traditional chess. Given a chess board with some chess pieces of the same color placed on it, the task is to capture all pieces but one using only moves that are allowed in chess. Moreover, in each move one piece has to be captured. We prove that deciding if a given instance of Solitaire Chess is solvable is NP-complete.

cs.CC↗

arXiv2014

The rectilinear Steiner tree problem with given topology and length restrictions

We consider the problem of embedding the Steiner points of a Steiner tree with given topology into the rectilinear plane. Thereby, the length of the path between a distinguished terminal and each other terminal must not exceed given length restrictions. We want to minimize the total length of the tree. The problem can be formulated as a linear program and therefore it is solvable in polynomial time. In this paper we analyze the structure of feasible embeddings and give a combinatorial polynomial time algorithm for the problem. Our algorithm combines a dynamic programming approach and binary search and relies on the total unimodularity of a matrix appearing in a sub-problem.

cs.DS↗

arXiv2014

Generalized Huffman Coding for Binary Trees with Choosable Edge Lengths

In this paper we study binary trees with choosable edge lengths, in particular rooted binary trees with the property that the two edges leading from every non-leaf to its two children are assigned integral lengths $l_1$ and $l_2$ with $l_1+l_2 =k$ for a constant $k\in\mathbb{N}$. The depth of a leaf is the total length of the edges of the unique root-leaf-path. We present a generalization of the Huffman Coding that can decide in polynomial time if for given values $d_1,...,d_n\in\mathbb{N}_{\geq 0}$ there exists a rooted binary tree with choosable edge lengths with $n$ leaves having depths at most $d_1,..., d_n$.

cs.IT↗

arXiv2014

The "Game about Squares" is NP-hard

In the "Game about Squares" the task is to push unit squares on an integer lattice onto corresponding dots. A square can only be moved into one given direction. When a square is pushed onto a lattice point with an arrow the direction of the square adopts the direction of the arrow. Moreover, squares can push other squares. In this paper we study the decision problem, whether all squares can be moved onto their corresponding dots by a finite number of pushes. We prove that this problem is NP-hard.

cs.CC↗