arXiv subjects

Gerold Jäger

Publications and source records attributed to Gerold Jäger.

13 recordsLinked to original sources

arXiv2026

Algorithms for Computing Set Tolerances: Theory, Computational Analysis and Applications

The regular set tolerance is an important term in sensitivity analysis. For combinatorial sum problems, e.g., the Traveling Salesman Problem, Shortest Path Problem and Minimum Spanning Tree Problem, it determines how much the sum of the costs of the elements of a set can be increased while ensuring that all current optimal solutions remain optimal. The regular set lower tolerance determines how much the sum of the costs of the elements of a set can be decreased while ensuring that the objective value of the optimal solution is not changed. We investigate general methods for computing regular (upper and lower) set tolerances in combinatorial sum problems. For the upper tolerance, we present a linear programming approach, and for the lower tolerances, three linear programming approaches, where the last two are novel and lead to recursive procedures for computation of the lower tolerances of all subsets of the given ground set. We give new upper bounds for set lower tolerances. For both upper and lower tolerances, we give an exact formula for sets of cardinality 2 and 3. Furthermore, we computationally compare and analyze the three different lower tolerance LPs. Finally, we consider the computation of tolerances for the Minimum Spanning Tree Problem, give a formula for single tolerances, a lower bound for regular set upper tolerances and an exact formula for regular set lower tolerances.

math.CO↗

arXiv2026

Exact number of flips required to sort a burnt stack of pancakes

In this work, we consider the burnt pancake problem, which is a well-studied problem going back to a work of Gates and Papadimitriou from 1979.The problem is to sort a stack of~$n$ one-sided burnt pancakes of different sizes, by a sequence of flips of the top pancakes, such that at the end of the flipping sequence the pancakes have increasing size and the burnt sides of all pancakes are face-down. The pancakes are denoted by $ 1,2,\dots,n$, and a number is multiplied by $-1$, if the corresponding pancake has burnt side face-up. Let $T(n)$ be the minimum number of flips to sort a special stack of $n$ pancakes, namely $\overline{I_n} := [-1,-2,...,-n]$. The instance $\overline{I_n} $ has strong relevance because of its easy structure and as it has been shown to be a worst-case instance for several small $n$. Heydari and Sudborough gave in 1997 the currently best upper bound of $T(n)$, namely $ \lfloor (3n+3)/2 \rfloor $ for $ n \equiv 3 \bmod 4$, which later has been shown to be exact by a work of Cibulka from 2011. Except these two works, no progress regarding lower and upper bounds has been made until now. In our work, we present that $ \lfloor (3n+3)/2 \rfloor $ is also an upper bound of $T(n)$ for $n \equiv 1 \bmod 4 $, which again matches the lower bound of Cibulka in 2011 and thus is exact. Furthermore, we show that our construction approach for $n \equiv 1 \bmod 4 $ and the one of Heydari and Sudborough for $n \equiv 3 \bmod 4 $ cannot be applied for even $n$. However, as there might be different construction approaches, the case of even $n$ remains an open problem, where two possible values for $T(n)$ are possible, namely $ (3/2)n + 1 $ or $ (3/2)n + 2 $. Finally, we found two values, namely $n=24$, $n=26$, where the lower bound is attained.

math.CO↗

arXiv2025

The Generalized Double Pouring Problem: Analysis, Bounds and Algorithms

We consider a logical puzzle which we call double pouring problem, which was original defined for $k=3$ vessels. We generalize this definition to $k \ge 2 $ as follows. Each of the $k$ vessels contains an integer amount of water, called its value, where the values are $a_i$ for $i=1,2,\dots,k$ and the sum of values is $n$. A pouring step means pouring water from one vessel with value $a_i$ to another vessel with value $a_j$, where $ 1 \le i \not= j \le k $ and $a_i \le a_j $. After this pouring step the first vessel has value $2a_i$ and the second one value $a_j-a_i$. Now the pouring problem is to find as few pourings steps as possible to empty at least one vessel, or to show that such an emptying is not possible (which is possible only in the case $k=2$). For $k=2$ each pouring step is unique. We give a necessary and sufficient condition, when for a given $ (a_1,a_2)$ with $a_1+a_2=n$ the pouring problem is solvable. For $k=3$ we improve the upper bound of the pouring problem for some special cases. For $k \ge 4 $ we extend the known lower bound for $k=3$ and improve the known upper bound $\mathcal{O}((\log n)^2)$ for $k=3$ to $\mathcal{O}(\log n\log\log n)$. Finally, for $k \ge 3$, we investigate values and bounds for some functions related to the pouring problem.

math.CO↗

arXiv2024

Extending the definition of set tolerances

Optimal solutions of combinatorial optimization problems can be sensitive to changes in the cost of one or more elements of the ground set E. Single and set tolerances measure the supremum / infimum possible change such that the current solution remains optimal for cost changes in one or more elements. The current definition does not apply to all elements of E or to all subsets of E. In this work, we broaden the definition to all elements for single tolerances and to all subsets of elements for set tolerances, while proving that key theoretical and computational properties still apply.

cs.DM↗

arXiv2024

On the Metric Dimension of $K_a \times K_b \times K_c$

In this work we determine the metric dimension of $ K_a \times K_b \times K_c$ for all $a,b,c\in \mathbb N$ with $ a \le b \le c $ as follows. For $3a c$, it is $\left \lfloor \frac{2}{3}(b+c-1) \right \rfloor$, and for $3a=b+c$, it is $\left \lfloor \frac{a+b+c}{2} \right \rfloor -1 $. The only open case is $3a>b+c$, where two values are possible, namely $\left \lfloor \frac{a+b+c}{2} \right \rfloor -1 $ and $\left \lfloor \frac{a+b+c}{2} \right \rfloor $. This result extends previous results of Cácere et al., who computed the metric dimension of $ K_a \times K_b$, and of Drewes and Jäger, who computed the metric dimension of $ K_a \times K_a \times K_a$. We prove our result by introducing and analyzing a new variant of Static Black-Peg Mastermind, in which each peg has its own permitted set of colors. For all cases, we present strategies which we prove to be both feasible and optimal. Our main result follows, as the number of questions of these strategies is equal to the metric dimension of $K_a \times K_b \times K_c$.

math.CO↗

arXiv2024

Enumeration of Row-Column Designs

We computationally completely enumerate a number of types of row-column designs up to isotopism, including double, sesqui and triple arrays as known from the literature, and two newly introduced types that we call mono arrays and AO-arrays. We calculate autotopism group sizes for the designs we generate. For larger parameter values, where complete enumeration is not feasible, we generate examples of some of the designs, and generate exhaustive lists of admissible parameters. For some admissible parameter sets, we prove non-existence results. We also give some explicit constructions of sesqui arrays, mono arrays and AO-arrays, and investigate connections to Youden rectangles and binary pseud Youden designs.

math.CO↗

arXiv2024

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR \emph{homogeneous}. In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR \emph{transitive}. We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix \emph{stepwise} to refer to sets of MOLR with this property. Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

math.CO↗

arXiv2023

Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs

In this paper we first study $k \times n$ Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where $k = n-1$, in a large scale computer search. In particular, we verify the previous counts for $(n,k) = (7,3), (7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call \emph{near Youden rectangles}. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.

math.CO↗

arXiv2022

Super Domination: Graph Classes, Products and Enumeration

The dominating set problem (DSP) is one of the most famous problems in combinatorial optimization. It is defined as follows. For a given simple graph $G=(V,E)$, a dominating set of $G$ is a subset $S\subseteq V$ such that every vertex in $ V \setminus S$ is adjacent to at least one vertex in $S$. Furthermore, the DSP is the problem of finding a minimum-size dominating set and the corresponding minimum size, the domination number of $G$. In this, work we investigate a variant of the DSP, the super dominating set problem (SDSP), which has attracted much attention during the last years. A dominating set $S$ is called a super dominating set of $G$, if for every vertex $u\in \overline{S}=V \setminus S$, there exists a $v\in S$ such that $N(v)\cap \overline{S}=\{u\}$. Analogously, the SDSP is to find a minimum-size super dominating set, and the corresponding minimum size, the super domination number of $G$. The decision variants of both the DSP and the SDSP have shown to be $\mathcal{NP}$-hard. In this paper, we present tight bounds for the super domination number of the neighbourhood corona product, $r$-gluing, and the Hajós sum of two graphs. Additionally, we present infinite families of graphs attaining our bounds. Finally, we give the exact number of minimum size super dominating sets for some graph classes. In particular, the number of super dominating sets for cycles has quite surprising properties as it varies between values of the set $\{4,n,2n,\frac{5n^2-10n}{8}\}$ based on $n\mod4$.

math.CO↗

arXiv2022

Optimal Strategies for Static Black-Peg AB Game With Two and Three Pegs

The AB~Game is a game similar to the popular game Mastermind. We study a version of this game called Static Black-Peg AB~Game. It is played by two players, the codemaker and the codebreaker. The codemaker creates a so-called secret by placing a color from a set of $c$ colors on each of $p \le c$ pegs, subject to the condition that every color is used at most once. The codebreaker tries to determine the secret by asking questions, where all questions are given at once and each question is a possible secret. As an answer the codemaker reveals the number of correctly placed colors for each of the questions. After that, the codebreaker only has one more try to determine the secret and thus to win the game. For given $p$ and $c$, our goal is to find the smallest number $k$ of questions the codebreaker needs to win, regardless of the secret, and the corresponding list of questions, called a $(k+1)$-strategy. We present a $\lceil 4c/3 \rceil-1)$-strategy for $p=2$ for all $c \ge 2$, and a $\lfloor (3c-1)/2 \rfloor$-strategy for $p=3$ for all $c \ge 4$ and show the optimality of both strategies, i.e., we prove that no $(k+1)$-strategy for a smaller $k$ exists.

math.CO↗

arXiv2018

Triples of Orthogonal Latin and Youden Rectangles For Small Orders

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of $k \times 8$ rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism group is non-trivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of $4 \times 8$ rectangles, formed by juxtaposing two selected copies of complete sets of MOLS of order 4.

math.CO↗

arXiv2014

An Effective Algorithm for and Phase Transitions of the Directed Hamiltonian Cycle Problem

The Hamiltonian cycle problem (HCP) is an important combinatorial problem with applications in many areas. It is among the first problems used for studying intrinsic properties, including phase transitions, of combinatorial problems. While thorough theoretical and experimental analyses have been made on the HCP in undirected graphs, a limited amount of work has been done for the HCP in directed graphs (DHCP). The main contribution of this work is an effective algorithm for the DHCP. Our algorithm explores and exploits the close relationship between the DHCP and the Assignment Problem (AP) and utilizes a technique based on Boolean satisfiability (SAT). By combining effective algorithms for the AP and SAT, our algorithm significantly outperforms previous exact DHCP algorithms, including an algorithm based on the award-winning Concorde TSP algorithm. The second result of the current study is an experimental analysis of phase transitions of the DHCP, verifying and refining a known phase transition of the DHCP.

cs.AI↗

arXiv2013

The Worst Case Number of Questions in Generalized AB Game with and without White-peg Answers

The AB game is a two-player game, where the codemaker has to choose a secret code and the codebreaker has to guess it in as few questions as possible. It is a variant of the famous Mastermind game, with the only difference that all pegs in both, the secret and the questions must have distinct colors. In this work, we consider the Generalized AB game, where for given arbitrary numbers $p$, $c$ with $p \le c$ the secret code consists of $p$ pegs each having one of $c$ colors and the answer consists only of a number of black and white pegs. There the number of black pegs equals the number of pegs matching in the corresponding question and the secret in position and color, and the number of white pegs equals the additional number of pegs matching in the corresponding question and the secret only in color. We consider also a variant of the Generalized AB game, where the information of white pegs is omitted. This variant is called Generalized Black-peg AB game. Let $\ab(p,c)$ and $\abb(p,c)$ be the worst case number of questions for Generalized AB game and Generalized Black-peg AB game, respectively. Combining a computer program with theoretical considerations, we confirm known exact values of $\ab(2,c)$ and $\ab(3,c)$ and prove tight bounds for $\ab(4,c)$. Furthermore, we present exact values for $\abb(2,c)$ and $\abb(3,c)$ and tight bounds for $\abb(4,c)$.

cs.GT↗