# Kim-Manuel KleinUniversität zu Lübeck · Institut für Theoretische Informatik

Kim-Manuel Klein

## About

42

Publications

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439

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## Publications

Publications (42)

The Euclidean algorithm is one of the oldest algorithms known to mankind. Given two integral numbers $a_1$ and $a_2$, it computes the greatest common divisor (gcd) of $a_1$ and $a_2$ in a very elegant way. From a lattice perspective, it computes a basis of the lattice generated by $a_1$ and $a_2$ as $\gcd(a_1,a_2) \mathbb{Z} = a_1 \mathbb{Z} + a_2...

We study the general integer programming problem where the number of variables n is a variable part of the input. We consider two natural parameters of the constraint matrix A: its numeric measure a and its sparsity measure d. We present an algorithm for solving integer programming in time [Formula: see text], where g is some computable function of...

The Euclidean algorithm is one of the oldest algorithms known to mankind. Given two integral numbers a and b , it computes the greatest common divisor (gcd) of a and b in a very elegant way. From a lattice perspective, it computes a basis of the sum of two one-dimensional lattices a Z and b Z as gcd(a , b)Z = a Z + b Z. In this paper, we show that...

In this paper we study the computational complexity of solving a class of block structured integer programs (IPs) - so called multistage stochastic IPs. A multistage stochastic IP is an IP of the form min { c ⊺ x ∣ Ax = b , x ≥ 0 , x integral} where the constraint matrix A consists of small block matrices ordered on the diagonal line and for each s...

The starting point of this paper is the problem of scheduling $n$ jobs with processing times and due dates on a single machine so as to minimize the total processing time of tardy jobs, i.e., $1||\sum p_j U_j$. This problem was identified by Bringmann et al.~(Algorithmica 2022) as a natural subquadratic-time special case of the classic $1||\sum w_j...

We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called 2-stage stochastic. A 2-stage stochastic ILP is an integer program of the form min{cTx∣Ax=b,ℓ≤x≤u,x∈Zr+ns} where the constraint matrix A∈Znt×r+ns consists of n matrices Ai∈Zt×r on the vertical line an...

In this paper we study the computational complexity of solving a class of block structured integer programs (IPs) - so called multistage stochastic IPs. A multistage stochastic IP is an IP of the form $\max \{ c^T x \mid \mathcal{A} x = b, \,l \leq x \leq u,\, x\text{ integral} \}$ where the constraint matrix $\mathcal{A}$ consists of small block m...

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form $$\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \in {\mathbb {Z}}^{s + nt} \}$$ max { c T x ∣ A x = b , l ≤ x ≤ u , x ∈ Z s + n t } where the cons...

Consider positive integral solutions $x \in \mathbb{Z}^{n+1}$ to the equation $a_0 x_0 + \ldots + a_n x_n = t$. In the so called unbounded subset sum problem, the objective is to decide whether such a solution exists, whereas in the Frobenius problem, the objective is to compute the largest $t$ such that there is no such solution. In this paper we...

We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called 2-stage stochastic. A 2-stage stochastic ILP is an integer program of the form min{cTx∣Ax=b,ℓ≤x≤u,x∈Zr+ns} where the constraint matrix A∈Znt×r+ns consists of n matrices Ai∈Zt×r on the vertical line an...

We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called $2$-stage stochastic. A $2$-stage stochastic ILP is an integer program of the form $\min \{w^T x \mid \mathcal{A} x = b, L \leq x \leq U, x \in \mathbb{Z}^{s + nt} \}$ where the constraint matrix $\ma...

We consider the bin packing problem with d different item sizes and revisit the structure theorem given by Goemans and Rothvoß about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new...

The vertices of the integer hull are the integral equivalent to the well-studied basic feasible solutions of linear programs. In this paper we give new bounds on the number of non-zero components -- their support -- of these vertices matching either the best known bounds or improving upon them. While the best known bounds make use of deep technique...

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form of multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form where the constraint matrix consists roughly of n repetitions of a block matrix on the vertical line and n repetitions of a matrix on the diagonal. In this paper we impro...

We introduce a very natural generalization of the well-known problem of simultaneous congruences. Instead of searching for a positive integer $s$ that is specified by $n$ fixed remainders modulo integer divisors $a_1,\dots,a_n$ we consider remainder intervals $R_1,\dots,R_n$ such that $s$ is feasible if and only if $s$ is congruent to $r_i$ modulo...

Machine scheduling is a fundamental optimization problem in computer science. The task of scheduling a set of jobs on a given number of machines and minimizing the makespan is well studied and among other results, we know that EPTAS's for machine scheduling on identical machines exist. Das and Wiese initiated the research on a generalization of mak...

We study the general integer programming problem where the number of variables $n$ is a variable part of the input. We consider two natural parameters of the constraint matrix $A$: its numeric measure $a$ and its sparsity measure $d$. We show that integer programming can be solved in time $g(a,d)\textrm{poly}(n,L)$, where $g$ is some computable fun...

We consider so called $2$-stage stochastic integer programs (IPs) and their generalized form of multi-stage stochastic IPs. A $2$-stage stochastic IP is an integer program of the form $\max \{ c^T x \mid Ax = b, l \leq x \leq u, x \in \mathbb{Z}^n \}$ where the constraint matrix $A \in \mathbb{Z}^{s \times r}$ consists roughly of $n$ repetition of...

Machine scheduling is a fundamental optimization problem in computer science. The task of scheduling a set of jobs on a given number of machines and minimizing the makespan is well studied and among other results, we know that EPTAS's for machine scheduling on identical machines exist. Das and Wiese initiated the research on a generalization of mak...

Integer programs (IPs) are one of the fundamental tools used to solve combinatorial problems in theory and practice. Understanding the structure of solutions of IPs is thus helpful to argue about the existence of solutions with a certain simple structure, leading to significant algorithmic improvements. Typical examples for such structural properti...

Integer programs (IPs) are one of the fundamental tools used to solve combinatorial problems in theory and practice. Understanding the structure of solutions of IPs is thus helpful to argue about the existence of solutions with a certain simple structure, leading to significant algorithmic improvements. Typical examples for such structural properti...

We consider integer programming problems $\max \{ c^T x : \mathcal{A} x = b, l \leq x \leq u, x \in \mathbb{Z}^{nt}\}$ where $\mathcal{A}$ has a (recursive) block-structure generalizing "$n$-fold integer programs" which recently received considerable attention in the literature. An $n$-fold IP is an integer program where $\mathcal{A}$ consists of $...

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems, where a set of items has to be placed in multiple target locations. Herein a configuration describes a possible placement on one of the target locations, and the IP is used to chose suitable configur...

We consider the relaxed online strip packing problem, where rectangular items arrive online and have to be packed into a strip of fixed width such that the packing height is minimized. Thereby, repacking of previously packed items is allowed. The amount of repacking is measured by the migration factor, defined as the total size of repacked items di...

This paper studies Upper Domination, i.e., the problem of computing the maximum cardinality of a minimal dominating set in a graph with respect to classical and parameterised complexity as well as approximability.

We consider Upper Domination, the problem of finding a maximum cardinality minimal dominating set in a graph. We show that this problem does not admit an \(n^{1-\epsilon }\) approximation for any \(\epsilon >0\), making it significantly harder than Dominating Set, while it remains hard even on severely restricted special cases, such as cubic graphs...

This paper studies Upper Domination, i.e., the problem of computing the maximum cardinality of a minimal dominating set in a graph, with a focus on parameterised complexity. Our main results include W[1]-hardness for Upper Domination, contrasting FPT membership for the parameterised dual Co-Upper Domination. The study of structural properties also...

We consider the bin packing problem with $d$ different item sizes and revisit the structure theorem given by Goemans and Rothvo\ss [6] about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of...

Makespan scheduling on identical machines is one of the most basic and fundamental packing problems studied in the discrete optimization literature. It asks for an assignment of $n$ jobs to a set of $m$ identical machines that minimizes the makespan. The problem is strongly NP-hard, and thus we do not expect a $(1+\epsilon)$-approximation algorithm...

We consider the fully dynamic bin packing problem, where items arrive and
depart in an online fashion. The goal is to minimize the number of used bins at
every timestep while repacking of already packed items is allowed. Ivkovic and
Lloyd [IL98] have developed an algorithm with asymptotic competitive ratio of
$5/4$ using $O(\log n)$ (amortized) shi...

In this paper we develop general LP and ILP techniques to find an approximate
solution with improved objective value close to an existing solution. The task
of improving an approximate solution is closely related to a classical theorem
of Cook et al. in the sensitivity analysis for LPs and ILPs. This result is
often applied in designing robust algo...

Traversierende Baumautomaten wurden bereits 1971 von Aho und Ullman untersucht. Die Berechnungsstärke dieses Modells konnte jedoch erst in den letzten Jahren besser eingeordnet werden und mit klassischen Baumautomaten-Modellen verglichen werden.
Um die Ausdrucksmächtigkeit von traversierenden Baumautomaten weiter zu erhöhen, kann man verschiedene...