arXiv subjects

Klaus Jansen

Publications and source records attributed to Klaus Jansen.

At least 19 recordsLinked to original sources

arXiv2026

A Fixed-Parameter Algorithm for 4-Block Integer Programming

We give a fixed-parameter tractable (FPT) algorithm with running time $f(k,\barΔ)\cdot {|I|}^{O(1)}$ for integer linear programs with 4-block structure, parameterized by the maximum block dimension $k$ and the largest absolute entry $\barΔ$ in the local-column blocks. This result resolves a long-standing open question in parameterized complexity, and answers a conjecture by Eisenbrand and Rothvoss (2026) in the positive. This result has several implications for other block-structured integer programming models, including mixed fracture number and special cases of 4-block programs with large entries outside of the diagonal. Important tools for this algorithm are structural properties of generalized $n$-fold integer programs shown by Ligthart~(2026) and an algorithm by Veselov, Gribanov, Zolotykh and Chirkov~(2020) for optimizing discrete convic functions. Independently and shortly after the first version of our paper appeared, Koutecký, Lassota, and Ligthart (2026) obtained an FPT algorithm for 4-block integer programming.

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arXiv2026

A Constructive FPT Algorithm for Integer Cone Optimization with Convex Costs

We give a deterministic fixed-parameter algorithm for integer cone optimization with convex costs. Given $τ$ rational polytopes in $\R^{d+k}$, we combine generators whose first $d$ coordinates are integral and whose remaining $k$ coordinates may be fractional. Each generator has a convex cost depending only on its integer coordinates. We minimize the total cost of nonnegative integer combinations subject to arbitrary rational linear constraints on the full typewise aggregates and generator counts. Our algorithm runs in time \[ 2^{τ\log(τ+1)\,2^{O(d)}}\cdot |I|^{O(1)}, \] where $|I|$ is the total binary input length. It constructs an optimal representation using at most $2^{2d+1}$ distinct generators per type. The number $k$ of continuous coordinates contributes only to the polynomial input factor. For a single generator type, the running time becomes $2^{2^{O(d)}}\cdot |I|^{O(1)}$. Applications include constructive FPT algorithms for Cone and Polytope Intersection, high-multiplicity bin packing, vector bin packing, and makespan minimization on uniform and unrelated machines. The double-exponential dependence on $d$ is tight for Point-in-Cone and both bin-packing problems assuming $\mathrm{NP}\nsubseteq\mathrm{SUBEXP}$. The mixed-integer formulation also yields additive EPTASs for makespan minimization, machine covering, and envy minimization on identical machines, and additive schemes fixed-parameter tractable in $1/\varepsilon+τ$ on uniform machines.

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arXiv2026

Compressing the Lawler-Moore Dynamic Program for Weighted Tardy Jobs

The classic scheduling problem of minimizing the total weight of tardy jobs ($P_m||\sum_jw_jU_j$) asks for a schedule of jobs each with a processing time, weight, and due date on a constant number of parallel identical machines. A job is late in a schedule if it completes after its due date, and an optimal schedule is one where the total weight of all late jobs is minimized. The classic dynamic programming algorithm of Lawler and Moore [Manag. Sci., 1969] solves this problem in $O(nd_{\max}^m)$ where $n$ is the number of jobs, $d_{\max}$ is the maximum due date of any job, and $m$ is the number of identical machines. We show that it is sufficient to only compute part of the dynamic program and obtain the running time $\tilde{O}(n + d_{\max}^{m + 1})$ where $\tilde{O}$ suppresses polylogarithmic factors. This affirmatively answers the open question posed by Hermelin, Molter, and Shabtay [INFORMS J. Comput., 2024], whether a running time of $\tilde{O}(n + d_{\max}^2)$ is possible on a single machine.

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arXiv2026

Minimizing the Weighted Makespan with Restarts on a Single Machine

We consider the problem of minimizing the weighted makespan on a single machine with restarts. Restarts are similar to preemptions but weaker: a job can be interrupted, but then it has to be run again from the start instead of resuming at the point of interruption later. The objective is to minimize the weighted makespan, defined as the maximum weighted completion time of jobs. We establish a lower bound of 1.4656 on the competitive ratio achievable by deterministic online algorithms. For the case where all jobs have identical processing times, we design and analyze a deterministic online algorithm that improves the competitive ratio to better than 1.274. Finally, we prove a lower bound of 1.238 for this case.

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arXiv2026

An Analysis of Decision Problems for Relational Pattern Languages under Various Constraints

Patterns are words with terminals and variables. The language of a pattern is the set of words obtained by uniformly substituting all variables with words that contain only terminals. In their original definition, patterns only allow for multiple distinct occurrences of some variables to be related by the equality relation, represented by using the same variable multiple times. In an extended notion, called relational patterns and relational pattern languages, variables may be related by arbitrary other relations, achieved by using regular patterns and relating individual variables independently from the patterns structure separately. We extend the ongoing investigation of the main decision problems for patterns (namely, the equivalence problem, the inclusion problem, and the membership problem) to relational pattern languages under a wide range of relevant individual relations, providing a comprehensive foundation in all three research directions.

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arXiv2026

An Analysis of Decision Problems for Relational Pattern Languages under Various Constraints

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arXiv2026

Improved Approximation Algorithms for Parallel Task Scheduling and Multiple Cluster Scheduling

In the problem of Parallel Task Scheduling (PTS), we are asked to schedule $n$ jobs, each with a fixed processing time and machine requirement, such that the completion time of the last job is minimized. Jansen and Rau (2019) presented an algorithm for PTS that achieves an approximation ratio of $(3/2)\text{OPT} + p_{\max}$. They additionally posed the open question whether an approximation ratio of $(4/3)\text{OPT} + p_{\max}$ is possible. In this work, we present such an algorithm with a running time of $O(n\log n)$. The problem of Multiple Cluster Scheduling (MCS) is a natural extension of PTS where we are given $N$ clusters each of $m$ machines to schedule jobs. Jansen and Rau (2019) adapted their PTS algorithm to MCS with the following results: (1) a 2 approximation, and (2) a near-linear 9/4 approximation if $N$ is divisible by 3. We improve the running time of their 2-approximation and generalize the 9/4 approximation to the general case. The 2-approximation for MCS is tight, since one cannot hope for an approximation ratio better than 2, unless P=NP [Zhuk, 2006]. In addition to our theoretical results, we implement our algorithm and show its practical applicability.

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arXiv2026

A Tight Double-Exponentially Lower Bound for High-Multiplicity Bin Packing

Consider a high-multiplicity Bin Packing instance $I$ with $d$ distinct item types. In 2014, Goemans and Rothvoss gave an algorithm with runtime ${{|I|}^2}^{O(d)}$ for this problem~[SODA'14], where $|I|$ denotes the encoding length of the instance $I$. Although Jansen and Klein~[SODA'17] later developed an algorithm that improves upon this runtime in a special case, it has remained a major open problem by Goemans and Rothvoss~[J.ACM'20] whether the doubly exponential dependency on $d$ is necessary. We solve this open problem by showing that unless the ETH fails, there is no algorithm solving the high-multiplicity Bin Packing problem in time ${{|I|}^2}^{o(d)}$. To prove this, we introduce a novel reduction from 3-SAT. The core of our construction is efficiently encoding all information from a 3-SAT instance with $n$ variables into an ILP with $O(\log(n))$ variables and constraints. This result confirms that the Goemans and Rothvoss algorithm is essentially best-possible for Bin Packing parameterized by the number $d$ of item sizes in the context of XP time algorithms.

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arXiv2026

The Support of Bin Packing is Exponential

Consider the classical Bin Packing problem with $d$ different item sizes $s_i$ and amounts of items $a_i.$ The support of a Bin Packing solution is the number of differently filled bins. In this work, we show that the lower bound on the support of this problem is $2^{Ω(d)}$. Our lower bound matches the upper bound of $2^d$ given by Eisenbrand and Shmonin [Oper.Research Letters '06] up to a constant factor. This result has direct implications for the time complexity of several Bin Packing algorithms, such as Goemans and Rothvoss [SODA '14], Jansen and Klein [SODA '17] and Jansen and Solis-Oba [IPCO '10]. To achieve our main result, we develop a technique to aggregate equality constrained ILPs with many constraints into an equivalent ILP with one constraint. Our technique contrasts existing aggregation techniques as we manage to integrate upper bounds on variables into the resulting constraint. We believe this technique can be useful for solving general ILPs or the $d$-dimensional knapsack problem.

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arXiv2026

A Practical 73/50 Approximation for Contiguous Monotone Moldable Job Scheduling

In moldable job scheduling, we are provided $m$ identical machines and $n$ jobs that can be executed on a variable number of machines. The execution time of each job depends on the number of machines assigned to execute that job. For the specific problem of monotone moldable job scheduling, jobs are assumed to have a processing time that is non-increasing in the number of machines. The previous best-known algorithms are: (1) a polynomial-time approximation scheme with time complexity $Ω(n^{g(1/\varepsilon)})$, where $g(\cdot)$ is a super-exponential function [Jansen and Thöle '08; Jansen and Land '18], (2) a fully polynomial approximation scheme for the case of $m \geq 8\frac{n}{\varepsilon}$ [Jansen and Land '18], and (3) a $\frac{3}{2}$ approximation with time complexity $O(nm\log(mn))$ [Wu, Zhang, and Chen '23]. We present a new practically efficient algorithm with an approximation ratio of $\approx (1.4593 + \varepsilon)$ and a time complexity of $O(nm \log \frac{1}{\varepsilon})$. Our result also applies to the contiguous variant of the problem. In addition to our theoretical results, we implement the presented algorithm and show that the practical performance is significantly better than the theoretical worst-case approximation ratio.

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arXiv2025

Approximation algorithms for integer programming with resource augmentation

The classic algorithm [Papadimitriou, J.ACM '81] for IPs has a running time $n^{O(m)}(m\cdot\max\{Δ,\|\textbf{b}\|_{\infty}\})^{O(m^2)}$, where $m$ is the number of constraints, $n$ is the number of variables, and $Δ$ and $\|\textbf{b}\|_{\infty}$ are, respectively, the largest absolute values among the entries in the constraint matrix and the right-hand side vector of the constraint. The running time is exponential in $m$, and becomes pseudo-polynomial if $m$ is a constant. In recent years, there has been extensive research on FPT (fixed parameter tractable) algorithms for the so-called $n$-fold IPs, which may possess a large number of constraints, but the constraint matrix satisfies a specific block structure. It is remarkable that these FPT algorithms take as parameters $Δ$ and the number of rows and columns of some small submatrices. If $Δ$ is not treated as a parameter, then the running time becomes pseudo-polynomial even if all the other parameters are taken as constants. This paper explores the trade-off between time and accuracy in solving an IP. We show that, for arbitrary small $\varepsilon>0$, there exists an algorithm for IPs with $m$ constraints that runs in ${f(m,\varepsilon)}\cdot\textnormal{poly}(|I|)$ time, and returns a near-feasible solution that violates the constraints by at most $\varepsilonΔ$. Furthermore, for $n$-fold IPs, we establish a similar result -- our algorithm runs in time that depends on the number of rows and columns of small submatrices together with $1/\varepsilon$, and returns a solution that slightly violates the constraints. Meanwhile, both solutions guarantee that their objective values are no worse than the corresponding optimal objective values satisfying the constraints. As applications, our results can be used to obtain additive approximation schemes for multidimensional knapsack as well as scheduling.

math.OC↗

arXiv2025

Equivalent Instances for Scheduling and Packing Problems

Two instances $(I,k)$ and $(I',k')$ of a parameterized problem $P$ are equivalent if they have the same set of solutions (static equivalent) or if the set of solutions of $(I,k)$ can be constructed by the set of solutions for $(I',k')$ and some computable pre-solutions. If the algorithm constructing such a (static) equivalent instance whose size is polynomial bounded runs in fixed-parameter tractable (FPT) time, we say that there exists a (static) equivalent instance for this problem. In this paper we present (static) equivalent instances for Scheduling and Knapsack problems. We improve the bound for the $\ell_1$-norm of an equivalent vector given by Eisenbrand, Hunkenschröder, Klein, Koutecký, Levin, and Onn and show how this yields equivalent instances for integer linear programs (ILPs) and related problems. We obtain an $O(MN^2\log(NU))$ static equivalent instance for feasibility ILPs where $M$ is the number of constraints, $N$ is the number of variables and $U$ is an upper bound for the $\ell_\infty$-norm of the smallest feasible solution. With this, we get an $O(n^2\log(n))$ static equivalent instance for Knapsack where $n$ is the number of items. Moreover, we give an $O(M^2N\log(NMΔ))$ kernel for feasibility ILPs where $Δ$ is an upper bound for the $\ell_\infty$-norm of the given constraint matrix. Using balancing results by Knop et al., the ConfILP and a proximity result by Eisenbrand and Weismantel we give an $O(d^2\log(p_{\max}))$ equivalent instance for LoadBalancing, a generalization of scheduling problems. Here $d$ is the number of different processing times and $p_{\max}$ is the largest processing time.

cs.CC↗

arXiv2025

Hardness of SetCover Reoptimization

We study hardness of reoptimization of the fundamental and hard to approximate SetCover problem. Reoptimization considers an instance together with a solution and a modified instance where the goal is to approximate the modified instance while utilizing the information gained by solution to the related instance. We study four different types of reoptimization for (weighted) SetCover: adding a set, removing a set, adding an element to the universe, and removing an element from the universe. A few of these cases are known to be easier to approximate than the classic SetCover problem. We show that all the other cases are essentially as hard to approximate as SetCover. The reoptimization problem of adding and removing an element in the unweighted case is known to admit a PTAS. For these settings we show that there is no EPTAS under common hardness assumptions via a novel combination of the classic way to show that a reoptimization problem is NP-hard and the relation between EPTAS and FPT.

cs.CC↗

arXiv2025

New Algorithm for Combinatorial $n$-folds and Applications

Block-structured integer linear programs (ILPs) play an important role in various application fields. We address $n$-fold ILPs where the matrix $\mathcal{A}$ has a specific structure, i.e., where the blocks in the lower part of $\mathcal{A}$ consist only of the row vectors $(1,\dots,1)$. In this paper, we propose an approach tailored to exactly these combinatorial $n$-folds. We utilize a divide and conquer approach to separate the original problem such that the right-hand side iteratively decreases in size. We show that this decrease in size can be calculated such that we only need to consider a bounded amount of possible right-hand sides. This, in turn, lets us efficiently combine solutions of the smaller right-hand sides to solve the original problem. We can decide the feasibility of, and also optimally solve, such problems in time $(n r Δ)^{O(r)} \log(\|b\|_\infty),$ where $n$ is the number of blocks, $r$ the number of rows in the upper blocks and $Δ=\|A\|_\infty$. We complement the algorithm by discussing applications of the $n$-fold ILPs with the specific structure we require. We consider the problems of (i) scheduling on uniform machines, (ii) closest string and (iii) (graph) imbalance. Regarding (i), our algorithm results in running times of $p_{\max}^{O(d)}|I|^{O(1)},$ matching a lower bound derived via ETH. For (ii) we achieve running times matching the current state-of-the-art in the general case. In contrast to the state-of-the-art, our result can leverage a bounded number of column-types to yield an improved running time. For (iii), we improve the parameter dependency on the size of the vertex cover.

cs.DS↗

arXiv2025

A $(4/3+\varepsilon)$-Approximation for Preemptive Scheduling with Batch Setup Times

We consider the $\mathcal{NP}$-hard problem $\mathrm{P} \mathbf{\vert} \mathrm{pmtn, setup=s_i} \mathbf{\vert} \mathrm{C_{\max}}$, the problem of scheduling $n$ jobs, which are divided into $c$ classes, on $m$ identical parallel machines while allowing preemption. For each class $i$ of the $c$ classes, we are given a setup time $s_i$ that is required to be scheduled whenever a machine switches from processing a job of one class to a job from another class. The goal is to find a schedule that minimizes the makespan. We give a $(4/3+\varepsilon)$-approximate algorithm with run time in $\mathcal{O}(n^2 \log(1/\varepsilon))$. For any $\varepsilon < 1/6$, this improves upon the previously best known approximation ratio of $3/2$ for this problem. Our main technical contributions are as follows. We first partition any instance into an "easy" and a "hard" part, such that a $4/3 T$-approximation for the former is easy to compute for some given makespan $T$. We then proceed to show our main structural result, namely that there always exists a $4/3 T$-approximation for any instance that has a solution with makespan $T$, where the hard part has some easy to compute properties. Finally, we obtain an algorithm that computes a $(4/3+\varepsilon)$-approximation in time n $\mathcal{O}(n^2 \log(1/\varepsilon))$ for general instances by computing solutions with the previously shown structural properties.

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arXiv2025

Robust Scheduling on Uniform Machines -- New Results Using a Relaxed Approximation Guarantee

We consider the problem of scheduling $n$ jobs on $m$ uniform machines while minimizing the makespan ($Q||C_{\max}$) and maximizing the minimum completion time ($Q||C_{\min}$) in an online setting with migration of jobs. In this online setting, the jobs are inserted or deleted over time, and at each step, the goal is to compute a near-optimal solution while reassigning some jobs, such that the overall processing time of reassigned jobs, called migration, is bounded by some factor $β$ times the processing time of the job added or removed. We propose Efficient Polynomial Time Approximation Schemes (EPTASs) with an additional load error of $\mathcal{O}(\varepsilon p_{\max})$ for both problems, with constant amortized migration factor $β$, where $p_{\max}$ is the maximum processing time in the instance over all steps. As an intermediate step, we obtain Efficient Parameterized Approximation Schemes (EPASs) for both problems, $(1+\varepsilon)$-competitive algorithms parameterized by $p_{\max}$ and the number of different processing times $d$ in an instance, with $β$ bounded in a function of $p_{\max}$, $d$ and $\varepsilon$. This is the first result in the direction of a polynomial time approximation scheme in the field of online scheduling with bounded reassignment on uniform machines; before, such results were known only for the considered problems on identical machines. Crucial to our result is a division of the machines into large and small machines depending on the current approximate objective value, allowing for different approaches on either machine set, as well as a new way of rounding the instance that does not depend on the current objective value.

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arXiv2025

Convolution and Knapsack in Higher Dimensions

In the Knapsack problem, one is given the task of packing a knapsack of a given size with items in order to gain a packing with a high profit value. An important connection to the $(\max,+)$-convolution problem has been established, where knapsack solutions can be combined by building the convolution of two sequences. This observation has been used in recent years to give conditional lower bounds but also parameterized algorithms. In this paper we carry these results into higher dimensions. We consider Knapsack where items are characterized by multiple properties -- given through a vector -- and a knapsack that has a capacity vector. The packing must not exceed any of the given capacity constraints. In order to show a similar sub-quadratic lower bound we consider a multidimensional version of $(\max, +)$-convolution. We then consider variants of this problem introduced by Cygan et al. and prove that they are all equivalent in terms of algorithms that allow for a running time sub-quadratic in the number of entries of the array. We develop a parameterized algorithm to solve higher dimensional Knapsack. The techniques we apply are inspired by an algorithm introduced by Axiotis and Tzamos. We will show that even for higher dimensional Knapsack, we can reduce the problem to convolution on one-dimensional, concave sequences, leading to an $\mathcal{O}(dn + dD \cdot \max\{Π_{i=1}^d{t_i}, t_{\max}\log t_{\max}\})$ algorithm, where $D$ is the number of different weight vectors, $t$ the capacity vector and $d$ is the dimension of the problem. Then, we use the techniques to improve the approach of Eisenbrand and Weismantel to obtain an algorithm for Integer Linear Programming with upper bounds with running time $\mathcal{O}(dn) + D \cdot \mathcal{O}(d Δ)^{d(d+1)} + T_{\mathrm{LP}}$.

cs.DS↗

arXiv2025

Hardness and Tight Approximations of Demand Strip Packing

We settle the pseudo-polynomial complexity of the Demand Strip Packing (DSP) problem: Given a strip of fixed width and a set of items with widths and heights, the items must be placed inside the strip with the objective of minimizing the peak height. This problem has gained significant scientific interest due to its relevance in smart grids[Deppert et al.\ APPROX'21, Gálvez et al.\ APPROX'21]. Smart Grids are a modern form of electrical grid that provide opportunities for optimization. They are forecast to impact the future of energy provision significantly. Algorithms running in pseudo-polynomial time lend themselves to these applications as considered time intervals, such as days, are small. Moreover, such algorithms can provide superior approximation guarantees over those running in polynomial time. Consequently, they evoke scientific interest in related problems. We prove that Demand Strip Packing is strongly NP-hard for approximation ratios below $5/4$. Through this proof, we provide novel insights into the relation of packing and scheduling problems. Using these insights, we show a series of frameworks that solve both Demand Strip Packing and Parallel Task Scheduling optimally when increasing the strip's width or number of machines. Such alterations to problems are known as resource augmentation. Applications are found when penalty costs are prohibitively large. Finally, we provide a pseudo-polynomial time approximation algorithm for DSP with an approximation ratio of $(5/4+\varepsilon)$, which is nearly optimal assuming $P\neq NP$. The construction of this algorithm provides several insights into the structure of DSP solutions and uses novel techniques to restructure optimal solutions.

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