arXiv subjects

Malin Rau

Publications and source records attributed to Malin Rau.

17 recordsLinked to original sources

arXiv2025

Improved Approximation Algorithms for Three-Dimensional Bin Packing

We study two fundamental three-dimensional (3D) geometric packing problems: 3D (Geometric) Bin Packing (3D-BP), and 3D Minimum Volume Bounding Box (3D-MVBB), where given a set of 3D (rectangular) cuboids, the goal is to find an axis-aligned nonoverlapping packing of all cuboids. In 3D-BP, we need to pack the given cuboids into the minimum number of unit cube bins. In 3D-MVBB, the goal is to pack them into a cuboid box of minimum volume. It is NP-hard to even decide whether a set of rectangles can be packed into a unit square bin -- giving an (absolute) approximation hardness of $2$ for 3D-BP. The previous best (absolute) approximation for both the problems follows from a result of Buchwald and Scheithauer (Int.~Trans.~Oper.~Res., 2016), yielding approximation ratios of $11$, and $5+\varepsilon$, respectively, for 3D-BP and 3D-MVBB. We provide improved approximation ratios of $6$, and $3+\varepsilon$, respectively, for the two problems, for any constant $\varepsilon > 0$. For 3D-BP, in the asymptotic regime, Bansal, Correa, Kenyon, and Sviridenko (Math.~Oper.~Res., 2006) showed that there is no asymptotic polynomial-time approximation scheme (APTAS) even when all items have the same height. Caprara (Math.~Oper.~Res., 2008) gave an asymptotic approximation ratio of $T_{\infty}^2 + \varepsilon\approx 2.86$, where $T_{\infty}$ is the well-known Harmonic constant in Bin Packing. We provide an algorithm with an improved asymptotic approximation ratio of $3T_{\infty}/2 +\varepsilon \approx 2.54$. Further, we show that unlike 3D-BP, 3D-MVBB admits an APTAS.

cs.CG↗

arXiv2024

A Tight ($3/2 + \varepsilon$)-Approximation Algorithm for Demand Strip Packing

We consider the Demand Strip Packing problem (DSP), in which we are given a set of jobs, each specified by a processing time and a demand. The task is to schedule all jobs such that they are finished before some deadline $D$ while minimizing the peak demand, i.e., the maximum total demand of tasks executed at any point in time. DSP is closely related to the Strip Packing problem (SP), in which we are given a set of axis-aligned rectangles that must be packed into a strip of fixed width while minimizing the maximum height. DSP and SP are known to be NP-hard to approximate to within a factor below $\frac{3}{2}$. To achieve the essentially best possible approximation guarantee, we prove a structural result. Any instance admits a solution with peak demand at most $\big(\frac32+\varepsilon\big)OPT$ satisfying one of two properties. Either (i) the solution leaves a gap for a job with demand $OPT$ and processing time $\mathcal O(\varepsilon D)$ or (ii) all jobs with demand greater than $\frac{OPT}2$ appear sorted by demand in immediate succession. We then provide two efficient algorithms that find a solution with maximum demand at most $\big(\frac32+\varepsilon\big)OPT$ in the respective case. A central observation, which sets our approach apart from previous ones for DSP, is that the properties (i) and (ii) need not be efficiently decidable: We can simply run both algorithms and use whichever solution is the better one.

cs.DS↗

arXiv2024

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, G\'alvez 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.

cs.DS↗

arXiv2023

Efficient Approximate Recovery from Pooled Data Using Doubly Regular Pooling Schemes

In the pooled data problem we are given $n$ agents with hidden state bits, either $0$ or $1$. The hidden states are unknown and can be seen as the underlying ground truth $σ$. To uncover that ground truth, we are given a querying method that queries multiple agents at a time. Each query reports the sum of the states of the queried agents. Our goal is to learn the hidden state bits using as few queries as possible. So far, most literature deals with exact reconstruction of all hidden state bits. We study a more relaxed variant in which we allow a small fraction of agents to be classified incorrectly. This becomes particularly relevant in the noisy variant of the pooled data problem where the queries' results are subject to random noise. In this setting, we provide a doubly regular test design that assigns agents to queries. For this design we analyze an approximate reconstruction algorithm that estimates the hidden bits in a greedy fashion. We give a rigorous analysis of the algorithm's performance, its error probability, and its approximation quality. As a main technical novelty, our analysis is uniform in the degree of noise and the sparsity of $σ$. Finally, simulations back up our theoretical findings and provide strong empirical evidence that our algorithm works well for realistic sample sizes.

cs.LG↗

arXiv2023

Dynamic Averaging Load Balancing on Arbitrary Graphs

In this paper we study dynamic averaging load balancing on general graphs. We consider infinite time and dynamic processes, where in every step new load items are assigned to randomly chosen nodes. A matching is chosen, and the load is averaged over the edges of that matching. We analyze the discrete case where load items are indivisible, moreover our results also carry over to the continuous case where load items can be split arbitrarily. For the choice of the matchings we consider three different models, random matchings of linear size, random matchings containing only single edges, and deterministic sequences of matchings covering the whole graph. We bound the discrepancy, which is defined as the difference between the maximum and the minimum load. Our results cover a broad range of graph classes and, to the best of our knowledge, our analysis is the first result for discrete and dynamic averaging load balancing processes. As our main technical contribution we develop a drift result that allows us to apply techniques based on the effective resistance in an electrical network to the setting of dynamic load balancing.

cs.DC↗

arXiv2022

Scheduling with Many Shared Resources

Consider the many shared resource scheduling problem where jobs have to be scheduled on identical parallel machines with the goal of minimizing the makespan. However, each job needs exactly one additional shared resource in order to be executed and hence prevents the execution of jobs that need the same resource while being processed. Previously a $(2m/(m+1))$-approximation was the best known result for this problem. Furthermore, a $6/5$-approximation for the case with only two machines was known as well as a PTAS for the case with a constant number of machines. We present a simple and fast 5/3-approximation and a much more involved but still reasonable 1.5-approximation. Furthermore, we provide a PTAS for the case with only a constant number of machines, which is arguably simpler and faster than the previously known one, as well as a PTAS with resource augmentation for the general case. The approximation schemes make use of the N-fold integer programming machinery, which has found more and more applications in the field of scheduling recently. It is plausible that the latter results can be improved and extended to more general cases. Lastly, we give a $5/4 - \varepsilon$ inapproximability result for the natural problem extension where each job may need up to a constant number (in particular $3$) of different resources.

cs.DS↗

arXiv2022

Inference of a Rumor's Source in the Independent Cascade Model

We consider the so-called Independent Cascade Model for rumor spreading or epidemic processes popularized by Kempe et al.\ [2003]. In this model, a small subset of nodes from a network are the source of a rumor. In discrete time steps, each informed node "infects" each of its uninformed neighbors with probability $p$. While many facets of this process are studied in the literature, less is known about the inference problem: given a number of infected nodes in a network, can we learn the source of the rumor? In the context of epidemiology this problem is often referred to as patient zero problem. It belongs to a broader class of problems where the goal is to infer parameters of the underlying spreading model, see, e.g., Lokhov [NeurIPS'16] or Mastakouri et al. [NeurIPS'20]. In this work we present a maximum likelihood estimator for the rumor's source, given a snapshot of the process in terms of a set of active nodes $X$ after $t$ steps. Our results show that, for cycle-free graphs, the likelihood estimator undergoes a non-trivial phase transition as a function $t$. We provide a rigorous analysis for two prominent classes of acyclic network, namely $d$-regular trees and Galton-Watson trees, and verify empirically that our heuristics work well in various general networks.

cs.SI↗

arXiv2022

On the Hierarchy of Distributed Majority Protocols

We study the Consensus problem among $n$ agents, defined as follows. Initially, each agent holds one of two possible opinions. The goal is to reach a consensus configuration in which every agent shares the same opinion. To this end, agents randomly sample other agents and update their opinion according to a simple update function depending on the sampled opinions. We consider two communication models: the gossip model and a variant of the population model. In the gossip model, agents are activated in parallel, synchronous rounds. In the population model, one agent is activated after the other in a sequence of discrete time steps. For both models we analyze the following natural family of majority processes called $j$-Majority: when activated, every agent samples $j$ other agents uniformly at random (with replacement) and adopts the majority opinion among the sample (breaking ties uniformly at random). As our main result we show a hierarchy among majority protocols: $(j+1)$-Majority (for $j > 1$) converges stochastically faster than $j$-Majority for any initial opinion configuration. In our analysis we use Strassen's Theorem to prove the existence of a coupling. This gives an affirmative answer for the case of two opinions to an open question asked by Berenbrink et al. [2017].

cs.DC↗

arXiv2022

Asynchronous Opinion Dynamics in Social Networks

Opinion spreading in a society decides the fate of elections, the success of products, and the impact of political or social movements. The model by Hegselmann and Krause is a well-known theoretical model to study such opinion formation processes in social networks. In contrast to many other theoretical models, it does not converge towards a situation where all agents agree on the same opinion. Instead, it assumes that people find an opinion reasonable if and only if it is close to their own. The system converges towards a stable situation where agents sharing the same opinion form a cluster, and agents in different clusters do not \mbox{influence each other.} We focus on the social variant of the Hegselmann-Krause model where agents are connected by a social network and their opinions evolve in an iterative process. When activated, an agent adopts the average of the opinions of its neighbors having a similar opinion. By this, the set of influencing neighbors of an agent may change over time. To the best of our knowledge, social Hegselmann-Krause systems with asynchronous opinion updates have only been studied with the complete graph as social network. We show that such opinion dynamics with random agent activation are guaranteed to converge for any social network. We provide an upper bound of $\mathcal{O}(n|E|^2 (\varepsilon/\delta)^2)$ on the expected number of opinion updates until convergence, where $|E|$ is the number of edges of the social network. For the complete social network we show a bound of $\mathcal{O}(n^3(n^2 + (\varepsilon/\delta)^2))$ that represents a major improvement over the previously best upper bound of $\mathcal{O}(n^9 (\varepsilon/\delta)^2)$. Our bounds are complemented by simulations that indicate asymptotically matching lower bounds.

cs.DS↗

arXiv2021

Closing the gap for single resource constraint scheduling

In the problem called single resource constraint scheduling, we are given $m$ identical machines and a set of jobs, each needing one machine to be processed as well as a share of a limited renewable resource $R$. A schedule of these jobs is feasible if, at each point in the schedule, the number of machines and resources required by jobs processed at this time is not exceeded. It is NP-hard to approximate this problem with a ratio better than $3/2$. On the other hand, the best algorithm so far has an absolute approximation ratio of $2+\varepsilon$. This paper presents an algorithm with absolute approximation ratio~$(3/2+\varepsilon)$, which closes the gap between inapproximability and best algorithm except for a negligible small~$\varepsilon$.

cs.DS↗

arXiv2021

Peak Demand Minimization via Sliced Strip Packing

We study Nonpreemptive Peak Demand Minimization (NPDM) problem, where we are given a set of jobs, specified by their processing times and energy requirements. The goal is to schedule all jobs within a fixed time period such that the peak load (the maximum total energy requirement at any time) is minimized. This problem has recently received significant attention due to its relevance in smart-grids. Theoretically, the problem is related to the classical strip packing problem (SP). In SP, a given set of axis-aligned rectangles must be packed into a fixed-width strip, such that the height of the strip is minimized. NPDM can be modeled as strip packing with slicing and stacking constraint: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions where two slices of the same rectangle are intersected by the same vertical line. Nonpreemption enforces the slices to be placed in contiguous horizontal locations (but may be placed at different vertical locations). We obtain a $(5/3+ε)$-approximation algorithm for the problem. We also provide an asymptotic efficient polynomial-time approximation scheme (AEPTAS) which generates a schedule for almost all jobs with energy consumption $(1+ε)OPT$. The remaining jobs fit into a thin container of height $1$. The previous best for NPDM was 2.7 approximation based on FFDH [Ranjan et al. 2015]. One of our key ideas is providing several new lower bounds on the optimal solution of a geometric packing, which could be useful in other related problems. These lower bounds help us to obtain approximative solutions based on Steinberg's algorithm in many cases. In addition, we show how to split schedules generated by the AEPTAS into few segments and to rearrange the corresponding jobs to insert the thin container mentioned above.

cs.DS↗

arXiv2020

Solving Packing Problems with Few Small Items Using Rainbow Matchings

An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number $k$ of small items. Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by $k$. The algorithms are randomized with one-sided error and run in time $4^{k} \cdot k! \cdot n^{O(1)}$. To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. We also present a deterministic fixed-parameter for Bin Packing with run time $(k!)^{2}\cdot k \cdot 2^{k}\cdot n\cdot \log(n)$.

cs.DS↗

arXiv2019

Linear Time Algorithms for Multiple Cluster Scheduling and Multiple Strip Packing

We study the Multiple Cluster Scheduling problem and the Multiple Strip Packing problem. For both problems, there is no algorithm with approximation ratio better than $2$ unless $P = NP$. In this paper, we present an algorithm with approximation ratio $2$ and running time $O(n)$ for both problems. While a $2$ approximation was known before, the running time of the algorithm is at least $Ω(n^{256})$ in the worst case. Therefore, an $O(n)$ algorithm is surprising and the best possible. We archive this result by calling an AEPTAS with approximation guarantee $(1+\varepsilon)OPT +p_{\max}$ and running time of the form $O(n\log(1/\varepsilon)+ f(1/\varepsilon))$ with a constant $\varepsilon$ to schedule the jobs on a single cluster. This schedule is then distributed on the $N$ clusters in $O(n)$. Moreover, this distribution technique can be applied to any variant of of Multi Cluster Scheduling for which there exists an AEPTAS with additive term $p_{\max}$. While the above result is strong from a theoretical point of view, it might not be very practical due to a large hidden constant caused by calling an AEPTAS with a constant $\varepsilon \geq 1/8$ as subroutine. Nevertheless, we point out that the general approach of finding first a schedule on one cluster and then distributing it onto the other clusters might come in handy in practical approaches. We demonstrate this by presenting a practical algorithm with running time $O(n\log(n))$, with out hidden constants, that is a $9/4$-approximation for one third of all possible instances, i.e, all instances where the number of clusters is dividable by $3$, and has an approximation ratio of at most $2.3$ for all instances with at least $9$ clusters.

cs.DS↗

arXiv2019

Closing the gap for pseudo-polynomial strip packing

The set of 2-dimensional packing problems builds an important class of optimization problems and Strip Packing together with 2-dimensional Bin Packing and 2-dimensional Knapsack is one of the most famous of these problems. Given a set of rectangular axis parallel items and a strip with bounded width and infinite height the objective is to find a packing of the items into the strip which minimizes the packing height. We speak of pseudo-polynomial Strip Packing if we consider algorithms with pseudo-polynomial running time with respect to the width of the strip. It is known that there is no pseudo-polynomial algorithm for Strip Packing with a ratio better than $5/4$ unless $\mathrm{P} = \mathrm{NP}$. The best algorithm so far has a ratio of $(4/3 + \varepsilon)$. In this paper, we close this gap between inapproximability result and best known algorithm by presenting an algorithm with approximation ratio $(5/4 + \varepsilon)$ and thus categorize the problem accurately. The algorithm uses a structural result which states that each optimal solution can be transformed such that it has one of a polynomial number of different forms. The strength of this structural result is that it applies to other problem settings as well for example to Strip Packing with rotations (90 degrees) and Contiguous Moldable Task Scheduling. This fact enabled us to present algorithms with approximation ratio $(5/4 + \varepsilon)$ for these problems as well.

cs.DM↗

arXiv2018

Empowering the Configuration-IP $-$ New PTAS Results for Scheduling with Setups Times

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 configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and therefore be solved efficiently. As an application, we consider scheduling problems with setup times, in which a set of jobs has to be scheduled on a set of identical machines, with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time $f(1/\varepsilon)\times \mathrm{poly}(|I|)$ with a single exponential term in $f$ for the first and a double exponential one for the second case. Previously, only constant factor approximations of $5/3$ and $4/3 + \varepsilon$ respectively were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.

cs.DS↗

arXiv2017

Improved approximation for two dimensional strip packing with polynomial bounded width

We study the well-known two-dimensional strip packing problem. Given is a set of rectangular axis-parallel items and a strip of width $W$ with infinite height. The objective is to find a packing of these items into the strip, which minimizes the packing height. Lately, it has been shown that the lower bound of $3/2$ of the absolute approximation ratio can be beaten when we allow a pseudo-polynomial running-time of type $(n W)^{f(1/\varepsilon)}$. If $W$ is polynomially bounded by the number of items, this is a polynomial running-time. We present a pseudo-polynomial algorithm with approximation ratio $4/3 +\varepsilon$ and running time $(n W)^{1/\varepsilon^{\mathcal{O}(2^{1/\varepsilon})}}$.

cs.DS↗

arXiv2017

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition.

cs.CC↗