arXiv subjects

Malte Tutas

Publications and source records attributed to Malte Tutas.

6 recordsLinked to original sources

arXiv2025

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 $2^{\Omega(d)}$. 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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arXiv2025

Improved Approximation Algorithms for Three-Dimensional Knapsack

We study the three-dimensional Knapsack (3DK) problem, in which we are given a set of axis-aligned cuboids with associated profits and an axis-aligned cube knapsack. The objective is to find a non-overlapping axis-aligned packing (by translation) of the maximum profit subset of cuboids into the cube. The previous best approximation algorithm is due to Diedrich, Harren, Jansen, Th\"{o}le, and Thomas (2008), who gave a $(7+\varepsilon)$-approximation algorithm for 3DK and a $(5+\varepsilon)$-approximation algorithm for the variant when the items can be rotated by 90 degrees around any axis, for any constant $\varepsilon>0$. Chleb\'{\i}k and Chleb\'{\i}kov\'{a} (2009) showed that the problem does not admit an asymptotic polynomial-time approximation scheme. We provide an improved polynomial-time $(139/29+\varepsilon) \approx 4.794$-approximation algorithm for 3DK and $(30/7+\varepsilon) \approx 4.286$-approximation when rotations by 90 degrees are allowed. We also provide improved approximation algorithms for several variants such as the cardinality case (when all items have the same profit) and uniform profit-density case (when the profit of an item is equal to its volume). Our key technical contribution is container packing -- a structured packing in 3D such that all items are assigned into a constant number of containers, and each container is packed using a specific strategy based on its type. We first show the existence of highly profitable container packings. Thereafter, we show that one can find near-optimal container packing efficiently using a variant of the Generalized Assignment Problem (GAP).

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arXiv2024

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 $(n r \Delta)^{O(r)} \log(\|b\|_\infty),$ where $n$ is the number of blocks, $r$ the number of rows in the upper blocks and $\Delta=\|A\|_\infty$. 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.

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arXiv2024

FPT Algorithms using Minimal Parameters for a Generalized Version of Maximin Shares

We study the computational complexity of fairly allocating indivisible, mixed-manna items. For basic measures of fairness, this problem is hard in general. Thus, research has flourished concerning input classes where efficient algorithms exist, both for the purpose of establishing theoretical boundaries and for the purpose of designing practical algorithms for real-world instances. Notably, the paradigm of fixed-parameter tractability (FPT) has lead to new insights and improved algorithms for a variety of fair allocation problems; see, for example, Bleim et al. (IJCAI 16), Aziz et al. (AAAI 17), Bredereck et al. (EC 19) and Kulkarni et al. (EC 21). Our focus is the fairness measure maximin shares (MMS). Motivated by the general non-existence of MMS allocations, Aziz et al. (AAAI 17) studied optimal MMS allocations, namely solutions that achieve the best $\alpha$-approximation for the maximin share value of every agent. These allocations are guaranteed to exist, prompting the important open question of whether optimal MMS allocations can be computed efficiently. We answer this question affirmatively by providing FPT algorithms to output optimal MMS allocations. Furthermore, our techniques extend to find allocations that optimize alternative objectives, such as minimizing the additive approximation, and maximizing some variants of global welfare. In fact, all our algorithms are designed for a more general MMS problem in machine scheduling. Here, each mixed-manna item (job) must be assigned to an agent (machine) and has a processing time and a deadline. We develop efficient algorithms running in FPT time. Formally, we present polynomial time algorithms w.r.t. the input size times some function dependent on the parameters that yield optimal maximin-value approximations (among others) when parameterized by a set of natural parameters

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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.

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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.

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