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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

We provide several novel algorithms and lower bounds in central settings of mixed-integer (non-)linear optimization, shedding new light on classic results in the field. This includes an improvement on record running time bounds obtained from a slight extension of Lenstra’s 1983 algorithm [Math. Oper. Res. '83] to optimizing under few constraints with small coefficients. This is important for ubiquitous tasks like knapsack-, subset sum- or scheduling problems [Eisenbrand and Weismantel, SODA'18, Jansen and Rohwedder, ITCS'19].
Further, we extend our algorithm to an intermediate linear optimization problem when the matrix has many rows that exhibit 2-stage stochastic structure, which adds to a prominent line of recent results on this and similarly restricted cases [Jansen et al. ICALP'19, Cslovjecsek et al. SODA'21, Brand et al. AAAI'21, Klein, Reuter SODA'22, Cslovjecsek et al. SODA'24]. We also show that the generalization of two fundamental classes of structured constraints from these works (n-fold and 2-stage stochastic programs) to separable-convex mixed-integer optimization are harder than their mixed-integer, linear counterparts. This counters a widespread belief popularized initially by an influential paper of Hochbaum and Shanthikumar, namely that "convex separable optimization is not much harder than linear optimization" [J. ACM '90].
To obtain our algorithms, we employ the mixed Graver basis introduced by Hemmecke [Math. Prog. '03], and our work is the first to give bounds on the norm of its elements. Importantly, we use these bounds differently from how purely-integer Graver bounds are exploited in related approaches, and prove that, surprisingly, this cannot be avoided.

Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak. Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{brand_et_al:LIPIcs.ESA.2024.32, author = {Brand, Cornelius and Kouteck\'{y}, Martin and Lassota, Alexandra and Ordyniak, Sebastian}, title = {{Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {32:1--32:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.32}, URN = {urn:nbn:de:0030-drops-211033}, doi = {10.4230/LIPIcs.ESA.2024.32}, annote = {Keywords: Mixed-Integer Programming, Separable Convex Optimization, Parameterized Algorithms, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

We develop algorithms for (min,+)-Convolution and related convolution problems such as Super Additivity Testing, Convolution 3-Sum and Minimum Consecutive Subsums which use the degree of convexity of the instance as a parameter. Assuming the min-plus conjecture (Künnemann-Paturi-Schneider, ICALP'17 and Cygan et al., ICALP'17), our results interpolate in an optimal manner between fully convex instances, which can be solved in near-linear time using Legendre transformations, and general non-convex sequences, where the trivial quadratic-time algorithm is conjectured to be best possible, up to subpolynomial factors.

Cornelius Brand and Alexandra Lassota. Fast Convolutions for Near-Convex Sequences. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{brand_et_al:LIPIcs.ISAAC.2023.16, author = {Brand, Cornelius and Lassota, Alexandra}, title = {{Fast Convolutions for Near-Convex Sequences}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.16}, URN = {urn:nbn:de:0030-drops-193188}, doi = {10.4230/LIPIcs.ISAAC.2023.16}, annote = {Keywords: (min,+)-convolution, fine-grained complexity, convex sequences} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We solve the Bin Packing problem in O^*(2^k) time, where k is the number of items less or equal to one third of the bin capacity. This parameter measures the distance from the polynomially solvable case of only large (i.e., greater than one third) items. Our algorithm is actually designed to work for a more general Vector Bin Packing problem, in which items are multidimensional vectors. We improve over the previous fastest O^*(k! ⋅ 4^k) time algorithm.
Our algorithm works by reducing the problem to finding an exact weight perfect matching in a (multi-)graph with O^*(2^k) edges, whose weights are integers of the order of O^*(2^k). To solve the matching problem in the desired time, we give a variant of the classic Mulmuley-Vazirani-Vazirani algorithm with only a linear dependence on the edge weights and the number of edges - which may be of independent interest.
Moreover, we give a tight lower bound, under the Strong Exponential Time Hypothesis (SETH), showing that the constant 2 in the base of the exponent cannot be further improved for Vector Bin Packing.
Our techniques also lead to improved algorithms for Vector Multiple Knapsack, Vector Bin Covering, and Perfect Matching with Hitting Constraints.

Alexandra Lassota, Aleksander Łukasiewicz, and Adam Polak. Tight Vector Bin Packing with Few Small Items via Fast Exact Matching in Multigraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 87:1-87:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lassota_et_al:LIPIcs.ICALP.2022.87, author = {Lassota, Alexandra and {\L}ukasiewicz, Aleksander and Polak, Adam}, title = {{Tight Vector Bin Packing with Few Small Items via Fast Exact Matching in Multigraphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {87:1--87:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.87}, URN = {urn:nbn:de:0030-drops-164286}, doi = {10.4230/LIPIcs.ICALP.2022.87}, annote = {Keywords: Bin Packing, Vector Bin Packing, Parameterized Complexity, Matching} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Makespan minimization on parallel identical machines is a classical and intensively studied problem in scheduling, and a classic example for online algorithm analysis with Graham’s famous list scheduling algorithm dating back to the 1960s. In this problem, jobs arrive over a list and upon an arrival, the algorithm needs to assign the job to a machine. The goal is to minimize the makespan, that is, the maximum machine load. In this paper, we consider the variant with an additional cardinality constraint: The algorithm may assign at most k jobs to each machine where k is part of the input. While the offline (strongly NP-hard) variant of cardinality constrained scheduling is well understood and an EPTAS exists here, no non-trivial results are known for the online variant. We fill this gap by making a comprehensive study of various different online models. First, we show that there is a constant competitive algorithm for the problem and further, present a lower bound of 2 on the competitive ratio of any online algorithm. Motivated by the lower bound, we consider a semi-online variant where upon arrival of a job of size p, we are allowed to migrate jobs of total size at most a constant times p. This constant is called the migration factor of the algorithm. Algorithms with small migration factors are a common approach to bridge the performance of online algorithms and offline algorithms. One can obtain algorithms with a constant migration factor by rounding the size of each incoming job and then applying an ordinal algorithm to the resulting rounded instance. With this in mind, we also consider the framework of ordinal algorithms and characterize the competitive ratio that can be achieved using the aforementioned approaches. More specifically, we show that in both cases, one can get a competitive ratio that is strictly lower than 2, which is the bound from the standard online setting. On the other hand, we prove that no PTAS is possible.

Leah Epstein, Alexandra Lassota, Asaf Levin, Marten Maack, and Lars Rohwedder. Cardinality Constrained Scheduling in Online Models. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{epstein_et_al:LIPIcs.STACS.2022.28, author = {Epstein, Leah and Lassota, Alexandra and Levin, Asaf and Maack, Marten and Rohwedder, Lars}, title = {{Cardinality Constrained Scheduling in Online Models}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.28}, URN = {urn:nbn:de:0030-drops-158385}, doi = {10.4230/LIPIcs.STACS.2022.28}, annote = {Keywords: Cardinality Constrained Scheduling, Makespan Minimization, Online Algorithms, Lower Bounds, Pure Online, Migration, Ordinal Algorithms} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

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⋅ k!⋅ 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 algorithm for Bin Covering with run time O((k!)² ⋅ k ⋅ 2^k ⋅ n log(n)).

Max Bannach, Sebastian Berndt, Marten Maack, Matthias Mnich, Alexandra Lassota, Malin Rau, and Malte Skambath. Solving Packing Problems with Few Small Items Using Rainbow Matchings. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bannach_et_al:LIPIcs.MFCS.2020.11, author = {Bannach, Max and Berndt, Sebastian and Maack, Marten and Mnich, Matthias and Lassota, Alexandra and Rau, Malin and Skambath, Malte}, title = {{Solving Packing Problems with Few Small Items Using Rainbow Matchings}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {11:1--11:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.11}, URN = {urn:nbn:de:0030-drops-126816}, doi = {10.4230/LIPIcs.MFCS.2020.11}, annote = {Keywords: Bin Packing, Knapsack, matching, fixed-parameter tractable} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We study an important case of ILPs max {c^Tx | Ax = b, l <= x <= u, x in Z^{n t}} with n * t variables and lower and upper bounds l, u in Z^{nt}. In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s x t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and Delta, where Delta is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Omega(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs Delta)^{O(r^2s + s^2)} L^2 * nt log^{O(1)}(nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L.

Klaus Jansen, Alexandra Lassota, and Lars Rohwedder. Near-Linear Time Algorithm for n-fold ILPs via Color Coding. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 75:1-75:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jansen_et_al:LIPIcs.ICALP.2019.75, author = {Jansen, Klaus and Lassota, Alexandra and Rohwedder, Lars}, title = {{Near-Linear Time Algorithm for n-fold ILPs via Color Coding}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {75:1--75:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.75}, URN = {urn:nbn:de:0030-drops-106518}, doi = {10.4230/LIPIcs.ICALP.2019.75}, annote = {Keywords: Near-Linear Time Algorithm, n-fold ILP, Color Coding} }

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