11 Search Results for "Hunkenschröder, Christoph"

Document
DOI: 10.4230/LIPIcs.IPEC.2025.15
New Algorithm for Combinatorial n-Folds and Applications

Authors: Klaus Jansen, Kai Kahler, Lis Pirotton, and Malte Tutas

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
Block-structured integer linear programs (ILPs) play an important role in various application fields. We address n-fold ILPs where the matrix 𝒜 has a specific structure, i.e., where the blocks in the lower part of 𝒜 consist only of the row vectors (1,… ,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‖_∞), where n is the number of blocks, r the number of rows in the upper blocks and Δ = ‖A‖_∞. 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.
Cite as

Klaus Jansen, Kai Kahler, Lis Pirotton, and Malte Tutas. New Algorithm for Combinatorial n-Folds and Applications. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2025.15,
  author =	{Jansen, Klaus and Kahler, Kai and Pirotton, Lis and Tutas, Malte},
  title =	{{New Algorithm for Combinatorial n-Folds and Applications}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.15},
  URN =		{urn:nbn:de:0030-drops-251472},
  doi =		{10.4230/LIPIcs.IPEC.2025.15},
  annote =	{Keywords: integer linear programming, n-fold, parameterized complexity, scheduling, uniform machines}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.IPEC.2025.1
A Brief History of Parameterized Algorithms for Block-Structured Integer Programs (Invited Talk)

Authors: Martin Koutecký

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
Integer Programming (IP) is a fundamental but computationally hard problem. Still, certain efficiently solvable subclasses have been identified over time, most notably totally unimodular IPs in the 1950s, and fixed-dimension IPs in the 1980s. Starting around the year 2000, a stream of research has identified block-structured IPs as yet another tractable subclass. In this paper, we give a brief and incomplete review of this history, with a focus on several of the author’s contributions.
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Martin Koutecký. A Brief History of Parameterized Algorithms for Block-Structured Integer Programs (Invited Talk). In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{koutecky:LIPIcs.IPEC.2025.1,
  author =	{Kouteck\'{y}, Martin},
  title =	{{A Brief History of Parameterized Algorithms for Block-Structured Integer Programs}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.1},
  URN =		{urn:nbn:de:0030-drops-251338},
  doi =		{10.4230/LIPIcs.IPEC.2025.1},
  annote =	{Keywords: Integer Programming, Parameterized Algorithm, Graver Basis, Treedepth, n-fold, tree-fold, 2-stage stochastic, multistage stochastic, Mixed-Integer Programming}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2025.16
Designing Compact ILPs via Fast Witness Verification

Authors: Michał Włodarczyk

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
The standard formalization of preprocessing in parameterized complexity is given by kernelization. In this work, we depart from this paradigm and study a different type of preprocessing for problems without polynomial kernels, still aiming at producing instances that are easily solvable in practice. Specifically, we ask for which parameterized problems an instance (I,k) can be reduced in polynomial time to an integer linear program (ILP) with poly(k) constraints. We show that this property coincides with the parameterized complexity class WK[1], previously studied in the context of Turing kernelization lower bounds. In turn, the class WK[1] enjoys an elegant characterization in terms of witness verification protocols: a yes-instance should admit a witness of size poly(k) that can be verified in time poly(k). By combining known data structures with new ideas, we design such protocols for several problems, such as r-Way Cut, Vertex Multiway Cut, Steiner Tree, and Minimum Common String Partition, thus showing that they can be modeled by compact ILPs. We also present explicit ILP and MILP formulations for Weighted Vertex Cover on graphs with small (unweighted) vertex cover number. We believe that these results will provide a background for a systematic study of ILP-oriented preprocessing procedures for parameterized problems.
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Michał Włodarczyk. Designing Compact ILPs via Fast Witness Verification. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wlodarczyk:LIPIcs.IPEC.2025.16,
  author =	{W{\l}odarczyk, Micha{\l}},
  title =	{{Designing Compact ILPs via Fast Witness Verification}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.16},
  URN =		{urn:nbn:de:0030-drops-251481},
  doi =		{10.4230/LIPIcs.IPEC.2025.16},
  annote =	{Keywords: integer programming, kernelization, nondeterminism, multiway cut}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2025.14
A Simple Algorithm for Combinatorial n-Fold ILPs Using the Steinitz Lemma

Authors: Sushmita Gupta, Pallavi Jain, Sanjay Seetharaman, and Meirav Zehavi

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
We present an algorithm for a class of n-fold ILPs whose existing algorithms in literature are often either (1) based on the augmentation framework where one starts with an arbitrary solution and then iteratively moves towards an optimal solution by solving appropriate programs; or (2) require solving a linear relaxation of the program; or (3) are based on decomposition/proximity based arguments. Combinatorial n-fold ILPs is a class of n-fold ILPs introduced and studied by Knop et al. [MP2020] that captures several other problems in a variety of domains. We present a simple and direct algorithm that solves combinatorial n-fold ILPs with unbounded non-negative variables via an application of the Steinitz lemma. Depending on the structure of the input ILP, we also improve upon the existing algorithms in the literature in terms of the running time, thereby showing an improvement that mirrors the one shown by Rohwedder [ICALP2025] contemporaneously and independently.
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Sushmita Gupta, Pallavi Jain, Sanjay Seetharaman, and Meirav Zehavi. A Simple Algorithm for Combinatorial n-Fold ILPs Using the Steinitz Lemma. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gupta_et_al:LIPIcs.IPEC.2025.14,
  author =	{Gupta, Sushmita and Jain, Pallavi and Seetharaman, Sanjay and Zehavi, Meirav},
  title =	{{A Simple Algorithm for Combinatorial n-Fold ILPs Using the Steinitz Lemma}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.14},
  URN =		{urn:nbn:de:0030-drops-251467},
  doi =		{10.4230/LIPIcs.IPEC.2025.14},
  annote =	{Keywords: n-fold integer linear program, parameterized algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.15
Parameterized Complexity of Directed Traveling Salesman Problem

Authors: Václav Blažej, Andreas Emil Feldmann, Foivos Fioravantes, Paweł Rzążewski, and Ondřej Suchý

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
The Directed Traveling Salesman Problem (DTSP) is a variant of the classical Traveling Salesman Problem in which the edges in the graph are directed and a vertex and edge can be visited multiple times. The goal is to find a directed closed walk of minimum length (or total weight) that visits every vertex of the given graph at least once. In a yet more general version, Directed Waypoint Routing Problem (DWRP), some vertices are marked as terminals and we are only required to visit all terminals. Furthermore, each edge has its capacity bounding the number of times this edge can be used by a solution. While both problems (and many other variants of TSP) were extensively investigated, mostly from the approximation point of view, there are surprisingly few results concerning the parameterized complexity. Our starting point is the result of Marx et al. [APPROX/RANDOM 2016] who proved that DTSP is W[1]-hard parameterized by distance to pathwidth 3. In this paper we aim to initiate the systematic complexity study of variants of Directed Traveling Salesman Problem with respect to various, mostly structural, parameters. We show that DWRP is FPT parameterized by the solution size, the feedback edge number and the vertex integrity of the underlying undirected graph. Furthermore, the problem is XP parameterized by treewidth. On the complexity side, we show that the problem is W[1]-hard parameterized by the distance to constant treedepth.
Cite as

Václav Blažej, Andreas Emil Feldmann, Foivos Fioravantes, Paweł Rzążewski, and Ondřej Suchý. Parameterized Complexity of Directed Traveling Salesman Problem. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blazej_et_al:LIPIcs.ISAAC.2025.15,
  author =	{Bla\v{z}ej, V\'{a}clav and Feldmann, Andreas Emil and Fioravantes, Foivos and Rz\k{a}\.{z}ewski, Pawe{\l} and Such\'{y}, Ond\v{r}ej},
  title =	{{Parameterized Complexity of Directed Traveling Salesman Problem}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.15},
  URN =		{urn:nbn:de:0030-drops-249231},
  doi =		{10.4230/LIPIcs.ISAAC.2025.15},
  annote =	{Keywords: Directed TSP, parameterized complexity, vertex integrity, treedepth}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.48
The Support of Bin Packing Is Exponential

Authors: Klaus Jansen, Lis Pirotton, and Malte Tutas

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
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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Klaus Jansen, Lis Pirotton, and Malte Tutas. The Support of Bin Packing Is Exponential. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jansen_et_al:LIPIcs.ESA.2025.48,
  author =	{Jansen, Klaus and Pirotton, Lis and Tutas, Malte},
  title =	{{The Support of Bin Packing Is Exponential}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{48:1--48:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.48},
  URN =		{urn:nbn:de:0030-drops-245167},
  doi =		{10.4230/LIPIcs.ESA.2025.48},
  annote =	{Keywords: Bin Packing, Integer Programming, Support}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.112
Parameterized Algorithms for Matching Integer Programs with Additional Rows and Columns

Authors: Alexandra Lassota and Koen Ligthart

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study integer linear programs (ILP) of the form min{c^⊤ x | Ax = b,l ≤ x ≤ u,x ∈ ℤⁿ} and analyze their parameterized complexity with respect to their distance to the generalized matching problem, following the well-established approach of capturing the hardness of a problem by the distance to triviality. The generalized matching problem is an ILP where each column of the constraint matrix has 1-norm of at most 2. It captures several well-known polynomial time solvable problems such as matching and flow problems. We parameterize by the size of variable and constraint backdoors, which measure the least number of columns or rows that must be deleted to obtain a generalized matching ILP. This extends generalized matching problems by allowing a parameterized number of additional arbitrary variables or constraints, yielding a novel parameter. We present the following results: (i) a fixed-parameter tractable (FPT) algorithm for ILPs parameterized by the size p of a minimum variable backdoor to generalized matching; (ii) a randomized slice-wise polynomial (XP) time algorithm for ILPs parameterized by the size h of a minimum constraint backdoor to generalized matching as long as c and A are encoded in unary; (iii) we complement (ii) by proving that solving an ILP is W[1]-hard when parameterized by h even when c,A,l,u have coefficients of constant size. To obtain (i), we prove a variant of lattice-convexity of the degree sequences of weighted b-matchings, which we study in the light of SBO jump M-convex functions. This allows us to model the matching part as a polyhedral constraint on the integer backdoor variables. The resulting ILP is solved in FPT time using an integer programming algorithm. For (ii), the randomized XP time algorithm is obtained by pseudo-polynomially reducing the problem to the exact matching problem. To prevent an exponential blowup in terms of the encoding length of b, we bound the Graver complexity of the constraint matrix and employ a Graver augmentation local search framework. The hardness result (iii) is obtained through a parameterized reduction from ILP with h constraints and coefficients encoded in unary.
Cite as

Alexandra Lassota and Koen Ligthart. Parameterized Algorithms for Matching Integer Programs with Additional Rows and Columns. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 112:1-112:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lassota_et_al:LIPIcs.ICALP.2025.112,
  author =	{Lassota, Alexandra and Ligthart, Koen},
  title =	{{Parameterized Algorithms for Matching Integer Programs with Additional Rows and Columns}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{112:1--112:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.112},
  URN =		{urn:nbn:de:0030-drops-234895},
  doi =		{10.4230/LIPIcs.ICALP.2025.112},
  annote =	{Keywords: Integer Programming, fixed-parameter Tractability, polyhedral Optimization, Matchings}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.126
ETH-Tight FPT Algorithm for Makespan Minimization on Uniform Machines

Authors: Lars Rohwedder

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Given n jobs with processing times p₁,...,p_n ∈ ℕ and m ≤ n machines with speeds s₁,...,s_m ∈ ℕ our goal is to allocate the jobs to machines minimizing the makespan. We present an algorithm that solves the problem in time p_{max}^{O(d)} ⋅ n, where p_{max} is the maximum processing time and d ≤ p_{max} is the number of distinct processing times. This is essentially the best possible due to a lower bound based on the exponential time hypothesis (ETH). Our result improves over prior works that had a quadratic term in d in the exponent and answers an open question by Koutecký and Zink. The algorithm is based on integer programming techniques combined with novel ideas from modular arithmetic. It can also be implemented efficiently for the more compact high-multiplicity instance encoding.
Cite as

Lars Rohwedder. ETH-Tight FPT Algorithm for Makespan Minimization on Uniform Machines. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 126:1-126:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{rohwedder:LIPIcs.ICALP.2025.126,
  author =	{Rohwedder, Lars},
  title =	{{ETH-Tight FPT Algorithm for Makespan Minimization on Uniform Machines}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{126:1--126:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.126},
  URN =		{urn:nbn:de:0030-drops-235037},
  doi =		{10.4230/LIPIcs.ICALP.2025.126},
  annote =	{Keywords: Scheduling, Integer Programming}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.83
Fine-Grained Equivalence for Problems Related to Integer Linear Programming

Authors: Lars Rohwedder and Karol Węgrzycki

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
Integer Linear Programming with n binary variables and m many 0/1-constraints can be solved in time 2^Õ(m²) poly(n) and it is open whether the dependence on m is optimal. Several seemingly unrelated problems, which include variants of Closest String, Discrepancy Minimization, Set Cover, and Set Packing, can be modelled as Integer Linear Programming with 0/1 constraints to obtain algorithms with the same running time for a natural parameter m in each of the problems. Our main result establishes through fine-grained reductions that these problems are equivalent, meaning that a 2^O(m^{2-ε}) poly(n) algorithm with ε > 0 for one of them implies such an algorithm for all of them. In the setting above, one can alternatively obtain an n^O(m) time algorithm for Integer Linear Programming using a straightforward dynamic programming approach, which can be more efficient if n is relatively small (e.g., subexponential in m). We show that this can be improved to {n'}^O(m) + O(nm), where n' is the number of distinct (i.e., non-symmetric) variables. This dominates both of the aforementioned running times.
Cite as

Lars Rohwedder and Karol Węgrzycki. Fine-Grained Equivalence for Problems Related to Integer Linear Programming. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 83:1-83:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{rohwedder_et_al:LIPIcs.ITCS.2025.83,
  author =	{Rohwedder, Lars and W\k{e}grzycki, Karol},
  title =	{{Fine-Grained Equivalence for Problems Related to Integer Linear Programming}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{83:1--83:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.83},
  URN =		{urn:nbn:de:0030-drops-227114},
  doi =		{10.4230/LIPIcs.ITCS.2025.83},
  annote =	{Keywords: Integer Programming, Fine-Grained Complexity, Fixed-Parameter Tractable Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.43
Approximate CVP_p in Time 2^{0.802 n}

Authors: Friedrich Eisenbrand and Moritz Venzin

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
We show that a constant factor approximation of the shortest and closest lattice vector problem w.r.t. any 𝓁_p-norm can be computed in time 2^{(0.802 +ε) n}. This matches the currently fastest constant factor approximation algorithm for the shortest vector problem w.r.t. 𝓁₂. To obtain our result, we combine the latter algorithm w.r.t. 𝓁₂ with geometric insights related to coverings.
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Friedrich Eisenbrand and Moritz Venzin. Approximate CVP_p in Time 2^{0.802 n}. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eisenbrand_et_al:LIPIcs.ESA.2020.43,
  author =	{Eisenbrand, Friedrich and Venzin, Moritz},
  title =	{{Approximate CVP\underlinep in Time 2^\{0.802 n\}}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{43:1--43:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.43},
  URN =		{urn:nbn:de:0030-drops-129097},
  doi =		{10.4230/LIPIcs.ESA.2020.43},
  annote =	{Keywords: Shortest and closest vector problem, approximation algorithm, sieving, covering convex bodies}
}
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DOI: 10.4230/LIPIcs.ICALP.2018.49
Faster Algorithms for Integer Programs with Block Structure

Authors: Friedrich Eisenbrand, Christoph Hunkenschröder, and Kim-Manuel Klein

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
We consider integer programming problems max {c^Tx : A x = b, l <= x <= u, x in Z^{nt}} where 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 A consists of n repetitions of submatrices A in Z^{r × t} on the top horizontal part and n repetitions of a matrix B in Z^{s × t} on the diagonal below the top part. Instead of allowing only two types of block matrices, one for the horizontal line and one for the diagonal, we generalize the n-fold setting to allow for arbitrary matrices in every block. We show that such an integer program can be solved in time n^2t^2 phi x (r s delta)^{O(rs^2+ sr^2)} (ignoring logarithmic factors). Here delta is an upper bound on the largest absolute value of an entry of A and phi is the largest binary encoding length of a coefficient of c. This improves upon the previously best algorithm of Hemmecke, Onn and Romanchuk that runs in time n^3t^3 phi x delta^{O(st(r+t))}. In particular, our algorithm is not exponential in the number t of columns of A and B. Our algorithm is based on a new upper bound on the l_1-norm of an element of the Graver basis of an integer matrix and on a proximity bound between the LP and IP optimal solutions tailored for IPs with block structure. These new bounds rely on the Steinitz Lemma. Furthermore, we extend our techniques to the recently introduced tree-fold IPs, where we again present a more efficient algorithm in a generalized setting.
Cite as

Friedrich Eisenbrand, Christoph Hunkenschröder, and Kim-Manuel Klein. Faster Algorithms for Integer Programs with Block Structure. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 49:1-49:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eisenbrand_et_al:LIPIcs.ICALP.2018.49,
  author =	{Eisenbrand, Friedrich and Hunkenschr\"{o}der, Christoph and Klein, Kim-Manuel},
  title =	{{Faster Algorithms for Integer Programs with Block Structure}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{49:1--49:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.49},
  URN =		{urn:nbn:de:0030-drops-90537},
  doi =		{10.4230/LIPIcs.ICALP.2018.49},
  annote =	{Keywords: n-fold, Tree-fold, Integer Programming}
}
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