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Documents authored by Koutecky, Martin

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Koutecký, Martin

Document
DOI: 10.4230/LIPIcs.ESA.2024.32
Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds

Authors: Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.18
Experimental Analysis of LP Scaling Methods Based on Circuit Imbalance Minimization

Authors: Jakub Komárek and Martin Koutecký

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
Linear programming (LP) is a fundamental problem with rich theory and wide applications. A ubiquitous technique in LP is scaling, where the input instance is transformed in some way to make its solution easier. Dadush et al. [STOC '20] have recently devised an algorithm which scales the columns of the constraint matrix of a linear program in a way that aims to minimize the circuit imbalance measure, a matrix condition number of growing theoretical interest. They show that this rescaling achieves favorable theoretical guarantees for certain LP algorithms. We follow up on their work in an experimental manner. First, we have implemented their algorithm, overcoming several engineering obstacles. Next, we have used our implementation to obtain a rescaling of 142 publicly available instances. Finally, we have performed experiments evaluating the effects of the obtained rescalings on the runtime of real-world LP solvers, and we have evaluated their quality with regard to the circuit imbalance measure.
Cite as

Jakub Komárek and Martin Koutecký. Experimental Analysis of LP Scaling Methods Based on Circuit Imbalance Minimization. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{komarek_et_al:LIPIcs.SEA.2024.18,
  author =	{Kom\'{a}rek, Jakub and Kouteck\'{y}, Martin},
  title =	{{Experimental Analysis of LP Scaling Methods Based on Circuit Imbalance Minimization}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{18:1--18:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.18},
  URN =		{urn:nbn:de:0030-drops-203832},
  doi =		{10.4230/LIPIcs.SEA.2024.18},
  annote =	{Keywords: Linear programming, scaling, circuit imbalance measure}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.73
Scheduling Kernels via Configuration LP

Authors: Dušan Knop and Martin Koutecký

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Makespan minimization (on parallel identical or unrelated machines) is arguably the most natural and studied scheduling problem. A common approach in practical algorithm design is to reduce the size of a given instance by a fast preprocessing step while being able to recover key information even after this reduction. This notion is formally studied as kernelization (or simply, kernel) - a polynomial time procedure which yields an equivalent instance whose size is bounded in terms of some given parameter. It follows from known results that makespan minimization parameterized by the longest job processing time p_max has a kernelization yielding a reduced instance whose size is exponential in p_max. Can this be reduced to polynomial in p_max? We answer this affirmatively not only for makespan minimization, but also for the (more complicated) objective of minimizing the weighted sum of completion times, also in the setting of unrelated machines when the number of machine kinds is a parameter. Our algorithm first solves the Configuration LP and based on its solution constructs a solution of an intermediate problem, called huge N-fold integer programming. This solution is further reduced in size by a series of steps, until its encoding length is polynomial in the parameters. Then, we show that huge N-fold IP is in NP, which implies that there is a polynomial reduction back to our scheduling problem, yielding a kernel. Our technique is highly novel in the context of kernelization, and our structural theorem about the Configuration LP is of independent interest. Moreover, we show a polynomial kernel for huge N-fold IP conditional on whether the so-called separation subproblem can be solved in polynomial time. Considering that integer programming does not admit polynomial kernels except for quite restricted cases, our "conditional kernel" provides new insight.
Cite as

Dušan Knop and Martin Koutecký. Scheduling Kernels via Configuration LP. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{knop_et_al:LIPIcs.ESA.2022.73,
  author =	{Knop, Du\v{s}an and Kouteck\'{y}, Martin},
  title =	{{Scheduling Kernels via Configuration LP}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{73:1--73:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.73},
  URN =		{urn:nbn:de:0030-drops-170118},
  doi =		{10.4230/LIPIcs.ESA.2022.73},
  annote =	{Keywords: Scheduling, Kernelization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.29
Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming

Authors: Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder

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

Abstract
An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the 𝓁₁-norm of the Graver basis is bounded by a function of the maximum 𝓁₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the 𝓁₁-norm of the Graver basis of the constraint matrix, when parameterized by the 𝓁₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.
Cite as

Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder. Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brianski_et_al:LIPIcs.ICALP.2022.29,
  author =	{Bria\'{n}ski, Marcin and Kouteck\'{y}, Martin and Kr\'{a}l', Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na and Schr\"{o}der, Felix},
  title =	{{Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{29:1--29:20},
  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.2022.29},
  URN =		{urn:nbn:de:0030-drops-163702},
  doi =		{10.4230/LIPIcs.ICALP.2022.29},
  annote =	{Keywords: Integer programming, width parameters, matroids, Graver basis, tree-depth, fixed parameter tractability}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.18
Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines

Authors: Martin Koutecký and Johannes Zink

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector, are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines (Q|HM|C_max) is NP-hard already with 6 job types, and that the related Cutting Stock problem is NP-hard already with 8 item types. For the more general unrelated machines model (R|HM|C_max), we show that if either the largest job size p_max, or the number of jobs n are polynomially bounded in the instance size |I|, there are algorithms with complexity |I|^poly(k). Our main result is that this is unlikely to be improved, because Q||C_max is W[1]-hard parameterized by k already when n, p_max, and the numbers describing the speeds are polynomial in |I|; the same holds for R|HM|C_max (without speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives 𝓁₂-norm minimization of the load vector and, partially, sum of weighted completion times ∑ w_j C_j. Along the way, we answer affirmatively the question whether makespan minimization on identical machines (P||C_max) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for Q||C_max this implies that the complexity of P|HM|C_max is the only remaining open case.
Cite as

Martin Koutecký and Johannes Zink. Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{koutecky_et_al:LIPIcs.ISAAC.2020.18,
  author =	{Kouteck\'{y}, Martin and Zink, Johannes},
  title =	{{Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.18},
  URN =		{urn:nbn:de:0030-drops-133620},
  doi =		{10.4230/LIPIcs.ISAAC.2020.18},
  annote =	{Keywords: Scheduling, cutting stock, hardness, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.33
New Bounds on Augmenting Steps of Block-Structured Integer Programs

Authors: Lin Chen, Martin Koutecký, Lei Xu, and Weidong Shi

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

Abstract
Iterative augmentation has recently emerged as an overarching method for solving Integer Programs (IP) in variable dimension, in stark contrast with the volume and flatness techniques of IP in fixed dimension. Here we consider 4-block n-fold integer programs, which are the most general class considered so far. A 4-block n-fold IP has a constraint matrix which consists of n copies of small matrices A, B, and D, and one copy of C, in a specific block structure. Iterative augmentation methods rely on the so-called Graver basis of the constraint matrix, which constitutes a set of fundamental augmenting steps. All existing algorithms rely on bounding the 𝓁₁- or 𝓁_∞-norm of elements of the Graver basis. Hemmecke et al. [Math. Prog. 2014] showed that 4-block n-fold IP has Graver elements of 𝓁_∞-norm at most 𝒪_FPT(n^{2^{s_D}}), leading to an algorithm with a similar runtime; here, s_D is the number of rows of matrix D and 𝒪_FPT hides a multiplicative factor that is only dependent on the small matrices A,B,C,D, However, it remained open whether their bounds are tight, in particular, whether they could be improved to 𝒪_FPT(1), perhaps at least in some restricted cases. We prove that the 𝓁_∞-norm of the Graver elements of 4-block n-fold IP is upper bounded by 𝒪_FPT(n^{s_D}), improving significantly over the previous bound 𝒪_FPT(n^{2^{s_D}}). We also provide a matching lower bound of Ω(n^{s_D}) which even holds for arbitrary non-zero lattice elements, ruling out augmenting algorithm relying on even more restricted notions of augmentation than the Graver basis. We then consider a special case of 4-block n-fold in which C is a zero matrix, called 3-block n-fold IP. We show that while the 𝓁_∞-norm of its Graver elements is Ω(n^{s_D}), there exists a different decomposition into lattice elements whose 𝓁_∞-norm is bounded by 𝒪_FPT(1), which allows us to provide improved upper bounds on the 𝓁_∞-norm of Graver elements for 3-block n-fold IP. The key difference between the respective decompositions is that a Graver basis guarantees a sign-compatible decomposition; this property is critical in applications because it guarantees each step of the decomposition to be feasible. Consequently, our improved upper bounds let us establish faster algorithms for 3-block n-fold IP and 4-block IP, and our lower bounds strongly hint at parameterized hardness of 4-block and even 3-block n-fold IP. Furthermore, we show that 3-block n-fold IP is without loss of generality in the sense that 4-block n-fold IP can be solved in FPT oracle time by taking an algorithm for 3-block n-fold IP as an oracle.
Cite as

Lin Chen, Martin Koutecký, Lei Xu, and Weidong Shi. New Bounds on Augmenting Steps of Block-Structured Integer Programs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chen_et_al:LIPIcs.ESA.2020.33,
  author =	{Chen, Lin and Kouteck\'{y}, Martin and Xu, Lei and Shi, Weidong},
  title =	{{New Bounds on Augmenting Steps of Block-Structured Integer Programs}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{33:1--33:19},
  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.2020.33},
  URN =		{urn:nbn:de:0030-drops-128994},
  doi =		{10.4230/LIPIcs.ESA.2020.33},
  annote =	{Keywords: Integer Programming, Graver basis, Fixed parameter tractable}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.26
Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming

Authors: Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry Δ are solvable in time g(d,Δ) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and Δ. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth. We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width.
Cite as

Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková. Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.ICALP.2020.26,
  author =	{Chan, Timothy F. N. and Cooper, Jacob W. and Kouteck\'{y}, Martin and Kr\'{a}l', Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na},
  title =	{{Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{26:1--26:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.26},
  URN =		{urn:nbn:de:0030-drops-124339},
  doi =		{10.4230/LIPIcs.ICALP.2020.26},
  annote =	{Keywords: Matroid algorithms, width parameters, integer programming, fixed parameter tractability, branch-width, branch-depth}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2018.21
Integer Programming in Parameterized Complexity: Three Miniatures

Authors: Tomás Gavenciak, Dusan Knop, and Martin Koutecký

Published in: LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

Abstract
Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra's algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f(k) poly(n). We focus on: - Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used. - Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups. - Minding the poly(n): reducing f(k) often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds. Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, and Max-q-Cut by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, and indefinite quadratic programs in fixed dimension.
Cite as

Tomás Gavenciak, Dusan Knop, and Martin Koutecký. Integer Programming in Parameterized Complexity: Three Miniatures. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gavenciak_et_al:LIPIcs.IPEC.2018.21,
  author =	{Gavenciak, Tom\'{a}s and Knop, Dusan and Kouteck\'{y}, Martin},
  title =	{{Integer Programming in Parameterized Complexity: Three Miniatures}},
  booktitle =	{13th International Symposium on Parameterized and Exact Computation (IPEC 2018)},
  pages =	{21:1--21:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-084-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{115},
  editor =	{Paul, Christophe and Pilipczuk, Michal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.21},
  URN =		{urn:nbn:de:0030-drops-102225},
  doi =		{10.4230/LIPIcs.IPEC.2018.21},
  annote =	{Keywords: graph coloring, parameterized complexity, integer linear programming, integer convex programming}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.85
A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs

Authors: Martin Koutecký, Asaf Levin, and Shmuel Onn

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

Abstract
The theory of n-fold integer programming has been recently emerging as an important tool in parameterized complexity. The input to an n-fold integer program (IP) consists of parameter A, dimension n, and numerical data of binary encoding length L. It was known for some time that such programs can be solved in polynomial time using O(n^{g(A)}L) arithmetic operations where g is an exponential function of the parameter. In 2013 it was shown that it can be solved in fixed-parameter tractable time using O(f(A)n^3L) arithmetic operations for a single-exponential function f. This, and a faster algorithm for a special case of combinatorial n-fold IP, have led to several very recent breakthroughs in the parameterized complexity of scheduling, stringology, and computational social choice. In 2015 it was shown that it can be solved in strongly polynomial time using O(n^{g(A)}) arithmetic operations. Here we establish a result which subsumes all three of the above results by showing that n-fold IP can be solved in strongly polynomial fixed-parameter tractable time using O(f(A)n^6 log n) arithmetic operations. In fact, our results are much more general, briefly outlined as follows. - There is a strongly polynomial algorithm for integer linear programming (ILP) whenever a so-called Graver-best oracle is realizable for it. - Graver-best oracles for the large classes of multi-stage stochastic and tree-fold ILPs can be realized in fixed-parameter tractable time. Together with the previous oracle algorithm, this newly shows two large classes of ILP to be strongly polynomial; in contrast, only few classes of ILP were previously known to be strongly polynomial. - We show that ILP is fixed-parameter tractable parameterized by the largest coefficient |A |_infty and the primal or dual treedepth of A, and that this parameterization cannot be relaxed, signifying substantial progress in understanding the parameterized complexity of ILP.
Cite as

Martin Koutecký, Asaf Levin, and Shmuel Onn. A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 85:1-85:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{koutecky_et_al:LIPIcs.ICALP.2018.85,
  author =	{Kouteck\'{y}, Martin and Levin, Asaf and Onn, Shmuel},
  title =	{{A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{85:1--85:14},
  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.2018.85},
  URN =		{urn:nbn:de:0030-drops-90898},
  doi =		{10.4230/LIPIcs.ICALP.2018.85},
  annote =	{Keywords: integer programming, parameterized complexity, Graver basis, n-fold integer programming}
}
Document
DOI: 10.4230/LIPIcs.SEA.2018.10
Evaluating and Tuning n-fold Integer Programming

Authors: Katerina Altmanová, Dusan Knop, and Martin Koutecký

Published in: LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

Abstract
In recent years, algorithmic breakthroughs in stringology, computational social choice, scheduling, etc., were achieved by applying the theory of so-called n-fold integer programming. An n-fold integer program (IP) has a highly uniform block structured constraint matrix. Hemmecke, Onn, and Romanchuk [Math. Programming, 2013] showed an algorithm with runtime a^{O(rst + r^2s)} n^3, where a is the largest coefficient, r,s, and t are dimensions of blocks of the constraint matrix and n is the total dimension of the IP; thus, an algorithm efficient if the blocks are of small size and with small coefficients. The algorithm works by iteratively improving a feasible solution with augmenting steps, and n-fold IPs have the special property that augmenting steps are guaranteed to exist in a not-too-large neighborhood. However, this algorithm has never been implemented and evaluated. We have implemented the algorithm and learned the following along the way. The original algorithm is practically unusable, but we discover a series of improvements which make its evaluation possible. Crucially, we observe that a certain constant in the algorithm can be treated as a tuning parameter, which yields an efficient heuristic (essentially searching in a smaller-than-guaranteed neighborhood). Furthermore, the algorithm uses an overly expensive strategy to find a "best" step, while finding only an "approximatelly best" step is much cheaper, yet sufficient for quick convergence. Using this insight, we improve the asymptotic dependence on n from n^3 to n^2 log n which yields the currently asymptotically fastest algorithm for n-fold IP. Finally, we tested the behavior of the algorithm with various values of the tuning parameter and different strategies of finding improving steps. First, we show that decreasing the tuning parameter initially leads to an increased number of iterations needed for convergence and eventually to getting stuck in local optima, as expected. However, surprisingly small values of the parameter already exhibit good behavior. Second, our new strategy for finding "approximatelly best" steps wildly outperforms the original construction.
Cite as

Katerina Altmanová, Dusan Knop, and Martin Koutecký. Evaluating and Tuning n-fold Integer Programming. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{altmanova_et_al:LIPIcs.SEA.2018.10,
  author =	{Altmanov\'{a}, Katerina and Knop, Dusan and Kouteck\'{y}, Martin},
  title =	{{Evaluating and Tuning n-fold Integer Programming}},
  booktitle =	{17th International Symposium on Experimental Algorithms (SEA 2018)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-070-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{103},
  editor =	{D'Angelo, Gianlorenzo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.10},
  URN =		{urn:nbn:de:0030-drops-89454},
  doi =		{10.4230/LIPIcs.SEA.2018.10},
  annote =	{Keywords: n-fold integer programming, integer programming, analysis of algorithms, primal heuristic, local search}
}
Document
DOI: 10.4230/LIPIcs.ESA.2017.54
Combinatorial n-fold Integer Programming and Applications

Authors: Dusan Knop, Martin Koutecky, and Matthias Mnich

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract
Many fundamental NP-hard problems can be formulated as integer linear programs (ILPs). A famous algorithm by Lenstra allows to solve ILPs in time that is exponential only in the dimension of the program. That algorithm therefore became a ubiquitous tool in the design of fixed-parameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problem by some parameter. However, it was discovered that in many cases using Lenstra’s algorithm has two drawbacks: First, the run time of the resulting algorithms is often doubly-exponential in the parameter, and second, an ILP formulation in small dimension can not easily express problems which involve many different costs. Inspired by the work of Hemmecke, Onn and Romanchuk [Math. Prog. 2013], we develop a single-exponential algorithm for so-called combinatorial n-fold integer programs, which are remarkably similar to prior ILP formulations for various problems, but unlike them, also allow variable dimension. We then apply our algorithm to a few representative problems like Closest String, Swap Bribery, Weighted Set Multicover, and obtain exponential speedups in the dependence on the respective parameters, the input size, or both. Unlike Lenstra’s algorithm, which is essentially a bounded search tree algorithm, our result uses the technique of augmenting steps. At its heart is a deep result stating that in combinatorial n-fold IPs an existence of an augmenting step implies an existence of a “local” augmenting step, which can be found using dynamic programming. Our results provide an important insight into many problems by showing that they exhibit this phenomenon, and highlights the importance of augmentation techniques.
Cite as

Dusan Knop, Martin Koutecky, and Matthias Mnich. Combinatorial n-fold Integer Programming and Applications. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{knop_et_al:LIPIcs.ESA.2017.54,
  author =	{Knop, Dusan and Koutecky, Martin and Mnich, Matthias},
  title =	{{Combinatorial n-fold Integer Programming and Applications}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.54},
  URN =		{urn:nbn:de:0030-drops-78616},
  doi =		{10.4230/LIPIcs.ESA.2017.54},
  annote =	{Keywords: integer programming, closest strings, fixed-parameter algorithms}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.46
Voting and Bribing in Single-Exponential Time

Authors: Dusan Knop, Martin Koutecký, and Matthias Mnich

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
We introduce a general problem about bribery in voting systems. In the R-Multi-Bribery problem, the goal is to bribe a set of voters at minimum cost such that a desired candidate wins the manipulated election under the voting rule R. Voters assign prices for withdrawing their vote, for swapping the positions of two consecutive candidates in their preference order, and for perturbing their approval count for a candidate. As our main result, we show that R-Multi-Bribery is fixed-parameter tractable parameterized by the number of candidates for many natural voting rules R, including Kemeny rule, all scoring protocols, maximin rule, Bucklin rule, fallback rule, SP-AV, and any C1 rule. In particular, our result resolves the parameterized of R-Swap Bribery for all those voting rules, thereby solving a long-standing open problem and "Challenge #2" of the 9 Challenges in computational social choice by Bredereck et al. Further, our algorithm runs in single-exponential time for arbitrary cost; it thus improves the earlier double-exponential time algorithm by Dorn and Schlotter that is restricted to the unit-cost case for all scoring protocols, the maximin rule, and Bucklin rule.
Cite as

Dusan Knop, Martin Koutecký, and Matthias Mnich. Voting and Bribing in Single-Exponential Time. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{knop_et_al:LIPIcs.STACS.2017.46,
  author =	{Knop, Dusan and Kouteck\'{y}, Martin and Mnich, Matthias},
  title =	{{Voting and Bribing in Single-Exponential Time}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.46},
  URN =		{urn:nbn:de:0030-drops-69885},
  doi =		{10.4230/LIPIcs.STACS.2017.46},
  annote =	{Keywords: Parameterized algorithm, swap bribery, n-fold integer programming}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2016.18
Extension Complexity, MSO Logic, and Treewidth

Authors: Petr Kolman, Martin Koutecký, and Hans Raj Tiwary

Published in: LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Abstract
We consider the convex hull P_phi(G) of all satisfying assignments of a given MSO_2 formula phi on a given graph G. We show that there exists an extended formulation of the polytope P_phi(G) that can be described by f(|phi|,tau)*n inequalities, where n is the number of vertices in G, tau is the treewidth of G and f is a computable function depending only on phi and tau. In other words, we prove that the extension complexity of P_phi(G) is linear in the size of the graph G, with a constant depending on the treewidth of G and the formula phi. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs.
Cite as

Petr Kolman, Martin Koutecký, and Hans Raj Tiwary. Extension Complexity, MSO Logic, and Treewidth. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kolman_et_al:LIPIcs.SWAT.2016.18,
  author =	{Kolman, Petr and Kouteck\'{y}, Martin and Tiwary, Hans Raj},
  title =	{{Extension Complexity, MSO Logic, and Treewidth}},
  booktitle =	{15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-011-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{53},
  editor =	{Pagh, Rasmus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.18},
  URN =		{urn:nbn:de:0030-drops-60405},
  doi =		{10.4230/LIPIcs.SWAT.2016.18},
  annote =	{Keywords: Extension Complexity, FPT, Courcelle's Theorem, MSO Logic}
}

Koutecky, Martin

Document
DOI: 10.4230/LIPIcs.ESA.2024.32
Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds

Authors: Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.18
Experimental Analysis of LP Scaling Methods Based on Circuit Imbalance Minimization

Authors: Jakub Komárek and Martin Koutecký

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
Linear programming (LP) is a fundamental problem with rich theory and wide applications. A ubiquitous technique in LP is scaling, where the input instance is transformed in some way to make its solution easier. Dadush et al. [STOC '20] have recently devised an algorithm which scales the columns of the constraint matrix of a linear program in a way that aims to minimize the circuit imbalance measure, a matrix condition number of growing theoretical interest. They show that this rescaling achieves favorable theoretical guarantees for certain LP algorithms. We follow up on their work in an experimental manner. First, we have implemented their algorithm, overcoming several engineering obstacles. Next, we have used our implementation to obtain a rescaling of 142 publicly available instances. Finally, we have performed experiments evaluating the effects of the obtained rescalings on the runtime of real-world LP solvers, and we have evaluated their quality with regard to the circuit imbalance measure.
Cite as

Jakub Komárek and Martin Koutecký. Experimental Analysis of LP Scaling Methods Based on Circuit Imbalance Minimization. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{komarek_et_al:LIPIcs.SEA.2024.18,
  author =	{Kom\'{a}rek, Jakub and Kouteck\'{y}, Martin},
  title =	{{Experimental Analysis of LP Scaling Methods Based on Circuit Imbalance Minimization}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{18:1--18:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.18},
  URN =		{urn:nbn:de:0030-drops-203832},
  doi =		{10.4230/LIPIcs.SEA.2024.18},
  annote =	{Keywords: Linear programming, scaling, circuit imbalance measure}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.73
Scheduling Kernels via Configuration LP

Authors: Dušan Knop and Martin Koutecký

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Makespan minimization (on parallel identical or unrelated machines) is arguably the most natural and studied scheduling problem. A common approach in practical algorithm design is to reduce the size of a given instance by a fast preprocessing step while being able to recover key information even after this reduction. This notion is formally studied as kernelization (or simply, kernel) - a polynomial time procedure which yields an equivalent instance whose size is bounded in terms of some given parameter. It follows from known results that makespan minimization parameterized by the longest job processing time p_max has a kernelization yielding a reduced instance whose size is exponential in p_max. Can this be reduced to polynomial in p_max? We answer this affirmatively not only for makespan minimization, but also for the (more complicated) objective of minimizing the weighted sum of completion times, also in the setting of unrelated machines when the number of machine kinds is a parameter. Our algorithm first solves the Configuration LP and based on its solution constructs a solution of an intermediate problem, called huge N-fold integer programming. This solution is further reduced in size by a series of steps, until its encoding length is polynomial in the parameters. Then, we show that huge N-fold IP is in NP, which implies that there is a polynomial reduction back to our scheduling problem, yielding a kernel. Our technique is highly novel in the context of kernelization, and our structural theorem about the Configuration LP is of independent interest. Moreover, we show a polynomial kernel for huge N-fold IP conditional on whether the so-called separation subproblem can be solved in polynomial time. Considering that integer programming does not admit polynomial kernels except for quite restricted cases, our "conditional kernel" provides new insight.
Cite as

Dušan Knop and Martin Koutecký. Scheduling Kernels via Configuration LP. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{knop_et_al:LIPIcs.ESA.2022.73,
  author =	{Knop, Du\v{s}an and Kouteck\'{y}, Martin},
  title =	{{Scheduling Kernels via Configuration LP}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{73:1--73:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.73},
  URN =		{urn:nbn:de:0030-drops-170118},
  doi =		{10.4230/LIPIcs.ESA.2022.73},
  annote =	{Keywords: Scheduling, Kernelization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.29
Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming

Authors: Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder

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

Abstract
An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the 𝓁₁-norm of the Graver basis is bounded by a function of the maximum 𝓁₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the 𝓁₁-norm of the Graver basis of the constraint matrix, when parameterized by the 𝓁₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.
Cite as

Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder. Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brianski_et_al:LIPIcs.ICALP.2022.29,
  author =	{Bria\'{n}ski, Marcin and Kouteck\'{y}, Martin and Kr\'{a}l', Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na and Schr\"{o}der, Felix},
  title =	{{Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{29:1--29:20},
  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.2022.29},
  URN =		{urn:nbn:de:0030-drops-163702},
  doi =		{10.4230/LIPIcs.ICALP.2022.29},
  annote =	{Keywords: Integer programming, width parameters, matroids, Graver basis, tree-depth, fixed parameter tractability}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.18
Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines

Authors: Martin Koutecký and Johannes Zink

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector, are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines (Q|HM|C_max) is NP-hard already with 6 job types, and that the related Cutting Stock problem is NP-hard already with 8 item types. For the more general unrelated machines model (R|HM|C_max), we show that if either the largest job size p_max, or the number of jobs n are polynomially bounded in the instance size |I|, there are algorithms with complexity |I|^poly(k). Our main result is that this is unlikely to be improved, because Q||C_max is W[1]-hard parameterized by k already when n, p_max, and the numbers describing the speeds are polynomial in |I|; the same holds for R|HM|C_max (without speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives 𝓁₂-norm minimization of the load vector and, partially, sum of weighted completion times ∑ w_j C_j. Along the way, we answer affirmatively the question whether makespan minimization on identical machines (P||C_max) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for Q||C_max this implies that the complexity of P|HM|C_max is the only remaining open case.
Cite as

Martin Koutecký and Johannes Zink. Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{koutecky_et_al:LIPIcs.ISAAC.2020.18,
  author =	{Kouteck\'{y}, Martin and Zink, Johannes},
  title =	{{Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.18},
  URN =		{urn:nbn:de:0030-drops-133620},
  doi =		{10.4230/LIPIcs.ISAAC.2020.18},
  annote =	{Keywords: Scheduling, cutting stock, hardness, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.33
New Bounds on Augmenting Steps of Block-Structured Integer Programs

Authors: Lin Chen, Martin Koutecký, Lei Xu, and Weidong Shi

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

Abstract
Iterative augmentation has recently emerged as an overarching method for solving Integer Programs (IP) in variable dimension, in stark contrast with the volume and flatness techniques of IP in fixed dimension. Here we consider 4-block n-fold integer programs, which are the most general class considered so far. A 4-block n-fold IP has a constraint matrix which consists of n copies of small matrices A, B, and D, and one copy of C, in a specific block structure. Iterative augmentation methods rely on the so-called Graver basis of the constraint matrix, which constitutes a set of fundamental augmenting steps. All existing algorithms rely on bounding the 𝓁₁- or 𝓁_∞-norm of elements of the Graver basis. Hemmecke et al. [Math. Prog. 2014] showed that 4-block n-fold IP has Graver elements of 𝓁_∞-norm at most 𝒪_FPT(n^{2^{s_D}}), leading to an algorithm with a similar runtime; here, s_D is the number of rows of matrix D and 𝒪_FPT hides a multiplicative factor that is only dependent on the small matrices A,B,C,D, However, it remained open whether their bounds are tight, in particular, whether they could be improved to 𝒪_FPT(1), perhaps at least in some restricted cases. We prove that the 𝓁_∞-norm of the Graver elements of 4-block n-fold IP is upper bounded by 𝒪_FPT(n^{s_D}), improving significantly over the previous bound 𝒪_FPT(n^{2^{s_D}}). We also provide a matching lower bound of Ω(n^{s_D}) which even holds for arbitrary non-zero lattice elements, ruling out augmenting algorithm relying on even more restricted notions of augmentation than the Graver basis. We then consider a special case of 4-block n-fold in which C is a zero matrix, called 3-block n-fold IP. We show that while the 𝓁_∞-norm of its Graver elements is Ω(n^{s_D}), there exists a different decomposition into lattice elements whose 𝓁_∞-norm is bounded by 𝒪_FPT(1), which allows us to provide improved upper bounds on the 𝓁_∞-norm of Graver elements for 3-block n-fold IP. The key difference between the respective decompositions is that a Graver basis guarantees a sign-compatible decomposition; this property is critical in applications because it guarantees each step of the decomposition to be feasible. Consequently, our improved upper bounds let us establish faster algorithms for 3-block n-fold IP and 4-block IP, and our lower bounds strongly hint at parameterized hardness of 4-block and even 3-block n-fold IP. Furthermore, we show that 3-block n-fold IP is without loss of generality in the sense that 4-block n-fold IP can be solved in FPT oracle time by taking an algorithm for 3-block n-fold IP as an oracle.
Cite as

Lin Chen, Martin Koutecký, Lei Xu, and Weidong Shi. New Bounds on Augmenting Steps of Block-Structured Integer Programs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chen_et_al:LIPIcs.ESA.2020.33,
  author =	{Chen, Lin and Kouteck\'{y}, Martin and Xu, Lei and Shi, Weidong},
  title =	{{New Bounds on Augmenting Steps of Block-Structured Integer Programs}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{33:1--33:19},
  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.2020.33},
  URN =		{urn:nbn:de:0030-drops-128994},
  doi =		{10.4230/LIPIcs.ESA.2020.33},
  annote =	{Keywords: Integer Programming, Graver basis, Fixed parameter tractable}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.26
Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming

Authors: Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry Δ are solvable in time g(d,Δ) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and Δ. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth. We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width.
Cite as

Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková. Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.ICALP.2020.26,
  author =	{Chan, Timothy F. N. and Cooper, Jacob W. and Kouteck\'{y}, Martin and Kr\'{a}l', Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na},
  title =	{{Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{26:1--26:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.26},
  URN =		{urn:nbn:de:0030-drops-124339},
  doi =		{10.4230/LIPIcs.ICALP.2020.26},
  annote =	{Keywords: Matroid algorithms, width parameters, integer programming, fixed parameter tractability, branch-width, branch-depth}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2018.21
Integer Programming in Parameterized Complexity: Three Miniatures

Authors: Tomás Gavenciak, Dusan Knop, and Martin Koutecký

Published in: LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

Abstract
Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra's algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f(k) poly(n). We focus on: - Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used. - Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups. - Minding the poly(n): reducing f(k) often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds. Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, and Max-q-Cut by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, and indefinite quadratic programs in fixed dimension.
Cite as

Tomás Gavenciak, Dusan Knop, and Martin Koutecký. Integer Programming in Parameterized Complexity: Three Miniatures. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gavenciak_et_al:LIPIcs.IPEC.2018.21,
  author =	{Gavenciak, Tom\'{a}s and Knop, Dusan and Kouteck\'{y}, Martin},
  title =	{{Integer Programming in Parameterized Complexity: Three Miniatures}},
  booktitle =	{13th International Symposium on Parameterized and Exact Computation (IPEC 2018)},
  pages =	{21:1--21:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-084-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{115},
  editor =	{Paul, Christophe and Pilipczuk, Michal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.21},
  URN =		{urn:nbn:de:0030-drops-102225},
  doi =		{10.4230/LIPIcs.IPEC.2018.21},
  annote =	{Keywords: graph coloring, parameterized complexity, integer linear programming, integer convex programming}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.85
A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs

Authors: Martin Koutecký, Asaf Levin, and Shmuel Onn

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

Abstract
The theory of n-fold integer programming has been recently emerging as an important tool in parameterized complexity. The input to an n-fold integer program (IP) consists of parameter A, dimension n, and numerical data of binary encoding length L. It was known for some time that such programs can be solved in polynomial time using O(n^{g(A)}L) arithmetic operations where g is an exponential function of the parameter. In 2013 it was shown that it can be solved in fixed-parameter tractable time using O(f(A)n^3L) arithmetic operations for a single-exponential function f. This, and a faster algorithm for a special case of combinatorial n-fold IP, have led to several very recent breakthroughs in the parameterized complexity of scheduling, stringology, and computational social choice. In 2015 it was shown that it can be solved in strongly polynomial time using O(n^{g(A)}) arithmetic operations. Here we establish a result which subsumes all three of the above results by showing that n-fold IP can be solved in strongly polynomial fixed-parameter tractable time using O(f(A)n^6 log n) arithmetic operations. In fact, our results are much more general, briefly outlined as follows. - There is a strongly polynomial algorithm for integer linear programming (ILP) whenever a so-called Graver-best oracle is realizable for it. - Graver-best oracles for the large classes of multi-stage stochastic and tree-fold ILPs can be realized in fixed-parameter tractable time. Together with the previous oracle algorithm, this newly shows two large classes of ILP to be strongly polynomial; in contrast, only few classes of ILP were previously known to be strongly polynomial. - We show that ILP is fixed-parameter tractable parameterized by the largest coefficient |A |_infty and the primal or dual treedepth of A, and that this parameterization cannot be relaxed, signifying substantial progress in understanding the parameterized complexity of ILP.
Cite as

Martin Koutecký, Asaf Levin, and Shmuel Onn. A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 85:1-85:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{koutecky_et_al:LIPIcs.ICALP.2018.85,
  author =	{Kouteck\'{y}, Martin and Levin, Asaf and Onn, Shmuel},
  title =	{{A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{85:1--85:14},
  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.2018.85},
  URN =		{urn:nbn:de:0030-drops-90898},
  doi =		{10.4230/LIPIcs.ICALP.2018.85},
  annote =	{Keywords: integer programming, parameterized complexity, Graver basis, n-fold integer programming}
}
Document
DOI: 10.4230/LIPIcs.SEA.2018.10
Evaluating and Tuning n-fold Integer Programming

Authors: Katerina Altmanová, Dusan Knop, and Martin Koutecký

Published in: LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

Abstract
In recent years, algorithmic breakthroughs in stringology, computational social choice, scheduling, etc., were achieved by applying the theory of so-called n-fold integer programming. An n-fold integer program (IP) has a highly uniform block structured constraint matrix. Hemmecke, Onn, and Romanchuk [Math. Programming, 2013] showed an algorithm with runtime a^{O(rst + r^2s)} n^3, where a is the largest coefficient, r,s, and t are dimensions of blocks of the constraint matrix and n is the total dimension of the IP; thus, an algorithm efficient if the blocks are of small size and with small coefficients. The algorithm works by iteratively improving a feasible solution with augmenting steps, and n-fold IPs have the special property that augmenting steps are guaranteed to exist in a not-too-large neighborhood. However, this algorithm has never been implemented and evaluated. We have implemented the algorithm and learned the following along the way. The original algorithm is practically unusable, but we discover a series of improvements which make its evaluation possible. Crucially, we observe that a certain constant in the algorithm can be treated as a tuning parameter, which yields an efficient heuristic (essentially searching in a smaller-than-guaranteed neighborhood). Furthermore, the algorithm uses an overly expensive strategy to find a "best" step, while finding only an "approximatelly best" step is much cheaper, yet sufficient for quick convergence. Using this insight, we improve the asymptotic dependence on n from n^3 to n^2 log n which yields the currently asymptotically fastest algorithm for n-fold IP. Finally, we tested the behavior of the algorithm with various values of the tuning parameter and different strategies of finding improving steps. First, we show that decreasing the tuning parameter initially leads to an increased number of iterations needed for convergence and eventually to getting stuck in local optima, as expected. However, surprisingly small values of the parameter already exhibit good behavior. Second, our new strategy for finding "approximatelly best" steps wildly outperforms the original construction.
Cite as

Katerina Altmanová, Dusan Knop, and Martin Koutecký. Evaluating and Tuning n-fold Integer Programming. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{altmanova_et_al:LIPIcs.SEA.2018.10,
  author =	{Altmanov\'{a}, Katerina and Knop, Dusan and Kouteck\'{y}, Martin},
  title =	{{Evaluating and Tuning n-fold Integer Programming}},
  booktitle =	{17th International Symposium on Experimental Algorithms (SEA 2018)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-070-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{103},
  editor =	{D'Angelo, Gianlorenzo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.10},
  URN =		{urn:nbn:de:0030-drops-89454},
  doi =		{10.4230/LIPIcs.SEA.2018.10},
  annote =	{Keywords: n-fold integer programming, integer programming, analysis of algorithms, primal heuristic, local search}
}
Document
DOI: 10.4230/LIPIcs.ESA.2017.54
Combinatorial n-fold Integer Programming and Applications

Authors: Dusan Knop, Martin Koutecky, and Matthias Mnich

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract
Many fundamental NP-hard problems can be formulated as integer linear programs (ILPs). A famous algorithm by Lenstra allows to solve ILPs in time that is exponential only in the dimension of the program. That algorithm therefore became a ubiquitous tool in the design of fixed-parameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problem by some parameter. However, it was discovered that in many cases using Lenstra’s algorithm has two drawbacks: First, the run time of the resulting algorithms is often doubly-exponential in the parameter, and second, an ILP formulation in small dimension can not easily express problems which involve many different costs. Inspired by the work of Hemmecke, Onn and Romanchuk [Math. Prog. 2013], we develop a single-exponential algorithm for so-called combinatorial n-fold integer programs, which are remarkably similar to prior ILP formulations for various problems, but unlike them, also allow variable dimension. We then apply our algorithm to a few representative problems like Closest String, Swap Bribery, Weighted Set Multicover, and obtain exponential speedups in the dependence on the respective parameters, the input size, or both. Unlike Lenstra’s algorithm, which is essentially a bounded search tree algorithm, our result uses the technique of augmenting steps. At its heart is a deep result stating that in combinatorial n-fold IPs an existence of an augmenting step implies an existence of a “local” augmenting step, which can be found using dynamic programming. Our results provide an important insight into many problems by showing that they exhibit this phenomenon, and highlights the importance of augmentation techniques.
Cite as

Dusan Knop, Martin Koutecky, and Matthias Mnich. Combinatorial n-fold Integer Programming and Applications. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{knop_et_al:LIPIcs.ESA.2017.54,
  author =	{Knop, Dusan and Koutecky, Martin and Mnich, Matthias},
  title =	{{Combinatorial n-fold Integer Programming and Applications}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.54},
  URN =		{urn:nbn:de:0030-drops-78616},
  doi =		{10.4230/LIPIcs.ESA.2017.54},
  annote =	{Keywords: integer programming, closest strings, fixed-parameter algorithms}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.46
Voting and Bribing in Single-Exponential Time

Authors: Dusan Knop, Martin Koutecký, and Matthias Mnich

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
We introduce a general problem about bribery in voting systems. In the R-Multi-Bribery problem, the goal is to bribe a set of voters at minimum cost such that a desired candidate wins the manipulated election under the voting rule R. Voters assign prices for withdrawing their vote, for swapping the positions of two consecutive candidates in their preference order, and for perturbing their approval count for a candidate. As our main result, we show that R-Multi-Bribery is fixed-parameter tractable parameterized by the number of candidates for many natural voting rules R, including Kemeny rule, all scoring protocols, maximin rule, Bucklin rule, fallback rule, SP-AV, and any C1 rule. In particular, our result resolves the parameterized of R-Swap Bribery for all those voting rules, thereby solving a long-standing open problem and "Challenge #2" of the 9 Challenges in computational social choice by Bredereck et al. Further, our algorithm runs in single-exponential time for arbitrary cost; it thus improves the earlier double-exponential time algorithm by Dorn and Schlotter that is restricted to the unit-cost case for all scoring protocols, the maximin rule, and Bucklin rule.
Cite as

Dusan Knop, Martin Koutecký, and Matthias Mnich. Voting and Bribing in Single-Exponential Time. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{knop_et_al:LIPIcs.STACS.2017.46,
  author =	{Knop, Dusan and Kouteck\'{y}, Martin and Mnich, Matthias},
  title =	{{Voting and Bribing in Single-Exponential Time}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.46},
  URN =		{urn:nbn:de:0030-drops-69885},
  doi =		{10.4230/LIPIcs.STACS.2017.46},
  annote =	{Keywords: Parameterized algorithm, swap bribery, n-fold integer programming}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2016.18
Extension Complexity, MSO Logic, and Treewidth

Authors: Petr Kolman, Martin Koutecký, and Hans Raj Tiwary

Published in: LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Abstract
We consider the convex hull P_phi(G) of all satisfying assignments of a given MSO_2 formula phi on a given graph G. We show that there exists an extended formulation of the polytope P_phi(G) that can be described by f(|phi|,tau)*n inequalities, where n is the number of vertices in G, tau is the treewidth of G and f is a computable function depending only on phi and tau. In other words, we prove that the extension complexity of P_phi(G) is linear in the size of the graph G, with a constant depending on the treewidth of G and the formula phi. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs.
Cite as

Petr Kolman, Martin Koutecký, and Hans Raj Tiwary. Extension Complexity, MSO Logic, and Treewidth. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kolman_et_al:LIPIcs.SWAT.2016.18,
  author =	{Kolman, Petr and Kouteck\'{y}, Martin and Tiwary, Hans Raj},
  title =	{{Extension Complexity, MSO Logic, and Treewidth}},
  booktitle =	{15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-011-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{53},
  editor =	{Pagh, Rasmus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.18},
  URN =		{urn:nbn:de:0030-drops-60405},
  doi =		{10.4230/LIPIcs.SWAT.2016.18},
  annote =	{Keywords: Extension Complexity, FPT, Courcelle's Theorem, MSO Logic}
}
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