25 Search Results for "Knop, Dušan"

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.ESA.2024.85
Tree Decompositions Meet Induced Matchings: Beyond Max Weight Independent Set

Authors: Paloma T. Lima, Martin Milanič, Peter Muršič, Karolina Okrasa, Paweł Rzążewski, and Kenny Štorgel

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

Abstract
For a tree decomposition 𝒯 of a graph G, by μ(𝒯) we denote the size of a largest induced matching in G all of whose edges intersect one bag of 𝒯. The induced matching treewidth of a graph G is the minimum value of μ(𝒯) over all tree decompositions 𝒯 of G. Yolov [SODA 2018] proved that for graphs of bounded induced matching treewidth, tree decompositions with bounded μ(𝒯) can be computed in polynomial time and Max Weight Independent Set can be solved in polynomial time. In this paper we explore what other problems are tractable in such classes of graphs. As our main result, we give a polynomial-time algorithm for Min Weight Feedback Vertex Set. We also provide some positive results concerning packing induced subgraphs, which in particular imply a PTAS for the problem of finding a largest induced subgraph of bounded treewidth. These results suggest that in graphs of bounded induced matching treewidth, one could find in polynomial time a maximum-weight induced subgraph of bounded treewidth satisfying a given CMSO₂ formula. We conjecture that such a result indeed holds and prove it for graphs of bounded tree-independence number, which form a rich and important family of subclasses of graphs of bounded induced matching treewidth. We complement these algorithmic results with a number of complexity and structural results concerning induced matching treewidth, including a linear relation to treewidth for graphs with bounded degree.
Cite as

Paloma T. Lima, Martin Milanič, Peter Muršič, Karolina Okrasa, Paweł Rzążewski, and Kenny Štorgel. Tree Decompositions Meet Induced Matchings: Beyond Max Weight Independent Set. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 85:1-85:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lima_et_al:LIPIcs.ESA.2024.85,
  author =	{Lima, Paloma T. and Milani\v{c}, Martin and Mur\v{s}i\v{c}, Peter and Okrasa, Karolina and Rz\k{a}\.{z}ewski, Pawe{\l} and \v{S}torgel, Kenny},
  title =	{{Tree Decompositions Meet Induced Matchings: Beyond Max Weight Independent Set}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{85:1--85:17},
  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.2024.85},
  URN =		{urn:nbn:de:0030-drops-211569},
  doi =		{10.4230/LIPIcs.ESA.2024.85},
  annote =	{Keywords: induced matching treewidth, tree-independence number, feedback vertex set, induced packing, algorithmic meta-theorem}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.96
Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM

Authors: Tim Randolph and Karol Węgrzycki

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

Abstract
We study the parameterized complexity of algorithmic problems whose input is an integer set A in terms of the doubling constant 𝒞 := |A+A| / |A|, a fundamental measure of additive structure. We present evidence that this new parameterization is algorithmically useful in the form of new results for two difficult, well-studied problems: Integer Programming and Subset Sum. First, we show that determining the feasibility of bounded Integer Programs is a tractable problem when parameterized in the doubling constant. Specifically, we prove that the feasibility of an integer program ℐ with n polynomially-bounded variables and m constraints can be determined in time n^{O_𝒞(1)} ⋅ poly(|ℐ|) when the column set of the constraint matrix has doubling constant 𝒞. Second, we show that the Subset Sum and Unbounded Subset Sum problems can be solved in time n^{O_C(1)} and n^{O_𝒞(log log log n)}, respectively, where the O_C notation hides functions that depend only on the doubling constant 𝒞. We also show the equivalence of achieving an FPT algorithm for Subset Sum with bounded doubling and achieving a milestone result for the parameterized complexity of Box ILP. Finally, we design near-linear time algorithms for k-SUM as well as tight lower bounds for 4-SUM and nearly tight lower bounds for k-SUM, under the k-SUM conjecture. Several of our results rely on a new proof that Freiman’s Theorem, a central result in additive combinatorics, can be made efficiently constructive. This result may be of independent interest.
Cite as

Tim Randolph and Karol Węgrzycki. Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 96:1-96:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{randolph_et_al:LIPIcs.ESA.2024.96,
  author =	{Randolph, Tim and W\k{e}grzycki, Karol},
  title =	{{Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{96:1--96:19},
  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.2024.96},
  URN =		{urn:nbn:de:0030-drops-211672},
  doi =		{10.4230/LIPIcs.ESA.2024.96},
  annote =	{Keywords: Parameterized algorithms, parameterized complexity, additive combinatorics, Subset Sum, integer programming, doubling constant}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.29
Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra

Authors: Václav Blažej, Dušan Knop, Jan Pokorný, and Šimon Schierreich

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
In the Equitable Connected Partition (ECP for short) problem, we are given a graph G = (V,E) together with an integer p ∈ ℕ, and our goal is to find a partition of V into p parts such that each part induces a connected sub-graph of G and the size of each two parts differs by at most 1. On the one hand, the problem is known to be NP-hard in general and W[1]-hard with respect to the path-width, the feedback-vertex set, and the number of parts p combined. On the other hand, fixed-parameter algorithms are known for parameters the vertex-integrity and the max leaf number. In this work, we systematically study ECP with respect to various structural restrictions of the underlying graph and provide a clear dichotomy of its parameterised complexity. Specifically, we show that the problem is in FPT when parameterized by the modular-width and the distance to clique. Next, we prove W[1]-hardness with respect to the distance to cluster, the 4-path vertex cover number, the distance to disjoint paths, and the feedback-edge set, and NP-hardness for constant shrub-depth graphs. Our hardness results are complemented by matching algorithmic upper-bounds: we give an XP algorithm for parameterisation by the tree-width and the distance to cluster. We also give an improved FPT algorithm for parameterisation by the vertex integrity and the first explicit FPT algorithm for the 3-path vertex cover number. The main ingredient of these algorithms is a formulation of ECP as N-fold IP, which clearly indicates that such formulations may, in certain scenarios, significantly outperform existing algorithms based on the famous algorithm of Lenstra.
Cite as

Václav Blažej, Dušan Knop, Jan Pokorný, and Šimon Schierreich. Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blazej_et_al:LIPIcs.MFCS.2024.29,
  author =	{Bla\v{z}ej, V\'{a}clav and Knop, Du\v{s}an and Pokorn\'{y}, Jan and Schierreich, \v{S}imon},
  title =	{{Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.29},
  URN =		{urn:nbn:de:0030-drops-205857},
  doi =		{10.4230/LIPIcs.MFCS.2024.29},
  annote =	{Keywords: Equitable Connected Partition, structural parameters, fixed-parameter tractability, N-fold integer programming, tree-width, shrub-depth, modular-width}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.58
Parameterized Vertex Integrity Revisited

Authors: Tesshu Hanaka, Michael Lampis, Manolis Vasilakis, and Kanae Yoshiwatari

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are very structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of structural parameterizations. We present a number of new results, which also answer some recently posed open questions from the literature. Specifically, we show that unweighted vertex integrity is W[1]-hard parameterized by treedepth; we show that the problem remains W[1]-hard if we parameterize by feedback edge set size (via a reduction from a Bin Packing variant which may be of independent interest); and complementing this we show that the problem is FPT by max-leaf number. Furthermore, for weighted vertex integrity, we show that the problem admits a single-exponential FPT algorithm parameterized by vertex cover or by modular width, the latter result improving upon a previous algorithm which required weights to be polynomially bounded.
Cite as

Tesshu Hanaka, Michael Lampis, Manolis Vasilakis, and Kanae Yoshiwatari. Parameterized Vertex Integrity Revisited. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hanaka_et_al:LIPIcs.MFCS.2024.58,
  author =	{Hanaka, Tesshu and Lampis, Michael and Vasilakis, Manolis and Yoshiwatari, Kanae},
  title =	{{Parameterized Vertex Integrity Revisited}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{58:1--58:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.58},
  URN =		{urn:nbn:de:0030-drops-206141},
  doi =		{10.4230/LIPIcs.MFCS.2024.58},
  annote =	{Keywords: Parameterized Complexity, Treedepth, Vertex Integrity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.51
Computing Tree Decompositions with Small Independence Number

Authors: Clément Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Martin Milanič

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum weight independent set, can be solved in time n^𝒪(k) if the input n-vertex graph is given together with a tree decomposition of independence number k. Yolov in [SODA 2018] gave an algorithm that given an n-vertex graph G and an integer k, in time n^𝒪(k³) either constructs a tree decomposition of G whose independence number is 𝒪(k³) or correctly reports that the tree-independence number of G is larger than k. In this paper, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time 2^𝒪(k²) n^𝒪(k) and either outputs a tree decomposition of G with independence number at most 8k, or determines that the tree-independence number of G is larger than k. This implies 2^𝒪(k²) n^𝒪(k)-time algorithms for various problems, like maximum weight independent set, parameterized by the tree-independence number k without needing the decomposition as an input. Assuming Gap-ETH, an n^Ω(k) factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant k ≥ 4 it is NP-hard to decide if a given graph has the tree-independence number at most k.
Cite as

Clément Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Martin Milanič. Computing Tree Decompositions with Small Independence Number. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dallard_et_al:LIPIcs.ICALP.2024.51,
  author =	{Dallard, Cl\'{e}ment and Fomin, Fedor V. and Golovach, Petr A. and Korhonen, Tuukka and Milani\v{c}, Martin},
  title =	{{Computing Tree Decompositions with Small Independence Number}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{51:1--51:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.51},
  URN =		{urn:nbn:de:0030-drops-201945},
  doi =		{10.4230/LIPIcs.ICALP.2024.51},
  annote =	{Keywords: tree-independence number, approximation, parameterized algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.53
Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy

Authors: Argyrios Deligkas, Eduard Eiben, Robert Ganian, Iyad Kanj, and M. S. Ramanujan

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We study the parameterized complexity of a generalization of the coordinated motion planning problem on graphs, where the goal is to route a specified subset of a given set of k robots to their destinations with the aim of minimizing the total energy (i.e., the total length traveled). We develop novel techniques to push beyond previously-established results that were restricted to solid grids. We design a fixed-parameter additive approximation algorithm for this problem parameterized by k alone. This result, which is of independent interest, allows us to prove the following two results pertaining to well-studied coordinated motion planning problems: (1) A fixed-parameter algorithm, parameterized by k, for routing a single robot to its destination while avoiding the other robots, which is related to the famous Rush-Hour Puzzle; and (2) a fixed-parameter algorithm, parameterized by k plus the treewidth of the input graph, for the standard Coordinated Motion Planning (CMP) problem in which we need to route all the k robots to their destinations. The latter of these results implies, among others, the fixed-parameter tractability of CMP parameterized by k on graphs of bounded outerplanarity, which include bounded-height subgrids. We complement the above results with a lower bound which rules out the fixed-parameter tractability for CMP when parameterized by the total energy. This contrasts the recently-obtained tractability of the problem on solid grids under the same parameterization. As our final result, we strengthen the aforementioned fixed-parameter tractability to hold not only on solid grids but all graphs of bounded local treewidth - a class including, among others, all graphs of bounded genus.
Cite as

Argyrios Deligkas, Eduard Eiben, Robert Ganian, Iyad Kanj, and M. S. Ramanujan. Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{deligkas_et_al:LIPIcs.ICALP.2024.53,
  author =	{Deligkas, Argyrios and Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ramanujan, M. S.},
  title =	{{Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{53:1--53:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.53},
  URN =		{urn:nbn:de:0030-drops-201968},
  doi =		{10.4230/LIPIcs.ICALP.2024.53},
  annote =	{Keywords: coordinated motion planning, multi-agent path finding, parameterized complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.61
Parameterized Algorithms for Steiner Forest in Bounded Width Graphs

Authors: Andreas Emil Feldmann and Michael Lampis

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In this paper we reassess the parameterized complexity and approximability of the well-studied Steiner Forest problem in several graph classes of bounded width. The problem takes an edge-weighted graph and pairs of vertices as input, and the aim is to find a minimum cost subgraph in which each given vertex pair lies in the same connected component. It is known that this problem is APX-hard in general, and NP-hard on graphs of treewidth 3, treedepth 4, and feedback vertex set size 2. However, Bateni, Hajiaghayi and Marx [JACM, 2011] gave an approximation scheme with a runtime of n^O(k²/ε) on graphs of treewidth k. Our main result is a much faster efficient parameterized approximation scheme (EPAS) with a runtime of 2^O(k²/ε log k/ε)⋅n^O(1). If k instead is the vertex cover number of the input graph, we show how to compute the optimum solution in 2^O(k log k)⋅n^O(1) time, and we also prove that this runtime dependence on k is asymptotically best possible, under ETH. Furthermore, if k is the size of a feedback edge set, then we obtain a faster 2^O(k)⋅n^O(1) time algorithm, which again cannot be improved under ETH.
Cite as

Andreas Emil Feldmann and Michael Lampis. Parameterized Algorithms for Steiner Forest in Bounded Width Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 61:1-61:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{feldmann_et_al:LIPIcs.ICALP.2024.61,
  author =	{Feldmann, Andreas Emil and Lampis, Michael},
  title =	{{Parameterized Algorithms for Steiner Forest in Bounded Width Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{61:1--61:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.61},
  URN =		{urn:nbn:de:0030-drops-202048},
  doi =		{10.4230/LIPIcs.ICALP.2024.61},
  annote =	{Keywords: Steiner Forest, Approximation Algorithms, FPT algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.67
Subexponential Parameterized Directed Steiner Network Problems on Planar Graphs: A Complete Classification

Authors: Esther Galby, Sándor Kisfaludi-Bak, Dániel Marx, and Roohani Sharma

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In the Directed Steiner Network problem, the input is a directed graph G, a set T ⊆ V(G) of k terminals, and a demand graph D on T. The task is to find a subgraph H ⊆ G with the minimum number of edges such that for every (s,t) ∈ E(D), the solution H contains a directed s → t path. The goal of this paper is to investigate how the complexity of the problem depends on the demand pattern in planar graphs. Formally, if 𝒟 is a class of directed graphs, then the 𝒟-Steiner Network (𝒟-DSN) problem is the special case where the demand graph D is restricted to be from 𝒟. We give a complete characterization of the behavior of every 𝒟-DSN problem on planar graphs. We classify every class 𝒟 closed under transitive equivalence and identification of vertices into three cases: assuming ETH, either the problem is 1) solvable in time 2^O(k)⋅n^O(1), i.e., FPT parameterized by the number k of terminals, but not solvable in time 2^o(k)⋅n^O(1), 2) solvable in time f(k)⋅n^O(√k), but cannot be solved in time f(k)⋅n^o(√k), or 3) solvable in time f(k)⋅n^O(k), but cannot be solved in time f(k)⋅n^o(k). Our result is a far-reaching generalization and unification of earlier results on Directed Steiner Tree, Directed Steiner Network, and Strongly Connected Steiner Subgraph on planar graphs. As an important step of our lower bound proof, we discover a rare example of a genuinely planar problem (i.e., described by a planar graph and two sets of vertices) that cannot be solved in time f(k)⋅n^o(k): given two sets of terminals S and T with |S|+|T| = k, find a subgraph with minimum number of edges such that every vertex of T is reachable from every vertex of S.
Cite as

Esther Galby, Sándor Kisfaludi-Bak, Dániel Marx, and Roohani Sharma. Subexponential Parameterized Directed Steiner Network Problems on Planar Graphs: A Complete Classification. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{galby_et_al:LIPIcs.ICALP.2024.67,
  author =	{Galby, Esther and Kisfaludi-Bak, S\'{a}ndor and Marx, D\'{a}niel and Sharma, Roohani},
  title =	{{Subexponential Parameterized Directed Steiner Network Problems on Planar Graphs: A Complete Classification}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{67:1--67:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.67},
  URN =		{urn:nbn:de:0030-drops-202104},
  doi =		{10.4230/LIPIcs.ICALP.2024.67},
  annote =	{Keywords: Directed Steiner Network, Sub-exponential algorithm}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2023.7
Treewidth Is NP-Complete on Cubic Graphs

Authors: Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondřej Suchý

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Abstract
In this paper, we show that Treewidth is NP-complete for cubic graphs, thereby improving the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9. We add a new and simpler proof of the NP-completeness of treewidth, and show that Treewidth remains NP-complete on subcubic induced subgraphs of the infinite 3-dimensional grid.
Cite as

Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondřej Suchý. Treewidth Is NP-Complete on Cubic Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2023.7,
  author =	{Bodlaender, Hans L. and Bonnet, \'{E}douard and Jaffke, Lars and Knop, Du\v{s}an and Lima, Paloma T. and Milani\v{c}, Martin and Ordyniak, Sebastian and Pandey, Sukanya and Such\'{y}, Ond\v{r}ej},
  title =	{{Treewidth Is NP-Complete on Cubic Graphs}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{7:1--7:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.7},
  URN =		{urn:nbn:de:0030-drops-194263},
  doi =		{10.4230/LIPIcs.IPEC.2023.7},
  annote =	{Keywords: Treewidth, cubic graphs, degree, NP-completeness}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.22
On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations

Authors: Václav Blažej, Pratibha Choudhary, Dušan Knop, Šimon Schierreich, Ondřej Suchý, and Tomáš Valla

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

Abstract
For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today’s computation, we employ one of the most successful models of such precomputation - the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations. We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also consider other structural parameters such as the vertex cover number and the size of a modulator to constant-sized paths. We complement our results from the negative side by showing that the existence of a polynomial-sized kernel with respect to the fractioning number, the combined parameter maximum degree and treewidth, and, in the case of {Subset TSP}, modulator to disjoint cycles (i.e., the treewidth two graphs) is unlikely.
Cite as

Václav Blažej, Pratibha Choudhary, Dušan Knop, Šimon Schierreich, Ondřej Suchý, and Tomáš Valla. On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blazej_et_al:LIPIcs.ESA.2022.22,
  author =	{Bla\v{z}ej, V\'{a}clav and Choudhary, Pratibha and Knop, Du\v{s}an and Schierreich, \v{S}imon and Such\'{y}, Ond\v{r}ej and Valla, Tom\'{a}\v{s}},
  title =	{{On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{22:1--22:16},
  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.2022.22},
  URN =		{urn:nbn:de:0030-drops-169600},
  doi =		{10.4230/LIPIcs.ESA.2022.22},
  annote =	{Keywords: Traveling Salesperson, Subset TSP, Waypoint Routing, Kernelization}
}
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
DOI: 10.4230/LIPIcs.IPEC.2020.8
Recognizing Proper Tree-Graphs

Authors: Steven Chaplick, Petr A. Golovach, Tim A. Hartmann, and Dušan Knop

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract
We investigate the parameterized complexity of the recognition problem for the proper H-graphs. The H-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph H, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of H-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper H-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-arc graphs. For these graph classes, H may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results. - For a tree T with t nodes, it can be decided in 2^{𝒪(t² log t)} ⋅ n³ time, whether an n-vertex graph G is a proper T-graph. For yes-instances, our algorithm outputs a proper T-representation. This proves that the recognition problem for proper H-graphs, where H required to be a tree, is fixed-parameter tractable when parameterized by the size of T. Previously only NP-completeness was known. - Contrasting to the first result, we prove that if H is not constrained to be a tree, then the recognition problem becomes much harder. Namely, we show that there is a multigraph H with 4 vertices and 5 edges such that it is NP-complete to decide whether G is a proper H-graph.
Cite as

Steven Chaplick, Petr A. Golovach, Tim A. Hartmann, and Dušan Knop. Recognizing Proper Tree-Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chaplick_et_al:LIPIcs.IPEC.2020.8,
  author =	{Chaplick, Steven and Golovach, Petr A. and Hartmann, Tim A. and Knop, Du\v{s}an},
  title =	{{Recognizing Proper Tree-Graphs}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{8:1--8:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.8},
  URN =		{urn:nbn:de:0030-drops-133118},
  doi =		{10.4230/LIPIcs.IPEC.2020.8},
  annote =	{Keywords: intersection graphs, H-graphs, recognition, fixed-parameter tractability}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.16
Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View

Authors: Radek Hušek, Dušan Knop, and Tomáš Masařk

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract
In the Steiner Tree problem, we are given an edge-weighted undirected graph G = (V,E) and a set of terminals R ⊆ V. The task is to find a connected subgraph of G containing R and minimizing the sum of weights of its edges. Steiner Tree is well known to be NP-complete and is undoubtedly one of the most studied problems in (applied) computer science. We observe that many approximation algorithms for Steiner Tree follow a similar scheme (meta-algorithm) and perform (exhaustively) a similar routine which we call star contraction. Here, by a star contraction, we mean finding a star-like subgraph in (the metric closure of) the input graph minimizing the ratio of its weight to the number of contained terminals minus one; and contract. It is not hard to see that the well-known MST-approximation seeks the best star to contract among those containing two terminals only. Zelikovsky’s approximation algorithm follows a similar workflow, finding the best star among those containing three terminals. We perform an empirical study of star contractions with the relaxed condition on the number of terminals in each star contraction motivated by a recent result of Dvořák et al. [Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices, STACS 2018]. Furthermore, we propose two improvements of Zelikovsky’s 11/6-approximation algorithm and we empirically confirm that the quality of the solution returned by any of these is better than the one returned by the former algorithm. However, such an improvement is exchanged for a slower running time (up to a multiplicative factor of the number of terminals).
Cite as

Radek Hušek, Dušan Knop, and Tomáš Masařk. Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{husek_et_al:LIPIcs.IPEC.2020.16,
  author =	{Hu\v{s}ek, Radek and Knop, Du\v{s}an and Masa\v{r}k, Tom\'{a}\v{s}},
  title =	{{Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.16},
  URN =		{urn:nbn:de:0030-drops-133193},
  doi =		{10.4230/LIPIcs.IPEC.2020.16},
  annote =	{Keywords: Steiner tree, approximation, star contractions, minimum spanning tree}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.36
Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

Authors: Matthias Bentert, Klaus Heeger, and Dušan Knop

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

Abstract
In the presented paper, we study the Length-Bounded Cut problem for special graph classes as well as from a parameterized-complexity viewpoint. Here, we are given a graph G, two vertices s and t, and positive integers β and λ. The task is to find a set F of edges of size at most β such that every s-t-path of length at most λ in G contains some edge in F. Bazgan et al. [Networks, 2019] conjectured that Length-Bounded Cut admits a polynomial-time algorithm if the input graph G is a proper interval graph. We confirm this conjecture by providing a dynamic-programming based polynomial-time algorithm. Moreover, we strengthen the W[1]-hardness result of Dvořák and Knop [Algorithmica, 2018] for Length-Bounded Cut parameterized by pathwidth. Our reduction is shorter, and the target of the reduction has stronger structural properties. Consequently, we give W[1]-hardness for the combined parameter pathwidth and maximum degree of the input graph. Finally, we prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both our hardness results complement known XP algorithms.
Cite as

Matthias Bentert, Klaus Heeger, and Dušan Knop. Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bentert_et_al:LIPIcs.ISAAC.2020.36,
  author =	{Bentert, Matthias and Heeger, Klaus and Knop, Du\v{s}an},
  title =	{{Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{36:1--36:14},
  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.2020.36},
  URN =		{urn:nbn:de:0030-drops-133800},
  doi =		{10.4230/LIPIcs.ISAAC.2020.36},
  annote =	{Keywords: Edge-disjoint paths, pathwidth, feedback vertex number}
}
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