# 20 Search Results for "Knop, Dušan"

Document
Track A: Algorithms, Complexity and Games
##### 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)

```@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},
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
##### 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)

```@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},
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
##### 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)

```@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},
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
##### 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)

```@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},
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
##### 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)

```@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},
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
##### 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)

```@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},
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
##### 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)

```@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},
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
##### 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)

```@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},
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
##### 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)

```@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},
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
##### 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)

```@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},
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}
}```
Document
##### Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters

Authors: Robert Bredereck, Klaus Heeger, Dušan Knop, and Rolf Niedermeier

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

##### Abstract
We continue and extend previous work on the parameterized complexity analysis of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby strengthening earlier results both on the side of parameterized hardness as well as on the side of fixed-parameter tractability. Other than for its famous sister problem Stable Marriage which focuses on a bipartite scenario, Stable Roommates with Incomplete Lists allows for arbitrary acceptability graphs whose edges specify the possible matchings of each two agents (agents are represented by graph vertices). Herein, incomplete lists and ties reflect the fact that in realistic application scenarios the agents cannot bring all other agents into a linear order. Among our main contributions is to show that it is W[1]-hard to compute a maximum-cardinality stable matching for acceptability graphs of bounded treedepth, bounded tree-cut width, and bounded feedback vertex number (these are each time the respective parameters). However, if we "only" ask for perfect stable matchings or the mere existence of a stable matching, then we obtain fixed-parameter tractability with respect to tree-cut width but not with respect to treedepth. On the positive side, we also provide fixed-parameter tractability results for the parameter feedback edge set number.

##### Cite as

Robert Bredereck, Klaus Heeger, Dušan Knop, and Rolf Niedermeier. Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

```@InProceedings{bredereck_et_al:LIPIcs.ISAAC.2019.44,
author =	{Bredereck, Robert and Heeger, Klaus and Knop, Du\v{s}an and Niedermeier, Rolf},
title =	{{Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters}},
booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
pages =	{44:1--44:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-130-6},
ISSN =	{1868-8969},
year =	{2019},
volume =	{149},
editor =	{Lu, Pinyan and Zhang, Guochuan},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.44},
URN =		{urn:nbn:de:0030-drops-115406},
doi =		{10.4230/LIPIcs.ISAAC.2019.44},
annote =	{Keywords: Stable matching, acceptability graph, fixed-parameter tractability, W\lbrack1\rbrack-hardness, treewidth, treedepth, tree-cut width, feedback set numbers}
}```
Document
##### Parameterized Complexity of Fair Vertex Evaluation Problems

Authors: Dušan Knop, Tomáš Masařík, and Tomáš Toufar

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

##### Abstract
A prototypical graph problem is centered around a graph-theoretic property for a set of vertices and a solution to it is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of a certain quality, where we use size as a quality measure. In this work, we are changing the measure to the fair measure (cf. Lin and Sahni [Li-Shin Lin and Sartaj Sahni, 1989]). The fair measure of a set of vertices S is (at most) k if the number of neighbors in the set S of any vertex (in the input graph) does not exceed k. One possible way to study graph problems is by defining the property in a certain logic. For a given objective, an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. More formally, we study the {MSO} Fair Vertex Evaluation, where the graph-theoretic property is described by an {MSO} formula. In the presented paper we show that there is an FPT algorithm for the {MSO} Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph and the size of the formula. One may define an extended variant of {MSO} Fair Vertex Evaluation for formulas with l free variables; here we measure a maximum number of neighbors in each of the l sets. However, such variant is {W[1]}-hard for parameter l even on graphs with twin cover one. Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is {W[1]}-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width.

##### Cite as

Dušan Knop, Tomáš Masařík, and Tomáš Toufar. Parameterized Complexity of Fair Vertex Evaluation Problems. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

```@InProceedings{knop_et_al:LIPIcs.MFCS.2019.33,
author =	{Knop, Du\v{s}an and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s}},
title =	{{Parameterized Complexity of Fair Vertex Evaluation Problems}},
booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages =	{33:1--33:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-117-7},
ISSN =	{1868-8969},
year =	{2019},
volume =	{138},
editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.33},
URN =		{urn:nbn:de:0030-drops-109775},
doi =		{10.4230/LIPIcs.MFCS.2019.33},
annote =	{Keywords: Fair objective, metatheorem, fair vertex cover, twin cover, modular width}
}```
Document
##### Complexity of the Steiner Network Problem with Respect to the Number of Terminals

Authors: Eduard Eiben, Dušan Knop, Fahad Panolan, and Ondřej Suchý

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

##### Abstract
In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T subseteq V(G) with |T|=q, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph H subseteq G of the minimum cost such that there is a directed path from s to t in H for all st in A(R). It is known that the problem can be solved in time |V(G)|^{O(|A(R)|)} [Feldman and Ruhl, SIAM J. Comput. 2006] and cannot be solved in time |V(G)|^{o(|A(R)|)} even if G is planar, unless the Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time |V(G)|^{o(q)}, unless ETH fails. Therefore, there is a significant gap in the complexity with respect to q in the exponent. We show that Directed Steiner Network is solvable in time f(q)* |V(G)|^{O(c_g * q)}, where c_g is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no f(q)* |V(G)|^{o(q^2/ log q)} algorithm for any function f for the problem on general graphs, unless ETH fails.

##### Cite as

Eduard Eiben, Dušan Knop, Fahad Panolan, and Ondřej Suchý. Complexity of the Steiner Network Problem with Respect to the Number of Terminals. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

```@InProceedings{eiben_et_al:LIPIcs.STACS.2019.25,
author =	{Eiben, Eduard and Knop, Du\v{s}an and Panolan, Fahad and Such\'{y}, Ond\v{r}ej},
title =	{{Complexity of the Steiner Network Problem with Respect to the Number of Terminals}},
booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
pages =	{25:1--25:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-100-9},
ISSN =	{1868-8969},
year =	{2019},
volume =	{126},
editor =	{Niedermeier, Rolf and Paul, Christophe},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.25},
URN =		{urn:nbn:de:0030-drops-102642},
doi =		{10.4230/LIPIcs.STACS.2019.25},
annote =	{Keywords: Directed Steiner Network, Planar Graphs, Parameterized Algorithms, Bounded Genus, Exponential Time Hypothesis}
}```
Document
##### Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Authors: Dušan Knop, Michał Pilipczuk, and Marcin Wrochna

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

##### Abstract
We consider the standard ILP Feasibility problem: given an integer linear program of the form {Ax = b, x >= 0}, where A is an integer matrix with k rows and l columns, x is a vector of l variables, and b is a vector of k integers, we ask whether there exists x in N^l that satisfies Ax = b. Each row of A specifies one linear constraint on x; our goal is to study the complexity of ILP Feasibility when both k, the number of constraints, and |A|_infty, the largest absolute value of an entry in A, are small. Papadimitriou [Christos H. Papadimitriou, 1981] was the first to give a fixed-parameter algorithm for ILP Feasibility under parameterization by the number of constraints that runs in time ((|A |_infty + |b|_infty) * k)^O(k^2). This was very recently improved by Eisenbrand and Weismantel [Friedrich Eisenbrand and Robert Weismantel, 2018], who used the Steinitz lemma to design an algorithm with running time (k |A|_infty)^{O(k)}* |b|_infty^2, which was subsequently improved by Jansen and Rohwedder [Klaus Jansen and Lars Rohwedder, 2019] to O(k |A |_infty)^k* log |b|_infty. We prove that for {0,1}-matrices A, the running time of the algorithm of Eisenbrand and Weismantel is probably optimal: an algorithm with running time 2^{o(k log k)}* (l+|{b}|_infty)^{o(k)} would contradict the Exponential Time Hypothesis (ETH). This improves previous non-tight lower bounds of Fomin et al. [Fedor V. Fomin et al., 2018]. We then consider integer linear programs that may have many constraints, but they need to be structured in a "shallow" way. Precisely, we consider the parameter {dual treedepth} of the matrix A, denoted td_D(A), which is the treedepth of the graph over the rows of A, where two rows are adjacent if in some column they simultaneously contain a non-zero entry. It was recently shown by Koutecký et al. [Martin Koutecký et al., 2018] that {ILP Feasibility} can be solved in time |A |_infty^{2^O(td_D(A))} * (k+l+log |b|_infty)^O(1). We present a streamlined proof of this fact and prove that, again, this running time is probably optimal: even assuming that all entries of A and {b} are in {-1,0,1}, the existence of an algorithm with running time 2^{2^o(td_D(A))} * (k+l)^O(1) would contradict the ETH.

##### Cite as

Dušan Knop, Michał Pilipczuk, and Marcin Wrochna. Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

```@InProceedings{knop_et_al:LIPIcs.STACS.2019.44,
author =	{Knop, Du\v{s}an and Pilipczuk, Micha{\l} and Wrochna, Marcin},
title =	{{Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints}},
booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
pages =	{44:1--44:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-100-9},
ISSN =	{1868-8969},
year =	{2019},
volume =	{126},
editor =	{Niedermeier, Rolf and Paul, Christophe},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.44},
URN =		{urn:nbn:de:0030-drops-102831},
doi =		{10.4230/LIPIcs.STACS.2019.44},
annote =	{Keywords: integer linear programming, fixed-parameter tractability, ETH}
}```
Document
##### 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)

```@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},
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}
}```
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