34 Search Results for "Woeginger, Gerhard"

Document
DOI: 10.4230/LIPIcs.ESA.2022.36
Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space

Authors: Jiehua Chen and Sanjukta Roy

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

Abstract
We investigate the Euclidean 𝖽-Dimensional Stable Roommates problem, which asks whether a given set V of 𝖽⋅ n points from the 2-dimensional Euclidean space can be partitioned into n disjoint (unordered) subsets Π = {V₁,…,V_{n}} with |V_i| = 𝖽 for each V_i ∈ Π such that Π is {stable}. Here, {stability} means that no point subset W ⊆ V is blocking Π, and W is said to be {blocking} Π if |W| = 𝖽 such that ∑_{w' ∈ W}δ(w,w') < ∑_{v ∈ Π(w)}δ(w,v) holds for each point w ∈ W, where Π(w) denotes the subset V_i ∈ Π which contains w and δ(a,b) denotes the Euclidean distance between points a and b. Complementing the existing known polynomial-time result for 𝖽 = 2, we show that such polynomial-time algorithms cannot exist for any fixed number 𝖽 ≥ 3 unless P=NP. Our result for 𝖽 = 3 answers a decade-long open question in the theory of Stable Matching and Hedonic Games [Iwama et al., 2007; Arkin et al., 2009; Vladimir G. Deineko and Gerhard J. Woeginger, 2013; Vladimir G. Deineko and Gerhard J. Woeginger, 2013; David F. Manlove, 2013].
Cite as

Jiehua Chen and Sanjukta Roy. Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chen_et_al:LIPIcs.ESA.2022.36,
  author =	{Chen, Jiehua and Roy, Sanjukta},
  title =	{{Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{36:1--36:16},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.36},
  URN =		{urn:nbn:de:0030-drops-169741},
  doi =		{10.4230/LIPIcs.ESA.2022.36},
  annote =	{Keywords: stable matchings, multidimensional stable roommates, Euclidean preferences, coalition formation games, stable cores, NP-hardness}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.12
Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds

Authors: Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
We consider the unlabeled motion-planning problem of m unit-disc robots moving in a simple polygonal workspace of n edges. The goal is to find a motion plan that moves the robots to a given set of m target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end. If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time O(n log n + mn + m²) if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance 4 apart, any two target positions are at least distance 4 apart, and any pair of a start and a target positions is at least distance 3 apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.
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Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot. Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{banyassady_et_al:LIPIcs.SoCG.2022.12,
  author =	{Banyassady, Bahareh and de Berg, Mark and Bringmann, Karl and Buchin, Kevin and Fernau, Henning and Halperin, Dan and Kostitsyna, Irina and Okamoto, Yoshio and Slot, Stijn},
  title =	{{Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{12:1--12:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.12},
  URN =		{urn:nbn:de:0030-drops-160203},
  doi =		{10.4230/LIPIcs.SoCG.2022.12},
  annote =	{Keywords: motion planning, computational geometry, simple polygon}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2021.19
An Investigation of the Recoverable Robust Assignment Problem

Authors: Dennis Fischer, Tim A. Hartmann, Stefan Lendl, and Gerhard J. Woeginger

Published in: LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

Abstract
We investigate the so-called recoverable robust assignment problem on complete bipartite graphs, a mainstream problem in robust optimization: For two given linear cost functions c₁ and c₂ on the edges and a given integer k, the goal is to find two perfect matchings M₁ and M₂ that minimize the objective value c₁(M₁)+c₂(M₂), subject to the constraint that M₁ and M₂ have at least k edges in common. We derive a variety of results on this problem. First, we show that the problem is W[1]-hard with respect to parameter k, and also with respect to the complementary parameter k' = n/2-k. This hardness result holds even in the highly restricted special case where both cost functions c₁ and c₂ only take the values 0 and 1. (On the other hand, containment of the problem in XP is straightforward to see.) Next, as a positive result we construct a polynomial time algorithm for the special case where one cost function is Monge, whereas the other one is Anti-Monge. Finally, we study the variant where matching M₁ is frozen, and where the optimization goal is to compute the best corresponding matching M₂. This problem variant is known to be contained in the randomized parallel complexity class RNC², and we show that it is at least as hard as the infamous problem Exact Red-Blue Matching in Bipartite Graphs whose computational complexity is a long-standing open problem.
Cite as

Dennis Fischer, Tim A. Hartmann, Stefan Lendl, and Gerhard J. Woeginger. An Investigation of the Recoverable Robust Assignment Problem. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fischer_et_al:LIPIcs.IPEC.2021.19,
  author =	{Fischer, Dennis and Hartmann, Tim A. and Lendl, Stefan and Woeginger, Gerhard J.},
  title =	{{An Investigation of the Recoverable Robust Assignment Problem}},
  booktitle =	{16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
  pages =	{19:1--19:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-216-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{214},
  editor =	{Golovach, Petr A. and Zehavi, Meirav},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.19},
  URN =		{urn:nbn:de:0030-drops-154025},
  doi =		{10.4230/LIPIcs.IPEC.2021.19},
  annote =	{Keywords: assignment problem, matchings, exact matching, robust optimization, fixed paramter tractablity, RNC}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.33
Dispersing Obnoxious Facilities on a Graph

Authors: Alexander Grigoriev, Tim A. Hartmann, Stefan Lendl, and Gerhard J. Woeginger

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

Abstract
We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance delta from each other. We investigate the complexity of this problem in terms of the rational parameter delta. The problem is polynomially solvable, if the numerator of delta is 1 or 2, while all other cases turn out to be NP-hard.
Cite as

Alexander Grigoriev, Tim A. Hartmann, Stefan Lendl, and Gerhard J. Woeginger. Dispersing Obnoxious Facilities on a Graph. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 33:1-33:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{grigoriev_et_al:LIPIcs.STACS.2019.33,
  author =	{Grigoriev, Alexander and Hartmann, Tim A. and Lendl, Stefan and Woeginger, Gerhard J.},
  title =	{{Dispersing Obnoxious Facilities on a Graph}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{33:1--33:11},
  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},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.33},
  URN =		{urn:nbn:de:0030-drops-102729},
  doi =		{10.4230/LIPIcs.STACS.2019.33},
  annote =	{Keywords: algorithms, complexity, optimization, graph theory, facility location}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.20
Graph Similarity and Approximate Isomorphism

Authors: Martin Grohe, Gaurav Rattan, and Gerhard J. Woeginger

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs G,H of the same order n with adjacency matrices A_G,A_H, a well-studied measure of similarity is the Frobenius distance dist(G,H):=min_{pi}|A_G^{pi}-A_H|_F, where pi ranges over all permutations of the vertex set of G, where A_G^pi denotes the matrix obtained from A_G by permuting rows and columns according to pi, and where |M |_F is the Frobenius norm of a matrix M. The (weighted) graph similarity problem, denoted by GSim (WSim), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases. It is known that GSim (WSim) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSim is best done in the framework of linear algebra. We show that WSim is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where the associated (weighted) adjacency graph for one of the matrices has a bounded number of twin equivalence classes. The key parameter underlying our algorithm is the clustering number of a graph; this parameter arises in context of the spectral graph drawing machinery.
Cite as

Martin Grohe, Gaurav Rattan, and Gerhard J. Woeginger. Graph Similarity and Approximate Isomorphism. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{grohe_et_al:LIPIcs.MFCS.2018.20,
  author =	{Grohe, Martin and Rattan, Gaurav and Woeginger, Gerhard J.},
  title =	{{Graph Similarity and Approximate Isomorphism}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{20:1--20:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.20},
  URN =		{urn:nbn:de:0030-drops-96021},
  doi =		{10.4230/LIPIcs.MFCS.2018.20},
  annote =	{Keywords: Graph Similarity, Quadratic Assignment Problem, Approximate Graph Isomorphism}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2017.14
The Dominating Set Problem in Geometric Intersection Graphs

Authors: Mark de Berg, Sándor Kisfaludi-Bak, and Gerhard Woeginger

Published in: LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

Abstract
We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that Dominating Set on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size). In two and higher dimensions, we prove that Dominating Set is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. This generalizes known results from the literature. Finally, we establish W[1]-hardness for a large class of intersection graphs.
Cite as

Mark de Berg, Sándor Kisfaludi-Bak, and Gerhard Woeginger. The Dominating Set Problem in Geometric Intersection Graphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{deberg_et_al:LIPIcs.IPEC.2017.14,
  author =	{de Berg, Mark and Kisfaludi-Bak, S\'{a}ndor and Woeginger, Gerhard},
  title =	{{The Dominating Set Problem in Geometric Intersection Graphs}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{14:1--14:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Lokshtanov, Daniel and Nishimura, Naomi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.14},
  URN =		{urn:nbn:de:0030-drops-85538},
  doi =		{10.4230/LIPIcs.IPEC.2017.14},
  annote =	{Keywords: dominating set, intersection graph, W-hierarchy}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.4
The Open Shop Scheduling Problem

Authors: Gerhard J. Woeginger

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
We discuss the computational complexity, the approximability, the algorithmics and the combinatorics of the open shop scheduling problem. We summarize the most important results from the literature and explain their main ideas, we sketch the most beautiful proofs, and we also list a number of open problems.
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Gerhard J. Woeginger. The Open Shop Scheduling Problem. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{woeginger:LIPIcs.STACS.2018.4,
  author =	{Woeginger, Gerhard J.},
  title =	{{The Open Shop Scheduling Problem}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{4:1--4:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.4},
  URN =		{urn:nbn:de:0030-drops-85375},
  doi =		{10.4230/LIPIcs.STACS.2018.4},
  annote =	{Keywords: Algorithms, Complexity, Scheduling, Approximation}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2017.26
Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

Authors: Mark de Berg, Tim Leijsen, Aleksandar Markovic, André van Renssen, Marcel Roeloffzen, and Gerhard Woeginger

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract
We introduce the fully-dynamic conflict-free coloring problem for a set S of intervals in R^1 with respect to points, where the goal is to maintain a conflict-free coloring for S under insertions and deletions. A coloring is conflict-free if for each point p contained in some interval, p is contained in an interval whose color is not shared with any other interval containing p. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: - a lower bound on the number of recolorings as a function of the number of colors, which implies that with O(1) recolorings per update the worst-case number of colors is Omega(log n/log log n), and that any strategy using O(1/epsilon) colors needs Omega(epsilon n^epsilon) recolorings; - a coloring strategy that uses O(log n) colors at the cost of O(log n) recolorings, and another strategy that uses O(1/epsilon) colors at the cost of O(n^epsilon/epsilon) recolorings; - stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.
Cite as

Mark de Berg, Tim Leijsen, Aleksandar Markovic, André van Renssen, Marcel Roeloffzen, and Gerhard Woeginger. Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 26:1-26:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2017.26,
  author =	{de Berg, Mark and Leijsen, Tim and Markovic, Aleksandar and van Renssen, Andr\'{e} and Roeloffzen, Marcel and Woeginger, Gerhard},
  title =	{{Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{26:1--26:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.26},
  URN =		{urn:nbn:de:0030-drops-82683},
  doi =		{10.4230/LIPIcs.ISAAC.2017.26},
  annote =	{Keywords: Conflict-free colorings, Dynamic data structures, Kinetic data structures}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.5
Fine-Grained Complexity Analysis of Two Classic TSP Variants

Authors: Mark de Berg, Kevin Buchin, Bart M. P. Jansen, and Gerhard Woeginger

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic tsp problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamicprogramming exercise to solve this problem in O(n^2) time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Fréchet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic tsp in O(n*log^2(n)) time and its bottleneck version in O(n*log^3(n)) time. In the more general pyramidal tsp problem, the points to be visited are labeled 1, ..., n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) tsp problem also work for the pyramidal tsp problem in the plane. Our second set of results concerns the popular k-opt heuristic for tsp in the graph setting. More precisely, we study the k-opt decision problem, which asks whether a given tour can be improved by a k-opt move that replaces k edges in the tour by k new edges. A simple algorithm solves k-opt in O(n^k) time for fixed k. For 2-opt, this is easily seen to be optimal. For k = 3 we prove that an algorithm with a runtime of the form ~O(n^{3-epsilon}) exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. For general k-opt, it is known that a runtime of f(k)*n^{o(k/log(k))} would contradict the Exponential Time Hypothesis. The results for k = 2, 3 may suggest that the actual time complexity of k-opt is Theta(n^k). We show that this is not the case, by presenting an algorithm that finds the best k-move in O(n^{lfoor 2k/3 rfloor +1}) time for fixed k >= 3. This implies that 4-opt can be solved in O(n^3) time, matching the best-known algorithm for 3-opt. Finally, we show how to beat the quadratic barrier for k = 2 in two important settings, namely for points in the plane and when we want to solve 2-opt repeatedly
Cite as

Mark de Berg, Kevin Buchin, Bart M. P. Jansen, and Gerhard Woeginger. Fine-Grained Complexity Analysis of Two Classic TSP Variants. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{deberg_et_al:LIPIcs.ICALP.2016.5,
  author =	{de Berg, Mark and Buchin, Kevin and Jansen, Bart M. P. and Woeginger, Gerhard},
  title =	{{Fine-Grained Complexity Analysis of Two Classic TSP Variants}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.5},
  URN =		{urn:nbn:de:0030-drops-62770},
  doi =		{10.4230/LIPIcs.ICALP.2016.5},
  annote =	{Keywords: Traveling salesman problem, fine-grained complexity, bitonic tours, k-opt}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2015.55
Scheduling Two Competing Agents When One Agent Has Significantly Fewer Jobs

Authors: Danny Hermelin, Judith-Madeleine Kubitza, Dvir Shabtay, Nimrod Talmon, and Gerhard Woeginger

Published in: LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Abstract
We study a scheduling problem where two agents (each equipped with a private set of jobs) compete to perform their respective jobs on a common single machine. Each agent wants to keep the weighted sum of completion times of his jobs below a given (agent-dependent) bound. This problem is known to be NP-hard, even for quite restrictive settings of the problem parameters. We consider parameterized versions of the problem where one of the agents has a small number of jobs (and where this small number constitutes the parameter). The problem becomes much more tangible in this case, and we present three positive algorithmic results for it. Our study is complemented by showing that the general problem is NP-complete even when one agent only has a single job.
Cite as

Danny Hermelin, Judith-Madeleine Kubitza, Dvir Shabtay, Nimrod Talmon, and Gerhard Woeginger. Scheduling Two Competing Agents When One Agent Has Significantly Fewer Jobs. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 55-65, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hermelin_et_al:LIPIcs.IPEC.2015.55,
  author =	{Hermelin, Danny and Kubitza, Judith-Madeleine and Shabtay, Dvir and Talmon, Nimrod and Woeginger, Gerhard},
  title =	{{Scheduling Two Competing Agents When One Agent Has Significantly Fewer Jobs}},
  booktitle =	{10th International Symposium on Parameterized and Exact Computation (IPEC 2015)},
  pages =	{55--65},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-92-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{43},
  editor =	{Husfeldt, Thore and Kanj, Iyad},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.55},
  URN =		{urn:nbn:de:0030-drops-55713},
  doi =		{10.4230/LIPIcs.IPEC.2015.55},
  annote =	{Keywords: Parameterized Complexity, Multiagent Scheduling}
}
Document
DOI: 10.4230/LIPIcs.STACS.2012.374
Motion planning with pulley, rope, and baskets

Authors: Christian E.J. Eggermont and Gerhard J. Woeginger

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Abstract
We study a motion planning problem where items have to be transported from the top room of a tower to the bottom of the tower, while simultaneously other items have to be transported into the opposite direction. Item sets are moved in two baskets hanging on a rope and pulley. To guarantee stability of the system, the weight difference between the contents of the two baskets must always stay below a given threshold. We prove that it is Pi-2-p-complete to decide whether some given initial situation of the underlying discrete system can lead to a given goal situation. Furthermore we identify several polynomially solvable special cases of this reachability problem, and we also settle the computational complexity of a number of related questions.
Cite as

Christian E.J. Eggermont and Gerhard J. Woeginger. Motion planning with pulley, rope, and baskets. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 374-383, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{eggermont_et_al:LIPIcs.STACS.2012.374,
  author =	{Eggermont, Christian E.J. and Woeginger, Gerhard J.},
  title =	{{Motion planning with pulley, rope, and baskets}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{374--383},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.374},
  URN =		{urn:nbn:de:0030-drops-33900},
  doi =		{10.4230/LIPIcs.STACS.2012.374},
  annote =	{Keywords: planning and scheduling; computational complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2011.484
Analysis of multi-stage open shop processing systems

Authors: Christian E.J. Eggermont, Alexander Schrijver, and Gerhard J. Woeginger

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Abstract
We study algorithmic problems in multi-stage open shop processing systems that are centered around reachability and deadlock detection questions. We characterize safe and unsafe system states. We show that it is easy to recognize system states that can be reached from the initial state (where the system is empty), but that in general it is hard to decide whether one given system state is reachable from another given system state. We show that the problem of identifying reachable deadlock states is hard in general open shop systems, but is easy in the special case where no job needs processing on more than two machines (by linear programming and matching theory), and in the special case where all machines have capacity one (by graph-theoretic arguments).
Cite as

Christian E.J. Eggermont, Alexander Schrijver, and Gerhard J. Woeginger. Analysis of multi-stage open shop processing systems. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 484-494, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{eggermont_et_al:LIPIcs.STACS.2011.484,
  author =	{Eggermont, Christian E.J. and Schrijver, Alexander and Woeginger, Gerhard J.},
  title =	{{Analysis of multi-stage open shop processing systems}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{484--494},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.484},
  URN =		{urn:nbn:de:0030-drops-30373},
  doi =		{10.4230/LIPIcs.STACS.2011.484},
  annote =	{Keywords: scheduling, resource allocation, deadlock, computational complexity}
}
Document
DOI: 10.4230/DagSemProc.10071.3
10071 Open Problems – Scheduling

Authors: Jim Anderson, Björn Andersson, Yossi Azar, Nikhil Bansal, Enrico Bini, Marek Chrobak, José Correa, Liliana Cucu-Grosjean, Rob Davis, Arvind Easwaran, Jeff Edmonds, Shelby Funk, Sathish Gopalakrishnan, Han Hoogeveen, Claire Mathieu, Nicole Megow, Seffi Naor, Kirk Pruhs, Maurice Queyranne, Adi Rosén, Nicolas Schabanel, Jiří Sgall, René Sitters, Sebastian Stiller, Marc Uetz, Tjark Vredeveld, and Gerhard J. Woeginger

Published in: Dagstuhl Seminar Proceedings, Volume 10071, Scheduling (2010)

Abstract
Collection of the open problems presented at the scheduling seminar.
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Jim Anderson, Björn Andersson, Yossi Azar, Nikhil Bansal, Enrico Bini, Marek Chrobak, José Correa, Liliana Cucu-Grosjean, Rob Davis, Arvind Easwaran, Jeff Edmonds, Shelby Funk, Sathish Gopalakrishnan, Han Hoogeveen, Claire Mathieu, Nicole Megow, Seffi Naor, Kirk Pruhs, Maurice Queyranne, Adi Rosén, Nicolas Schabanel, Jiří Sgall, René Sitters, Sebastian Stiller, Marc Uetz, Tjark Vredeveld, and Gerhard J. Woeginger. 10071 Open Problems – Scheduling. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{anderson_et_al:DagSemProc.10071.3,
  author =	{Anderson, Jim and Andersson, Bj\"{o}rn and Azar, Yossi and Bansal, Nikhil and Bini, Enrico and Chrobak, Marek and Correa, Jos\'{e} and Cucu-Grosjean, Liliana and Davis, Rob and Easwaran, Arvind and Edmonds, Jeff and Funk, Shelby and Gopalakrishnan, Sathish and Hoogeveen, Han and Mathieu, Claire and Megow, Nicole and Naor, Seffi and Pruhs, Kirk and Queyranne, Maurice and Ros\'{e}n, Adi and Schabanel, Nicolas and Sgall, Ji\v{r}{\'\i} and Sitters, Ren\'{e} and Stiller, Sebastian and Uetz, Marc and Vredeveld, Tjark and Woeginger, Gerhard J.},
  title =	{{10071 Open Problems – Scheduling}},
  booktitle =	{Scheduling},
  pages =	{1--24},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{10071},
  editor =	{Susanne Albers and Sanjoy K. Baruah and Rolf H. M\"{o}hring and Kirk Pruhs},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.3},
  URN =		{urn:nbn:de:0030-drops-25367},
  doi =		{10.4230/DagSemProc.10071.3},
  annote =	{Keywords: Open problems, scheduling}
}
Document
DOI: 10.4230/LIPIcs.STACS.2010.2458
The Traveling Salesman Problem under Squared Euclidean Distances

Authors: Fred van Nijnatten, René Sitters, Gerhard J. Woeginger, Alexander Wolff, and Mark de Berg

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract
Let $P$ be a set of points in $\Reals^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by \tsp($d,\alpha$). We design a 5-approximation algorithm for \tsp(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\,\alpha}\!/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-\tsp\ of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-\tsp$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-\tsp$(d, \alpha)$ is \apx-hard if $d\ge3$ and $\alpha>1$. The \apx-hardness proof carries over to \tsp$(d, \alpha)$ for the same parameter ranges.
Cite as

Fred van Nijnatten, René Sitters, Gerhard J. Woeginger, Alexander Wolff, and Mark de Berg. The Traveling Salesman Problem under Squared Euclidean Distances. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 239-250, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{vannijnatten_et_al:LIPIcs.STACS.2010.2458,
  author =	{van Nijnatten, Fred and Sitters, Ren\'{e} and Woeginger, Gerhard J. and Wolff, Alexander and de Berg, Mark},
  title =	{{The Traveling Salesman Problem under Squared Euclidean Distances}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{239--250},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2458},
  URN =		{urn:nbn:de:0030-drops-24580},
  doi =		{10.4230/LIPIcs.STACS.2010.2458},
  annote =	{Keywords: Geometric traveling salesman problem, power-assignment in wireless networks, distance-power gradient, NP-hard, APX-hard}
}
Document
DOI: 10.4230/DagSemProc.07261.1
07261 Abstracts Collection – Fair Division

Authors: Steven J. Brams and Kirk Pruhs

Published in: Dagstuhl Seminar Proceedings, Volume 7261, Fair Division (2007)

Abstract
From 24.06. to 29.06.2007, the Dagstuhl Seminar 07261 % generate automatically ``Fair Division'' % generate automatically was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Cite as

Steven J. Brams and Kirk Pruhs. 07261 Abstracts Collection – Fair Division. In Fair Division. Dagstuhl Seminar Proceedings, Volume 7261, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{brams_et_al:DagSemProc.07261.1,
  author =	{Brams, Steven J. and Pruhs, Kirk},
  title =	{{07261 Abstracts Collection – Fair Division}},
  booktitle =	{Fair Division},
  pages =	{1--16},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7261},
  editor =	{Steven Brams and Kirk Pruhs and Gerhard Woeginger},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07261.1},
  URN =		{urn:nbn:de:0030-drops-12444},
  doi =		{10.4230/DagSemProc.07261.1},
  annote =	{Keywords: Economics, Fairness, Allocation, Political Science}
}
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