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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Mixed-integer linear programming (MILP) is at the core of many advanced algorithms for solving fundamental problems in combinatorial optimization. The complexity of solving MILPs directly correlates with their support size, which is the minimum number of non-zero integer variables in an optimal solution. A hallmark result by Eisenbrand and Shmonin (Oper. Res. Lett. , 2006) shows that any feasible integer linear program (ILP) has a solution with support size s ≤ 2m⋅log(4mΔ), where m is the number of constraints, and Δ is the largest absolute coefficient in any constraint.
Our main combinatorial result are improved support size bounds for ILPs.
We show that any ILP has a solution with support size s ≤ m⋅(log(3A_max)+√{log(A_max)}), where A_max≔ ‖A‖₁ denotes the 1-norm of the constraint matrix A. Furthermore, we show support bounds in the linearized form s ≤ 2m⋅log(1.46 A_max). Our upper bounds also hold with A_max replaced by √mΔ, which improves on the previously best constants in the linearized form.
Our main algorithmic result are the fastest known approximation schemes for fundamental scheduling problems, which use the improved support bounds as one ingredient.
We design an efficient approximation scheme (EPTAS) for makespan minimization on uniformly related machines (Q||C_{max}). Our EPTAS yields a (1+ε)-approximation for Q||C_{max} on N jobs in time 2^𝒪(1/ε log³(1/ε)log(log(1/ε))) + 𝒪(N), which improves over the previously fastest algorithm by Jansen, Klein and Verschae (Math. Oper. Res., 2020) with run time 2^𝒪(1/ε log⁴(1/ε)) + N^𝒪(1). Arguably, our approximation scheme is also simpler than all previous EPTASes for Q||C_max, as we reduce the problem to a novel MILP formulation which greatly benefits from the small support.

Sebastian Berndt, Hauke Brinkop, Klaus Jansen, Matthias Mnich, and Tobias Stamm. New Support Size Bounds for Integer Programming, Applied to Makespan Minimization on Uniformly Related Machines. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berndt_et_al:LIPIcs.ISAAC.2023.13, author = {Berndt, Sebastian and Brinkop, Hauke and Jansen, Klaus and Mnich, Matthias and Stamm, Tobias}, title = {{New Support Size Bounds for Integer Programming, Applied to Makespan Minimization on Uniformly Related Machines}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.13}, URN = {urn:nbn:de:0030-drops-193155}, doi = {10.4230/LIPIcs.ISAAC.2023.13}, annote = {Keywords: Integer programming, scheduling algorithms, uniformly related machines, makespan minimization} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone.
Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels,
- Independent Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using 𝒪(dk²log n) space;
- Max Cut can be solved in time n^𝒪(dk) using 𝒪(dk log n) space; and
- Dominating Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using n^𝒪(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial.

Benjamin Bergougnoux, Vera Chekan, Robert Ganian, Mamadou Moustapha Kanté, Matthias Mnich, Sang-il Oum, Michał Pilipczuk, and Erik Jan van Leeuwen. Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.ESA.2023.18, author = {Bergougnoux, Benjamin and Chekan, Vera and Ganian, Robert and Kant\'{e}, Mamadou Moustapha and Mnich, Matthias and Oum, Sang-il and Pilipczuk, Micha{\l} and van Leeuwen, Erik Jan}, title = {{Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {18:1--18:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.18}, URN = {urn:nbn:de:0030-drops-186710}, doi = {10.4230/LIPIcs.ESA.2023.18}, annote = {Keywords: Parameterized complexity, shrubdepth, space complexity, algebraic methods} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

One of the most studied extensions of the famous Traveling Salesperson Problem (TSP) is the Multiple TSP: a set of m ≥ 1 salespersons collectively traverses a set of n cities by m non-trivial tours, to minimize the total length of their tours. This problem can also be considered to be a variant of Uncapacitated Vehicle Routing, where the objective is to minimize the sum of all tour lengths. When all m tours start from and end at a single common depot v₀, then the metric Multiple TSP can be approximated equally well as the standard metric TSP, as shown by Frieze (1983).
The metric Multiple TSP becomes significantly harder to approximate when there is a set D of d ≥ 1 depots that form the starting and end points of the m tours. For this case, only a (2-1/d)-approximation in polynomial time is known, as well as a 3/2-approximation for constant d which requires a prohibitive run time of n^Θ(d) (Xu and Rodrigues, INFORMS J. Comput., 2015). A recent work of Traub, Vygen and Zenklusen (STOC 2020) gives another approximation algorithm for metric Multiple TSP with run time n^Θ(d), which reduces the problem to approximating metric TSP.
In this paper we overcome the n^Θ(d) time barrier: we give the first efficient approximation algorithm for Multiple TSP with a variable number d of depots that yields a better-than-2 approximation. Our algorithm runs in time (1/ε)^O(dlog d) ⋅ n^O(1), and produces a (3/2+ε)-approximation with constant probability. For the graphic case, we obtain a deterministic 3/2-approximation in time 2^d ⋅ n^O(1).

Max Deppert, Matthias Kaul, and Matthias Mnich. A (3/2 + ε)-Approximation for Multiple TSP with a Variable Number of Depots. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{deppert_et_al:LIPIcs.ESA.2023.39, author = {Deppert, Max and Kaul, Matthias and Mnich, Matthias}, title = {{A (3/2 + \epsilon)-Approximation for Multiple TSP with a Variable Number of Depots}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {39:1--39:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.39}, URN = {urn:nbn:de:0030-drops-186925}, doi = {10.4230/LIPIcs.ESA.2023.39}, annote = {Keywords: Traveling salesperson problem, rural postperson problem, multiple TSP, vehicle routing} }

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APPROX

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph G which intersects all copies of subgraphs F from a fixed family F. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs G, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals with parity constraints, which have recently received significant attention, but for which no PTASs are known.
In the even-cycle transversal (ECT) problem, the goal is to find a minimum-weight hitting set for the set of even cycles in an undirected graph. For ECT, Fiorini et al. (IPCO 2010) showed that the integrality gap of the standard covering LP relaxation is Θ(log n), and that adding sparsity inequalities reduces the integrality gap to 10.
Our main result is a primal-dual algorithm that yields a 47/7 ≈ 6.71-approximation for ECT on node-weighted planar graphs, and an integrality gap of the same value for the standard LP relaxation on node-weighted planar graphs.

Alexander Göke, Jochen Koenemann, Matthias Mnich, and Hao Sun. Hitting Weighted Even Cycles in Planar Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 25:1-25:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goke_et_al:LIPIcs.APPROX/RANDOM.2021.25, author = {G\"{o}ke, Alexander and Koenemann, Jochen and Mnich, Matthias and Sun, Hao}, title = {{Hitting Weighted Even Cycles in Planar Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {25:1--25:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.25}, URN = {urn:nbn:de:0030-drops-147186}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.25}, annote = {Keywords: Even cycles, planar graphs, integrality gap} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number k of small items.
Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by k. The algorithms are randomized with one-sided error and run in time 4^k⋅ k!⋅ n^{O(1)}. To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. We also present a deterministic fixed-parameter algorithm for Bin Covering with run time O((k!)² ⋅ k ⋅ 2^k ⋅ n log(n)).

Max Bannach, Sebastian Berndt, Marten Maack, Matthias Mnich, Alexandra Lassota, Malin Rau, and Malte Skambath. Solving Packing Problems with Few Small Items Using Rainbow Matchings. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bannach_et_al:LIPIcs.MFCS.2020.11, author = {Bannach, Max and Berndt, Sebastian and Maack, Marten and Mnich, Matthias and Lassota, Alexandra and Rau, Malin and Skambath, Malte}, title = {{Solving Packing Problems with Few Small Items Using Rainbow Matchings}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {11:1--11:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.11}, URN = {urn:nbn:de:0030-drops-126816}, doi = {10.4230/LIPIcs.MFCS.2020.11}, annote = {Keywords: Bin Packing, Knapsack, matching, fixed-parameter tractable} }

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Track A: Algorithms, Complexity and Games

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

In the Directed Long Cycle Hitting Set problem we are given a directed graph G, and the task is to find a set S of at most k vertices/arcs such that G-S has no cycle of length longer than ℓ. We show that the problem can be solved in time 2^O(ℓ^6 + ℓ k^3 log k + k^5 log k log ℓ) ⋅ n^O(1), that is, it is fixed-parameter tractable (FPT) parameterized by k and ℓ. This algorithm can be seen as a far-reaching generalization of the fixed-parameter tractability of Mixed Graph Feedback Vertex Set [Bonsma and Lokshtanov WADS 2011], which is already a common generalization of the fixed-parameter tractability of (undirected) Feedback Vertex Set and the Directed Feedback Vertex Set problems, two classic results in parameterized algorithms. The algorithm requires significant insights into the structure of graphs without directed cycles of length longer than ℓ and can be seen as an exact version of the approximation algorithm following from the Erdős-Pósa property for long cycles in directed graphs proved by Kreutzer and Kawarabayashi [STOC 2015].

Alexander Göke, Dániel Marx, and Matthias Mnich. Hitting Long Directed Cycles Is Fixed-Parameter Tractable. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{goke_et_al:LIPIcs.ICALP.2020.59, author = {G\"{o}ke, Alexander and Marx, D\'{a}niel and Mnich, Matthias}, title = {{Hitting Long Directed Cycles Is Fixed-Parameter Tractable}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {59:1--59:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.59}, URN = {urn:nbn:de:0030-drops-124664}, doi = {10.4230/LIPIcs.ICALP.2020.59}, annote = {Keywords: Directed graphs, directed feedback vertex set, circumference} }

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**Published in:** LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Deciding feasibility of large systems of linear equations and inequalities is one of the most fundamental algorithmic tasks. However, due to inaccuracies of the data or modeling errors, in practical applications one often faces linear systems that are infeasible.
Extensive theoretical and practical methods have been proposed for post-infeasibility analysis of linear systems. This generally amounts to detecting a feasibility blocker of small size k, which is a set of equations and inequalities whose removal or perturbation from the large system of size m yields a feasible system. This motivates a parameterized approach towards post-infeasibility analysis, where we aim to find a feasibility blocker of size at most k in fixed-parameter time f(k)* m^{O(1)}.
On the one hand, we establish parameterized intractability (W[1]-hardness) results even in very restricted settings. On the other hand, we develop fixed-parameter algorithms parameterized by the number of perturbed inequalities and the number of positive/negative right-hand sides. Our algorithms capture the case of Directed Feedback Arc Set, a fundamental parameterized problem whose fixed-parameter tractability was shown by Chen et al. (STOC 2008).

Alexander Göke, Lydia Mirabel Mendoza Cadena, and Matthias Mnich. Resolving Infeasibility of Linear Systems: A Parameterized Approach. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goke_et_al:LIPIcs.IPEC.2019.17, author = {G\"{o}ke, Alexander and Mendoza Cadena, Lydia Mirabel and Mnich, Matthias}, title = {{Resolving Infeasibility of Linear Systems: A Parameterized Approach}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.17}, URN = {urn:nbn:de:0030-drops-114787}, doi = {10.4230/LIPIcs.IPEC.2019.17}, annote = {Keywords: Infeasible subsystems, linear programming, fixed-parameter algorithms} }

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

We give faster and simpler approximation algorithms for the (1,2)-TSP problem, a well-studied variant of the traveling salesperson problem where all distances between cities are either 1 or 2.
Our main results are two approximation algorithms for (1,2)-TSP, one with approximation factor 8/7 and run time O(n^3) and the other having an approximation guarantee of 7/6 and run time O(n^{2.5}). The 8/7-approximation matches the best known approximation factor for (1,2)-TSP, due to Berman and Karpinski (SODA 2006), but considerably improves the previous best run time of O(n^9). Thus, ours is the first improvement for the (1,2)-TSP problem in more than 10 years. The algorithm is based on combining three copies of a minimum-cost cycle cover of the input graph together with a relaxed version of a minimum weight matching, which allows using "half-edges". The resulting multigraph is then edge-colored with four colors so that each color class yields a collection of vertex-disjoint paths. The paths from one color class can then be extended to an 8/7-approximate traveling salesperson tour. Our algorithm, and in particular its analysis, is simpler than the previously best 8/7-approximation.
The 7/6-approximation algorithm is similar and even simpler, and has the advantage of not using Hartvigsen's complicated algorithm for computing a minimum-cost triangle-free cycle cover.

Anna Adamaszek, Matthias Mnich, and Katarzyna Paluch. New Approximation Algorithms for (1,2)-TSP. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{adamaszek_et_al:LIPIcs.ICALP.2018.9, author = {Adamaszek, Anna and Mnich, Matthias and Paluch, Katarzyna}, title = {{New Approximation Algorithms for (1,2)-TSP}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.9}, URN = {urn:nbn:de:0030-drops-90133}, doi = {10.4230/LIPIcs.ICALP.2018.9}, annote = {Keywords: Approximation algorithms, traveling salesperson problem, cycle cover} }

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

In this paper, we study the problem of deciding the winner of reachability switching games. We study zero-, one-, and two-player variants of these games. We show that the zero-player case is NL-hard, the one-player case is NP-complete, and that the two-player case is PSPACE-hard and in EXPTIME. For the zero-player case, we also show P-hardness for a succinctly-represented model that maintains the upper bound of NP n coNP. For the one- and two-player cases, our results hold in both the natural, explicit model and succinctly-represented model. We also study the structure of winning strategies in these games, and in particular we show that exponential memory is required in both the one- and two-player settings.

John Fearnley, Martin Gairing, Matthias Mnich, and Rahul Savani. Reachability Switching Games. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 124:1-124:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fearnley_et_al:LIPIcs.ICALP.2018.124, author = {Fearnley, John and Gairing, Martin and Mnich, Matthias and Savani, Rahul}, title = {{Reachability Switching Games}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {124:1--124:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.124}, URN = {urn:nbn:de:0030-drops-91282}, doi = {10.4230/LIPIcs.ICALP.2018.124}, annote = {Keywords: Deterministic Random Walks, Model Checking, Reachability, Simple Stochastic Game, Switching Systems} }

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

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

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

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

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Fixed-parameter algorithms and kernelization are two powerful methods to solve NP-hard problems. Yet, so far those algorithms have been largely restricted to static inputs. In this paper we provide fixed-parameter algorithms and kernelizations for fundamental NP-hard problems with dynamic inputs. We consider a variety of parameterized graph and hitting set problems which are known to have f(k)n^{1+o(1)} time algorithms on inputs of size n, and we consider the question of whether there is a data structure that supports small updates (such as edge/vertex/set/element insertions and deletions) with an update time of g(k)n^{o(1)}; such an update time would be essentially optimal. Update and query times independent of n are particularly desirable. Among many other results, we show that Feedback Vertex Set and k-Path admit dynamic algorithms with f(k)log O(1) n update and query times for some function f depending on the solution size k only.
We complement our positive results by several conditional and unconditional lower bounds. For example, we show that unlike their undirected counterparts, Directed Feedback Vertex Set and Directed k-Path do not admit dynamic algorithms with n^{o(1) } update and query times even for constant solution sizes k <= 3, assuming popular hardness hypotheses. We also show that unconditionally, in the cell probe model, Directed Feedback Vertex Set cannot be solved with update time that is purely a function of k.

Josh Alman, Matthias Mnich, and Virginia Vassilevska Williams. Dynamic Parameterized Problems and Algorithms. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{alman_et_al:LIPIcs.ICALP.2017.41, author = {Alman, Josh and Mnich, Matthias and Vassilevska Williams, Virginia}, title = {{Dynamic Parameterized Problems and Algorithms}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {41:1--41:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.41}, URN = {urn:nbn:de:0030-drops-74419}, doi = {10.4230/LIPIcs.ICALP.2017.41}, annote = {Keywords: Dynamic algorithms, fixed-parameter algorithms} }

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

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

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

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

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

We study feedback vertex sets (FVS) in tournaments, which are orientations of complete graphs. As our main result, we show that any tournament on n nodes has at most 1.5949^n minimal FVS. This significantly improves the previously best upper bound of 1.6667^n by Fomin et al. (STOC 2016). Our new upper bound almost matches the best known lower bound of 21^{n/7} approx 1.5448^n, due to Gaspers and Mnich (ESA 2010). Our proof is algorithmic, and shows that all minimal FVS of tournaments can be enumerated in time O(1.5949^n).

Matthias Mnich and Eva-Lotta Teutrine. Improved Bounds for Minimal Feedback Vertex Sets in Tournaments. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 24:1-24:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mnich_et_al:LIPIcs.IPEC.2016.24, author = {Mnich, Matthias and Teutrine, Eva-Lotta}, title = {{Improved Bounds for Minimal Feedback Vertex Sets in Tournaments}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {24:1--24:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.24}, URN = {urn:nbn:de:0030-drops-69236}, doi = {10.4230/LIPIcs.IPEC.2016.24}, annote = {Keywords: exponential-time algorithms, feedback vertex sets, tournaments} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph. In this paper we continue this line of research and considerably improve several of those results:
* We show that an algorithm by Crowston et al. [ICALP 2012] for (Signed) Max-Cut Above Edwards-Erdos Bound can be implemented in such a way that it runs in linear time 8^k · O(m); this significantly improves the previous analysis with run time 8^k · O(n^4).
* We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards-Erdos Bound with O(k) vertices, improving a kernel with O(k^3) vertices by Crowston et al. [COCOON 2013].
* We improve all known kernels for strongly lambda-extendable properties parameterized above tight lower bound by Crowston et al. [FSTTCS 2013] from O(k^3) vertices to O(k) vertices.
* As a consequence, Max Acyclic Subdigraph parameterized above Poljak-Turzik bound admits a kernel with O(k) vertices and can be solved in time 2^{O(k)} * n^{O(1)} ; this answers an open question by Crowston et al. [FSTTCS 2012].
All presented kernels can be computed in time O(km).

Michael Etscheid and Matthias Mnich. Linear Kernels and Linear-Time Algorithms for Finding Large Cuts. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 31:1-31:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{etscheid_et_al:LIPIcs.ISAAC.2016.31, author = {Etscheid, Michael and Mnich, Matthias}, title = {{Linear Kernels and Linear-Time Algorithms for Finding Large Cuts}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {31:1--31:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.31}, URN = {urn:nbn:de:0030-drops-68016}, doi = {10.4230/LIPIcs.ISAAC.2016.31}, annote = {Keywords: Max-Cut, fixed-parameter tractability, kernelization} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/NDP is currently not well understood; the best known lower bound is Omega(log^{1/2 - varepsilon} n), assuming NP not subseteq ZPTIME(n^{poly log n}). This constitutes a significant gap to the best known approximation upper bound of O(n^1/2) due to Chekuri et al. (2006) and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica, 1987) introduce the technique of randomized rounding for LPs; their technique gives an O(1)-approximation when edges (or nodes) may be used by O(log n/log log n) paths.
In this paper, we strengthen the above fundamental results. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following.
- For MaxEDP, we give an O(r^0.5 log^1.5 kr)-approximation algorithm. As r<=n, up to logarithmic factors, our result strengthens the best known ratio O(n^0.5) due to Chekuri et al.
- Further, we show how to route Omega(opt) pairs with congestion O(log(kr)/log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson.
- For MaxNDP, we give an algorithm that gives the optimal answer in time (k+r)^O(r)n. This is a substantial improvement on the run time of 2^kr^O(r)n, which can be obtained via an algorithm by Scheffler.
We complement these positive results by proving that MaxEDP is NP-hard even for r=1, and MaxNDP is W[1]-hard for parameter r. This shows that neither problem is fixed-parameter tractable in r unless FPT = W[1] and that our approximability results are relevant even for very small constant values of r.

Krzysztof Fleszar, Matthias Mnich, and Joachim Spoerhase. New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fleszar_et_al:LIPIcs.ESA.2016.42, author = {Fleszar, Krzysztof and Mnich, Matthias and Spoerhase, Joachim}, title = {{New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {42:1--42:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.42}, URN = {urn:nbn:de:0030-drops-63542}, doi = {10.4230/LIPIcs.ESA.2016.42}, annote = {Keywords: disjoint paths, approximation algorithms, feedback vertex set} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is NP-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first 7/3 approximation algorithm for this problem, improving on the previously best known ratio 5/2 given by Cai et al. [FOCS 1998, SICOMP 2001].

Matthias Mnich, Virginia Vassilevska Williams, and László A. Végh. A 7/3-Approximation for Feedback Vertex Sets in Tournaments. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mnich_et_al:LIPIcs.ESA.2016.67, author = {Mnich, Matthias and Vassilevska Williams, Virginia and V\'{e}gh, L\'{a}szl\'{o} A.}, title = {{A 7/3-Approximation for Feedback Vertex Sets in Tournaments}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {67:1--67:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.67}, URN = {urn:nbn:de:0030-drops-64098}, doi = {10.4230/LIPIcs.ESA.2016.67}, annote = {Keywords: Approximation algorithms, feedback vertex sets, tournaments} }

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

Map graphs generalize planar graphs and were introduced by Chen, Grigni and Papadimitriou [STOC 1998, J.ACM 2002]. They showed that the problem of recognizing map graphs is in NP by proving the existence of a planar witness graph W. Shortly after, Thorup [FOCS 1998] published a polynomial-time recognition algorithm for map graphs. However, the run time of this algorithm is estimated to be Omega(n^{120}) for n-vertex graphs, and a full description of its details remains unpublished.
We give a new and purely combinatorial algorithm that decides whether a graph G is a map graph having an outerplanar witness W. This is a step towards a first combinatorial recognition algorithm for general map graphs. The algorithm runs in time and space O(n+m). In contrast to Thorup's approach, it computes the witness graph W in the affirmative case.

Matthias Mnich, Ignaz Rutter, and Jens M. Schmidt. Linear-Time Recognition of Map Graphs with Outerplanar Witness. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mnich_et_al:LIPIcs.SWAT.2016.5, author = {Mnich, Matthias and Rutter, Ignaz and Schmidt, Jens M.}, title = {{Linear-Time Recognition of Map Graphs with Outerplanar Witness}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.5}, URN = {urn:nbn:de:0030-drops-60349}, doi = {10.4230/LIPIcs.SWAT.2016.5}, annote = {Keywords: Algorithms and data structures, map graphs, recognition, planar graphs} }

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

We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph G and a collection of k source-destination pairs M = (s_1, t_1), ..., (s_k, t_k). The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset M' of the pairs is a collection P of paths such that, for each pair (s_i, t_i) in M', there is a path in P connecting s_i to t_i. In the Maximum Edge Disjoint Paths (MaxEDP) problem, the graph G has capacities cap(e) on the edges and a routing P is feasible if each edge e is in at most cap(e) of the paths of P. The Maximum Node Disjoint Paths (MaxNDP) problem is the node-capacitated counterpart of MaxEDP.
In this paper we obtain an O(r^3) approximation for MaxEDP on graphs of treewidth at most r and a matching approximation for MaxNDP on graphs of pathwidth at most r. Our results build on and significantly improve the work by Chekuri et al. [ICALP 2013] who obtained an O(r * 3^r) approximation for MaxEDP.

Alina Ene, Matthias Mnich, Marcin Pilipczuk, and Andrej Risteski. On Routing Disjoint Paths in Bounded Treewidth Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ene_et_al:LIPIcs.SWAT.2016.15, author = {Ene, Alina and Mnich, Matthias and Pilipczuk, Marcin and Risteski, Andrej}, title = {{On Routing Disjoint Paths in Bounded Treewidth Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.15}, URN = {urn:nbn:de:0030-drops-60378}, doi = {10.4230/LIPIcs.SWAT.2016.15}, annote = {Keywords: Algorithms and data structures, disjoint paths, treewidth} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We consider the problem to find a set X of vertices (or arcs) with |X| <= k in a given digraph G such that D = G-X is an acyclic digraph. In its generality, this is DIRECTED FEEDBACK VERTEX SET or DIRECTED FEEDBACK ARC SET respectively. The existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made.
In this paper, we consider both deletion problems with an additional restriction on D, namely that D must be an out-forest, an out-tree, or a (directed) pumpkin. Our main results show that for each of these three restrictions the vertex deletion problem remains NP-hard, but we can obtain a kernel with k^{O(1)} vertices on general digraphs G. We also show that, in contrast to the vertex deletion problem, the arc deletion problem with each of the above restrictions can be solved in polynomial time.

Matthias Mnich and Erik Jan van Leeuwen. Polynomial Kernels for Deletion to Classes of Acyclic Digraphs. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mnich_et_al:LIPIcs.STACS.2016.55, author = {Mnich, Matthias and van Leeuwen, Erik Jan}, title = {{Polynomial Kernels for Deletion to Classes of Acyclic Digraphs}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {55:1--55:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.55}, URN = {urn:nbn:de:0030-drops-57569}, doi = {10.4230/LIPIcs.STACS.2016.55}, annote = {Keywords: directed feedback vertex/arc set, parameterized algorithms, kernels} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We consider the Steiner Multicut problem, which asks, given an undirected graph G, a collection T = \{T_{1},...,T_{t}}, T_i \subseteq V(G), of terminal sets of size at most p, and an integer k, whether there is a set S of at most k edges or nodes such that of each set T_{i} at least one pair of terminals is in different connected components of G \ S. This problem generalizes several well-studied graph cut problems, in particular the Multicut problem, which corresponds to the case p = 2. The Multicut problem was recently shown to be fixed-parameter tractable for parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. The question whether this result generalizes to Steiner Multicut motivates the present work.
We answer the question that motivated this work, and in fact provide a dichotomy of the parameterized complexity of Steiner Multicut on general graphs. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). Among the many results in the paper, we highlight that:
- The edge deletion version of Steiner Multicut is fixed-parameter tractable for parameter k+t on general graphs (but has no polynomial kernel, even on trees).
- In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs.
- All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node.
Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1) as well as a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).

Karl Bringmann, Danny Hermelin, Matthias Mnich, and Erik Jan van Leeuwen. Parameterized Complexity Dichotomy for Steiner Multicut. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 157-170, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bringmann_et_al:LIPIcs.STACS.2015.157, author = {Bringmann, Karl and Hermelin, Danny and Mnich, Matthias and van Leeuwen, Erik Jan}, title = {{Parameterized Complexity Dichotomy for Steiner Multicut}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {157--170}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.157}, URN = {urn:nbn:de:0030-drops-49115}, doi = {10.4230/LIPIcs.STACS.2015.157}, annote = {Keywords: graph cut problems, Steiner cut, fixed-parameter tractability} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Poljak and Turzík (Discrete Math. 1986) introduced the notion of lambda-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0 < lambda < 1 and lambda-extendible property Pi, any connected graph G on n vertices and m edges contains a spanning subgraph H in Pi with at least lambda m+ (1-lambda)/2 (n-1) edges. The property of being bipartite is lambda-extendible for lambda=1/2, and thus the Poljak-Turzík bound generalizes the well-known Edwards-Erdos bound for MAXCUT.
We define a variant, namely strong lambda-extendibility, to which the Poljak-Turzík bound applies. For a strong lambda-extendible graph property \Pi, we define the parameterized Above Poljak-Turzík problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H in Pi and H has at least lambda m+ (1-lambda)/2 (n-1)+k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzík bound.
We consider properties Pi for which the Above Poljak-Turzík problem is fixed-parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, Above Poljak-Turzík is FPT for all 0< lambda <1. Our results hold for properties of oriented graphs and graphs with edge labels.
Our results generalize the recent result of Crowston et al. (ICALP 2012) on MAXCUT parameterized above the Edwards-Erdos, and yield FPT algorithms for several graph problems parameterized above lower bounds. For instance, we get that the above-guarantee Max q-Colorable Subgraph problem is FPT. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

Matthias Mnich, Geevarghese Philip, Saket Saurabh, and Ondrej Suchy. Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 412-423, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{mnich_et_al:LIPIcs.FSTTCS.2012.412, author = {Mnich, Matthias and Philip, Geevarghese and Saurabh, Saket and Suchy, Ondrej}, title = {{Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {412--423}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.412}, URN = {urn:nbn:de:0030-drops-38776}, doi = {10.4230/LIPIcs.FSTTCS.2012.412}, annote = {Keywords: Algorithms and data structures; fixed-parameter algorithms; bipartite graphs; above-guarantee parameterization} }

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