Volume
LIPIcs, Volume 249
17th International Symposium on Parameterized and Exact Computation (IPEC 2022)
Event
IPEC 2022, September 7-9, 2022, Potsdam, GermanyEditors
Publication Details
- published at: 2022-12-14
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-260-0
- DBLP: db/conf/iwpec/ipec2022
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Complete Volume
LIPIcs, Volume 249, IPEC 2022, Complete Volume
Abstract
LIPIcs, Volume 249, IPEC 2022, Complete Volume
Cite as
17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 1-520, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@Proceedings{dell_et_al:LIPIcs.IPEC.2022,
title = {{LIPIcs, Volume 249, IPEC 2022, Complete Volume}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {1--520},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022},
URN = {urn:nbn:de:0030-drops-173553},
doi = {10.4230/LIPIcs.IPEC.2022},
annote = {Keywords: LIPIcs, Volume 249, IPEC 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dell_et_al:LIPIcs.IPEC.2022.0,
author = {Dell, Holger and Nederlof, Jesper},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {0:i--0:xviii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.0},
URN = {urn:nbn:de:0030-drops-173562},
doi = {10.4230/LIPIcs.IPEC.2022.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
A Finite Algorithm for the Realizabilty of a Delaunay Triangulation
Abstract
The Delaunay graph of a point set P ⊆ ℝ² is the plane graph with the vertex-set P and the edge-set that contains {p,p'} if there exists a disc whose intersection with P is exactly {p,p'}. Accordingly, a triangulated graph G is Delaunay realizable if there exists a triangulation of the Delaunay graph of some P ⊆ ℝ², called a Delaunay triangulation of P, that is isomorphic to G. The objective of Delaunay Realization is to compute a point set P ⊆ ℝ² that realizes a given graph G (if such a P exists). Known algorithms do not solve Delaunay Realization as they are non-constructive. Obtaining a constructive algorithm for Delaunay Realization was mentioned as an open problem by Hiroshima et al. [Hiroshima et al., 2000]. We design an n^𝒪(n)-time constructive algorithm for Delaunay Realization. In fact, our algorithm outputs sets of points with integer coordinates.
Cite as
Akanksha Agrawal, Saket Saurabh, and Meirav Zehavi. A Finite Algorithm for the Realizabilty of a Delaunay Triangulation. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2022.1,
author = {Agrawal, Akanksha and Saurabh, Saket and Zehavi, Meirav},
title = {{A Finite Algorithm for the Realizabilty of a Delaunay Triangulation}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {1:1--1:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.1},
URN = {urn:nbn:de:0030-drops-173573},
doi = {10.4230/LIPIcs.IPEC.2022.1},
annote = {Keywords: Delaunay Triangulation, Delaunay Realization, Finite Algorithm, Integer Coordinate Realization}
}
Document
Parameterized Complexity of Perfectly Matched Sets
Abstract
For an undirected graph G, a pair of vertex disjoint subsets (A, B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (= |B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2^O(√k)⋅ n^O(1), and ii) K_{b,b}-free graphs. We obtain a linear kernel for planar graphs and k^𝒪(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.
Cite as
Akanksha Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, and Abhishek Sahu. Parameterized Complexity of Perfectly Matched Sets. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2022.2,
author = {Agrawal, Akanksha and Bhattacharjee, Sutanay and Jana, Satyabrata and Sahu, Abhishek},
title = {{Parameterized Complexity of Perfectly Matched Sets}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {2:1--2:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.2},
URN = {urn:nbn:de:0030-drops-173580},
doi = {10.4230/LIPIcs.IPEC.2022.2},
annote = {Keywords: Perfectly Matched Sets, Parameterized Complexity, Apex-minor-free graphs, d-degenerate graphs, Planar graphs, Interval Graphs}
}
Document
On the Hardness of Generalized Domination Problems Parameterized by Mim-Width
Abstract
For nonempty σ, ρ ⊆ ℕ, a vertex set S in a graph G is a (σ, ρ)-dominating set if for all v ∈ S, |N(v) ∩ S| ∈ σ, and for all v ∈ V(G) ⧵ S, |N(v) ∩ S| ∈ ρ. The Min/Max (σ,ρ)-Dominating Set problems ask, given a graph G and an integer k, whether G contains a (σ, ρ)-dominating set of size at most k and at least k, respectively. This framework captures many well-studied graph problems related to independence and domination. Bui-Xuan, Telle, and Vatshelle [TCS 2013] showed that for finite or co-finite σ and ρ, the Min/Max (σ,ρ)-Dominating Set problems are solvable in XP time parameterized by the mim-width of a given branch decomposition of the input graph. In this work we consider the parameterized complexity of these problems and obtain the following: For minimization problems, we complete several scattered W[1]-hardness results in the literature to a full dichotomoy into polynomial-time solvable and W[1]-hard cases, and for maximization problems we obtain the same result under the additional restriction that σ and ρ are finite sets. All W[1]-hard cases hold assuming that a linear branch decomposition of bounded mim-width is given, and with the solution size being an additional part of the parameter. Furthermore, for all W[1]-hard cases we also rule out f(w)n^o(w/log w)-time algorithms assuming the Exponential Time Hypothesis, where f is any computable function, n is the number of vertices and w the mim-width of the given linear branch decomposition of the input graph.
Cite as
Brage I. K. Bakkane and Lars Jaffke. On the Hardness of Generalized Domination Problems Parameterized by Mim-Width. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 3:1-3:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bakkane_et_al:LIPIcs.IPEC.2022.3,
author = {Bakkane, Brage I. K. and Jaffke, Lars},
title = {{On the Hardness of Generalized Domination Problems Parameterized by Mim-Width}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {3:1--3:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.3},
URN = {urn:nbn:de:0030-drops-173597},
doi = {10.4230/LIPIcs.IPEC.2022.3},
annote = {Keywords: generalized domination, linear mim-width, W\lbrack1\rbrack-hardness, Exponential Time Hypothesis}
}
Document
FPT Approximation for Fair Minimum-Load Clustering
Abstract
In this paper, we consider the Minimum-Load k-Clustering/Facility Location (MLkC) problem where we are given a set P of n points in a metric space that we have to cluster and an integer k > 0 that denotes the number of clusters. Additionally, we are given a set F of cluster centers in the same metric space. The goal is to select a set C ⊆ F of k centers and assign each point in P to a center in C, such that the maximum load over all centers is minimized. Here the load of a center is the sum of the distances between it and the points assigned to it.
Although clustering/facility location problems have rich literature, the minimum-load objective has not been studied substantially, and hence MLkC has remained a poorly understood problem. More interestingly, the problem is notoriously hard even in some special cases including the one in line metrics as shown by Ahmadian et al. [APPROX 2014, ACM Trans. Algorithms 2018]. They also show APX-hardness of the problem in the plane. On the other hand, the best-known approximation factor for MLkC is O(k), even in the plane.
In this work, we study a fair version of MLkC inspired by the work of Chierichetti et al. [NeurIPS, 2017]. Here the input points are partitioned into 𝓁 protected groups, and only clusters that proportionally represent each group are allowed. MLkC is the special case with 𝓁 = 1. For the fair version, we are able to obtain a randomized 3-approximation algorithm in f(k,𝓁)⋅ n^O(1) time. Also, our scheme leads to an improved (1 + ε)-approximation in the case of Euclidean norm with the same running time (depending also linearly on the dimension d). Our results imply the same approximations for MLkC with running time f(k)⋅ n^O(1), achieving the first constant-factor FPT approximations for this problem in general and Euclidean metric spaces.
Cite as
Sayan Bandyapadhyay, Fedor V. Fomin, Petr A. Golovach, Nidhi Purohit, and Kirill Simonov. FPT Approximation for Fair Minimum-Load Clustering. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bandyapadhyay_et_al:LIPIcs.IPEC.2022.4,
author = {Bandyapadhyay, Sayan and Fomin, Fedor V. and Golovach, Petr A. and Purohit, Nidhi and Simonov, Kirill},
title = {{FPT Approximation for Fair Minimum-Load Clustering}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {4:1--4:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.4},
URN = {urn:nbn:de:0030-drops-173600},
doi = {10.4230/LIPIcs.IPEC.2022.4},
annote = {Keywords: fair clustering, load balancing, parameterized approximation}
}
Document
On Sparse Hitting Sets: From Fair Vertex Cover to Highway Dimension
Abstract
We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system (V,ℱ,ℬ) with two families ℱ,ℬ of subsets of the universe V. The task is to find a hitting set for ℱ that minimizes the maximum number of elements in any of the sets of ℬ. This generalizes several problems that have been studied in the literature. Our focus is on determining the complexity of some of these special cases of Sparse-HS with respect to the sparseness k, which is the optimum number of hitting set elements in any set of ℬ (i.e., the value of the objective function).
For the Sparse Vertex Cover (Sparse-VC) problem, the universe is given by the vertex set V of a graph, and ℱ is its edge set. We prove NP-hardness for sparseness k ≥ 2 and polynomial time solvability for k = 1. We also provide a polynomial-time 2-approximation algorithm for any k. A special case of Sparse-VC is Fair Vertex Cover (Fair-VC), where the family ℬ is given by vertex neighbourhoods. For this problem it was open whether it is FPT (or even XP) parameterized by the sparseness k. We answer this question in the negative, by proving NP-hardness for constant k. We also provide a polynomial-time (2-1/k)-approximation algorithm for Fair-VC, which is better than any approximation algorithm possible for Sparse-VC or the Vertex Cover problem (under the Unique Games Conjecture).
We then switch to a different set of problems derived from Sparse-HS related to the highway dimension, which is a graph parameter modelling transportation networks. In recent years a growing literature has shown interesting algorithms for graphs of low highway dimension. To exploit the structure of such graphs, most of them compute solutions to the r-Shortest Path Cover (r-SPC) problem, where r > 0, ℱ contains all shortest paths of length between r and 2r, and ℬ contains all balls of radius 2r. It is known that there is an XP algorithm that computes solutions to r-SPC of sparseness at most h if the input graph has highway dimension h. However it was not known whether a corresponding FPT algorithm exists as well. We prove that r-SPC and also the related r-Highway Dimension (r-HD) problem, which can be used to formally define the highway dimension of a graph, are both W[1]-hard. Furthermore, by the result of Abraham et al. [ICALP 2011] there is a polynomial-time O(log k)-approximation algorithm for r-HD, but for r-SPC such an algorithm is not known. We prove that r-SPC admits a polynomial-time O(log n)-approximation algorithm.
Cite as
Johannes Blum, Yann Disser, Andreas Emil Feldmann, Siddharth Gupta, and Anna Zych-Pawlewicz. On Sparse Hitting Sets: From Fair Vertex Cover to Highway Dimension. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{blum_et_al:LIPIcs.IPEC.2022.5,
author = {Blum, Johannes and Disser, Yann and Feldmann, Andreas Emil and Gupta, Siddharth and Zych-Pawlewicz, Anna},
title = {{On Sparse Hitting Sets: From Fair Vertex Cover to Highway Dimension}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {5:1--5:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.5},
URN = {urn:nbn:de:0030-drops-173612},
doi = {10.4230/LIPIcs.IPEC.2022.5},
annote = {Keywords: sparse hitting set, fair vertex cover, highway dimension}
}
Document
On the Complexity of Problems on Tree-Structured Graphs
Abstract
In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in f(k)n^O(1) time and f(k)log n space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on "tree-structured graphs" are complete for this class: we show that List Coloring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by log n, and Max Cut parameterized by cliquewidth are also XALP-complete.
Besides finding a "natural home" for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most f(k)n^O(1) and use f(k)log n space. Moreover, we introduce "tree-shaped" variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete.
Cite as
Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Marcin Pilipczuk, and Michał Pilipczuk. On the Complexity of Problems on Tree-Structured Graphs. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2022.6,
author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo and Pilipczuk, Marcin and Pilipczuk, Micha{\l}},
title = {{On the Complexity of Problems on Tree-Structured Graphs}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {6:1--6:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.6},
URN = {urn:nbn:de:0030-drops-173626},
doi = {10.4230/LIPIcs.IPEC.2022.6},
annote = {Keywords: Parameterized Complexity, Treewidth, XALP, XNLP}
}
Document
On the Parameterized Complexity of Computing Tree-Partitions
Abstract
We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree.
On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an n-vertex graph G and an integer k, constructs a tree-partition of width O(k⁷) for G or reports that G has tree-partition width more than k, in time k^O(1) n². We can improve slightly on the approximation factor by sacrificing the dependence on k, or on n.
On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is W[t]-hard for all t. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width.
Cite as
Hans L. Bodlaender, Carla Groenland, and Hugo Jacob. On the Parameterized Complexity of Computing Tree-Partitions. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2022.7,
author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo},
title = {{On the Parameterized Complexity of Computing Tree-Partitions}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {7:1--7:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.7},
URN = {urn:nbn:de:0030-drops-173633},
doi = {10.4230/LIPIcs.IPEC.2022.7},
annote = {Keywords: Parameterized algorithms, Tree partitions, tree-partition-width, Treewidth, Domino Treewidth, Approximation Algorithms, Parameterized Complexity}
}
Document
XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure
Abstract
In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space.
In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.
Cite as
Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T. Lima. XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2022.8,
author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo and Jaffke, Lars and Lima, Paloma T.},
title = {{XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {8:1--8:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.8},
URN = {urn:nbn:de:0030-drops-173640},
doi = {10.4230/LIPIcs.IPEC.2022.8},
annote = {Keywords: parameterized complexity, XNLP, linear clique-width, W-hierarchy, pathwidth, linear mim-width, bandwidth}
}
Document
Twin-Width VIII: Delineation and Win-Wins
Abstract
We introduce the notion of delineation. A graph class C is said delineated by twin-width (or simply, delineated) if for every hereditary closure D of a subclass of C, it holds that D has bounded twin-width if and only if D is monadically dependent. An effective strengthening of delineation for a class C implies that tractable FO model checking on C is perfectly understood: On hereditary closures of subclasses D of C, FO model checking on D is fixed-parameter tractable (FPT) exactly when D has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we observe or show that segment graphs, directed path graphs (with arbitrarily many roots), and visibility graphs of simple polygons are not delineated.
In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA '21]. We show that K_{t,t}-free segment graphs, and axis-parallel H_t-free unit segment graphs have bounded twin-width, where H_t is the half-graph or ladder of height t. In contrast, axis-parallel H₄-free two-lengthed segment graphs have unbounded twin-width. We leave as an open question whether unit segment graphs are delineated.
More broadly, we explore which structures (large bicliques, half-graphs, or independent sets) are responsible for making the twin-width large on the main classes of intersection and visibility graphs. Our new results, combined with the FPT algorithm for first-order model checking on graphs given with O(1)-sequences [BKTW, JACM '22], give rise to a variety of algorithmic win-win arguments. They all fall in the same framework: If p is an FO definable graph parameter that effectively functionally upperbounds twin-width on a class C, then p(G) ⩾ k can be decided in FPT time f(k) ⋅ |V(G)|^O(1). For instance, we readily derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains, and k-Independent Set on visibility graphs of simple polygons. This showcases that the theory of twin-width can serve outside of classes of bounded twin-width.
Cite as
Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Noleen Köhler, Raul Lopes, and Stéphan Thomassé. Twin-Width VIII: Delineation and Win-Wins. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2022.9,
author = {Bonnet, \'{E}douard and Chakraborty, Dibyayan and Kim, Eun Jung and K\"{o}hler, Noleen and Lopes, Raul and Thomass\'{e}, St\'{e}phan},
title = {{Twin-Width VIII: Delineation and Win-Wins}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {9:1--9:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.9},
URN = {urn:nbn:de:0030-drops-173650},
doi = {10.4230/LIPIcs.IPEC.2022.9},
annote = {Keywords: Twin-width, intersection graphs, visibility graphs, monadic dependence and stability, first-order model checking}
}
Document
Obstructions to Faster Diameter Computation: Asteroidal Sets
Abstract
An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let Ext_α be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than α pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every m-edge graph in Ext_{α} can be computed in deterministic O(α³ m^{3/2}) time. We then improve the runtime to O(α³m) for bipartite graphs, to O(α⁵m) for triangle-free graphs, O(α³Δm) for graphs with maximum degree Δ, and more generally to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive +1-approximation of all vertex eccentricities in deterministic O(α² m) time. This is in sharp contrast with general m-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in O(m^{2-ε}) time for any ε > 0.
As important special cases of our main result, we derive an O(m^{3/2})-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an O(k³m^{3/2})-time algorithm for this problem on graphs of asteroidal number at most k. Both results extend prior works on exact and approximate diameter computation within AT-free graphs. To the best of our knowledge, this is also the first deterministic subquadratic-time algorithm for computing the diameter within the subclasses of: chordal graphs of bounded leafage (generalizing the interval graphs), k-moplex graphs and k-polygon graphs (generalizing the permutation graphs) for any fixed k. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions.
Our approach is purely combinatorial, that differs from most prior recent works in this area which have relied on geometric primitives such as Voronoi diagrams or range queries. On our way, we uncover interesting connections between the diameter problem, Lexicographic Breadth-First Search, graph extremities and the asteroidal number.
Cite as
Guillaume Ducoffe. Obstructions to Faster Diameter Computation: Asteroidal Sets. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ducoffe:LIPIcs.IPEC.2022.10,
author = {Ducoffe, Guillaume},
title = {{Obstructions to Faster Diameter Computation: Asteroidal Sets}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {10:1--10:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.10},
URN = {urn:nbn:de:0030-drops-173668},
doi = {10.4230/LIPIcs.IPEC.2022.10},
annote = {Keywords: Diameter computation, Asteroidal number, LexBFS}
}
Document
On the Parameterized Complexity of Symmetric Directed Multicut
Abstract
We study the problem Symmetric Directed Multicut from a parameterized complexity perspective. In this problem, the input is a digraph D, a set of cut requests C = {(s₁,t₁),…,(s_l,t_l)} and an integer k, and the task is to find a set X ⊆ V(D) of size at most k such that for every 1 ≤ i ≤ l, X intersects either all (s_i,t_i)-paths or all (t_i,s_i)-paths. Equivalently, every strongly connected component of D-X contains at most one vertex out of s_i and t_i for every i. This problem is previously known from research in approximation algorithms, where it is known to have an O(log k log log k)-approximation. We note that the problem, parameterized by k, directly generalizes multiple interesting FPT problems such as (Undirected) Vertex Multicut and Directed Subset Feedback Vertex Set. We are not able to settle the existence of an FPT algorithm parameterized purely by k, but we give three partial results: An FPT algorithm parameterized by k+l; an FPT-time 2-approximation parameterized by k; and an FPT algorithm parameterized by k for the special case that the cut requests form a clique, Symmetric Directed Multiway Cut. The existence of an FPT algorithm parameterized purely by k remains an intriguing open possibility.
Cite as
Eduard Eiben, Clément Rambaud, and Magnus Wahlström. On the Parameterized Complexity of Symmetric Directed Multicut. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{eiben_et_al:LIPIcs.IPEC.2022.11,
author = {Eiben, Eduard and Rambaud, Cl\'{e}ment and Wahlstr\"{o}m, Magnus},
title = {{On the Parameterized Complexity of Symmetric Directed Multicut}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {11:1--11:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.11},
URN = {urn:nbn:de:0030-drops-173679},
doi = {10.4230/LIPIcs.IPEC.2022.11},
annote = {Keywords: Parameterized complexity, directed graphs, graph separation problems}
}
Document
Computing Generalized Convolutions Faster Than Brute Force
Abstract
In this paper, we consider a general notion of convolution. Let D be a finite domain and let Dⁿ be the set of n-length vectors (tuples) of D. Let f : D × D → D be a function and let ⊕_f be a coordinate-wise application of f. The f-Convolution of two functions g,h : Dⁿ → {-M,…,M} is (g ⊛_f h)(v) := ∑_{v_g,v_h ∈ D^n s.t. v = v_g ⊕_f v_h} g(v_g) ⋅ h(v_h) for every 𝐯 ∈ Dⁿ. This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function f and domain D we can compute f-Convolution via brute-force enumeration in 𝒪̃(|D|^{2n} ⋅ polylog(M)) time.
Our main result is an improvement over this naive algorithm. We show that f-Convolution can be computed exactly in 𝒪̃((c ⋅ |D|²)ⁿ ⋅ polylog(M)) for constant c := 5/6 when D has even cardinality. Our main observation is that a cyclic partition of a function f : D × D → D can be used to speed up the computation of f-Convolution, and we show that an appropriate cyclic partition exists for every f.
Furthermore, we demonstrate that a single entry of the f-Convolution can be computed more efficiently. In this variant, we are given two functions g,h : Dⁿ → {-M,…,M} alongside with a vector 𝐯 ∈ Dⁿ and the task of the f-Query problem is to compute integer (g ⊛_f h)(𝐯). This is a generalization of the well-known Orthogonal Vectors problem. We show that f-Query can be computed in 𝒪̃(|D|^{(ω/2)n} ⋅ polylog(M)) time, where ω ∈ [2,2.373) is the exponent of currently fastest matrix multiplication algorithm.
Cite as
Barış Can Esmer, Ariel Kulik, Dániel Marx, Philipp Schepper, and Karol Węgrzycki. Computing Generalized Convolutions Faster Than Brute Force. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{esmer_et_al:LIPIcs.IPEC.2022.12,
author = {Esmer, Bar{\i}\c{s} Can and Kulik, Ariel and Marx, D\'{a}niel and Schepper, Philipp and W\k{e}grzycki, Karol},
title = {{Computing Generalized Convolutions Faster Than Brute Force}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {12:1--12:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.12},
URN = {urn:nbn:de:0030-drops-173685},
doi = {10.4230/LIPIcs.IPEC.2022.12},
annote = {Keywords: Generalized Convolution, Fast Fourier Transform, Fast Subset Convolution}
}
Document
Exact Exponential Algorithms for Clustering Problems
Abstract
In this paper we initiate a systematic study of exact algorithms for some of the well known clustering problems, namely k-MEDIAN and k-MEANS. In k-MEDIAN, the input consists of a set X of n points belonging to a metric space, and the task is to select a subset C ⊆ X of k points as centers, such that the sum of the distances of every point to its nearest center is minimized. In k-MEANS, the objective is to minimize the sum of squares of the distances instead. It is easy to design an algorithm running in time max_{k ≤ n} {n choose k} n^𝒪(1) = 𝒪^*(2ⁿ) (here, 𝒪^*(⋅) notation hides polynomial factors in n). In this paper we design first non-trivial exact algorithms for these problems. In particular, we obtain an 𝒪^*((1.89)ⁿ) time exact algorithm for k-MEDIAN that works for any value of k. Our algorithm is quite general in that it does not use any properties of the underlying (metric) space - it does not even require the distances to satisfy the triangle inequality. In particular, the same algorithm also works for k-Means. We complement this result by showing that the running time of our algorithm is asymptotically optimal, up to the base of the exponent. That is, unless the Exponential Time Hypothesis fails, there is no algorithm for these problems running in time 2^o(n)⋅n^𝒪(1).
Finally, we consider the "facility location" or "supplier" versions of these clustering problems, where, in addition to the set X we are additionally given a set of m candidate centers (or facilities) F, and objective is to find a subset of k centers from F. The goal is still to minimize the k-Median/k-Means/k-Center objective. For these versions we give a 𝒪(2ⁿ (mn)^𝒪(1)) time algorithms using subset convolution. We complement this result by showing that, under the Set Cover Conjecture, the "supplier" versions of these problems do not admit an exact algorithm running in time 2^{(1-ε) n} (mn)^𝒪(1).
Cite as
Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Nidhi Purohit, and Saket Saurabh. Exact Exponential Algorithms for Clustering Problems. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{fomin_et_al:LIPIcs.IPEC.2022.13,
author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Purohit, Nidhi and Saurabh, Saket},
title = {{Exact Exponential Algorithms for Clustering Problems}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {13:1--13:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.13},
URN = {urn:nbn:de:0030-drops-173691},
doi = {10.4230/LIPIcs.IPEC.2022.13},
annote = {Keywords: clustering, k-median, k-means, exact algorithms}
}
Document
Domination and Cut Problems on Chordal Graphs with Bounded Leafage
Abstract
The leafage of a chordal graph G is the minimum integer 𝓁 such that G can be realized as an intersection graph of subtrees of a tree with 𝓁 leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2^𝒪(𝓁²) ⋅ n^𝒪(1). We present a conceptually much simpler algorithm that runs in time 2^𝒪(𝓁) ⋅ n^𝒪(1). We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple n^𝒪(𝓁)-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in n^O(1)-time.
Cite as
Esther Galby, Dániel Marx, Philipp Schepper, Roohani Sharma, and Prafullkumar Tale. Domination and Cut Problems on Chordal Graphs with Bounded Leafage. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{galby_et_al:LIPIcs.IPEC.2022.14,
author = {Galby, Esther and Marx, D\'{a}niel and Schepper, Philipp and Sharma, Roohani and Tale, Prafullkumar},
title = {{Domination and Cut Problems on Chordal Graphs with Bounded Leafage}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {14:1--14:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.14},
URN = {urn:nbn:de:0030-drops-173704},
doi = {10.4230/LIPIcs.IPEC.2022.14},
annote = {Keywords: Chordal Graphs, Leafage, FPT Algorithms, Dominating Set, MultiCut with Undeletable Terminals, Multiway Cut with Undeletable Terminals}
}
Document
Slim Tree-Cut Width
Abstract
Tree-cut width is a parameter that has been introduced as an attempt to obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its desirable structural properties, it turned out that tree-cut width falls short as an edge-cut based alternative to treewidth in algorithmic aspects. This has led to the very recent introduction of a simple edge-based parameter called edge-cut width [WG 2022], which has precisely the algorithmic applications one would expect from an analogue of treewidth for edge cuts, but does not have the desired structural properties.
In this paper, we study a variant of tree-cut width obtained by changing the threshold for so-called thin nodes in tree-cut decompositions from 2 to 1. We show that this "slim tree-cut width" satisfies all the requirements of an edge-cut based analogue of treewidth, both structural and algorithmic, while being less restrictive than edge-cut width. Our results also include an alternative characterization of slim tree-cut width via an easy-to-use spanning-tree decomposition akin to the one used for edge-cut width, a characterization of slim tree-cut width in terms of forbidden immersions as well as an approximation algorithm for computing the parameter.
Cite as
Robert Ganian and Viktoriia Korchemna. Slim Tree-Cut Width. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ganian_et_al:LIPIcs.IPEC.2022.15,
author = {Ganian, Robert and Korchemna, Viktoriia},
title = {{Slim Tree-Cut Width}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {15:1--15:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.15},
URN = {urn:nbn:de:0030-drops-173714},
doi = {10.4230/LIPIcs.IPEC.2022.15},
annote = {Keywords: tree-cut width, structural parameters, graph immersions}
}
Document
A Fixed-Parameter Algorithm for the Schrijver Problem
Abstract
The Schrijver graph S(n,k) is defined for integers n and k with n ≥ 2k as the graph whose vertices are all the k-subsets of {1,2,…,n} that do not include two consecutive elements modulo n, where two such sets are adjacent if they are disjoint. A result of Schrijver asserts that the chromatic number of S(n,k) is n-2k+2 (Nieuw Arch. Wiskd., 1978). In the computational Schrijver problem, we are given an access to a coloring of the vertices of S(n,k) with n-2k+1 colors, and the goal is to find a monochromatic edge. The Schrijver problem is known to be complete in the complexity class PPA. We prove that it can be solved by a randomized algorithm with running time n^O(1) ⋅ k^O(k), hence it is fixed-parameter tractable with respect to the parameter k.
Cite as
Ishay Haviv. A Fixed-Parameter Algorithm for the Schrijver Problem. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{haviv:LIPIcs.IPEC.2022.16,
author = {Haviv, Ishay},
title = {{A Fixed-Parameter Algorithm for the Schrijver Problem}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {16:1--16:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.16},
URN = {urn:nbn:de:0030-drops-173721},
doi = {10.4230/LIPIcs.IPEC.2022.16},
annote = {Keywords: Schrijver graph, Kneser graph, Fixed-parameter tractability}
}
Document
Towards Exact Structural Thresholds for Parameterized Complexity
Abstract
Parameterized complexity seeks to optimally use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth, i.e., graphs decomposable by small separators: Many problems admit fast algorithms relative to treewidth and many of them are optimal under the Strong Exponential-Time Hypothesis (SETH). Fewer such results are known for more general structure such as low clique-width (decomposition by large and dense but structured separators) and more restrictive structure such as low deletion distance to some sparse graph class.
Despite these successes, such results remain "islands" within the realm of possible structure. Rather than adding more islands, we seek to determine the transitions between them, that is, we aim for structural thresholds where the complexity increases as input structure becomes more general. Going from deletion distance to treewidth, is a single deletion set to a graph with simple components enough to yield the same lower bound as for treewidth or does it take many disjoint separators? Going from treewidth to clique-width, how much more density entails the same complexity as clique-width? Conversely, what is the most restrictive structure that yields the same lower bound?
For treewidth, we obtain both refined and new lower bounds that apply already to graphs with a single separator X such that G-X has treewidth at most r = 𝒪(1), while G has treewidth |X|+𝒪(1). We rule out algorithms running in time 𝒪^*((r+1-ε)^k) for Deletion to r-Colorable parameterized by k = |X|; this implies the same lower bound relative to treedepth and (hence) also to treewidth. It specializes to 𝒪^*((3-ε)^k) for Odd Cycle Transversal where tw(G-X) ≤ r = 2 is best possible. For clique-width, an extended version of the above reduction rules out time 𝒪^*((4-ε)^k), where X is allowed to be a possibly large separator consisting of k (true) twinclasses, while the treewidth of G - X remains r; this is proved also for the more general Deletion to r-Colorable and it implies the same lower bound relative to clique-width. Further results complement what is known for Vertex Cover, Dominating Set and Maximum Cut. All lower bounds are matched by existing and newly designed algorithms.
Cite as
Falko Hegerfeld and Stefan Kratsch. Towards Exact Structural Thresholds for Parameterized Complexity. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{hegerfeld_et_al:LIPIcs.IPEC.2022.17,
author = {Hegerfeld, Falko and Kratsch, Stefan},
title = {{Towards Exact Structural Thresholds for Parameterized Complexity}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {17:1--17:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.17},
URN = {urn:nbn:de:0030-drops-173734},
doi = {10.4230/LIPIcs.IPEC.2022.17},
annote = {Keywords: Parameterized complexity, lower bound, vertex cover, odd cycle transversal, SETH, modulator, treedepth, cliquewidth}
}
Document
Hardness of Interval Scheduling on Unrelated Machines
Abstract
We provide new (parameterized) computational hardness results for Interval Scheduling on Unrelated Machines. It is a classical scheduling problem motivated from just-in-time or lean manufacturing, where the goal is to complete jobs exactly at their deadline. We are given n jobs and m machines. Each job has a deadline, a weight, and a processing time that may be different on each machine. The goal is find a schedule that maximizes the total weight of jobs completed exactly at their deadline. Note that this uniquely defines a processing time interval for each job on each machine.
Interval Scheduling on Unrelated Machines is closely related to coloring interval graphs and has been thoroughly studied for several decades. However, as pointed out by Mnich and van Bevern [Computers & Operations Research, 2018], the parameterized complexity for the number m of machines as a parameter remained open. We resolve this by showing that Interval Scheduling on Unrelated Machines is W[1]-hard when parameterized by the number m of machines. To this end, we prove W[1]-hardness with respect to m of the special case where we have parallel machines with eligible machine sets for jobs. This answers Open Problem 8 of Mnich and van Bevern’s list of 15 open problems in the parameterized complexity of scheduling [Computers & Operations Research, 2018].
Furthermore, we resolve the computational complexity status of the unweighted version of Interval Scheduling on Unrelated Machines by proving that it is NP-complete. This answers an open question by Sung and Vlach [Journal of Scheduling, 2005].
Cite as
Danny Hermelin, Yuval Itzhaki, Hendrik Molter, and Dvir Shabtay. Hardness of Interval Scheduling on Unrelated Machines. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{hermelin_et_al:LIPIcs.IPEC.2022.18,
author = {Hermelin, Danny and Itzhaki, Yuval and Molter, Hendrik and Shabtay, Dvir},
title = {{Hardness of Interval Scheduling on Unrelated Machines}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {18:1--18:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.18},
URN = {urn:nbn:de:0030-drops-173748},
doi = {10.4230/LIPIcs.IPEC.2022.18},
annote = {Keywords: Just-in-time scheduling, Parallel machines, Eligible machine sets, W\lbrack1\rbrack-hardness, NP-hardness}
}
Document
Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees
Abstract
Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem.
Cite as
Leon Kellerhals, Tomohiro Koana, and Pascal Kunz. Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kellerhals_et_al:LIPIcs.IPEC.2022.19,
author = {Kellerhals, Leon and Koana, Tomohiro and Kunz, Pascal},
title = {{Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {19:1--19:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.19},
URN = {urn:nbn:de:0030-drops-173758},
doi = {10.4230/LIPIcs.IPEC.2022.19},
annote = {Keywords: parameterized complexity, vertex cover, feedback vertex set, above guarantee parameterization}
}
Document
Parameterized Local Search for Vertex Cover: When Only the Search Radius Is Crucial
Abstract
A k-swap W for a vertex cover S of a graph G is a vertex set of size at most k such that S' = (S ⧵ W) ∪ (W ⧵ S), the symmetric difference of S and W, is a vertex cover of G. If |S'| < |S|, then W is improving. In LS-Vertex Cover, one is given a vertex cover S of a graph G and wants to know if there is an improving k-swap for S in G. In applications of LS-Vertex Cover, k is a very small parameter that can be set by a user to determine the trade-off between running time and solution quality. Consequently, k can be considered to be a constant. Motivated by this and the fact that LS-Vertex Cover is W[1]-hard with respect to k, we aim for algorithms with running time 𝓁^f(k) ⋅ n^𝒪(1) where 𝓁 is a structural graph parameter upper-bounded by n. We say that such a running time grows mildly with respect to 𝓁 and strongly with respect to k. We obtain algorithms with such a running time for 𝓁 being the h-index of G, the treewidth of G, or the modular-width of G. In addition, we consider a novel parameter, the maximum degree over all quotient graphs in a modular decomposition of G. Moreover, we adapt these algorithms to the more general problem where each vertex is assigned a weight and where we want to find a d-improving k-swap, that is, a k-swap which decreases the weight of the vertex cover by at least d.
Cite as
Christian Komusiewicz and Nils Morawietz. Parameterized Local Search for Vertex Cover: When Only the Search Radius Is Crucial. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{komusiewicz_et_al:LIPIcs.IPEC.2022.20,
author = {Komusiewicz, Christian and Morawietz, Nils},
title = {{Parameterized Local Search for Vertex Cover: When Only the Search Radius Is Crucial}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {20:1--20:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.20},
URN = {urn:nbn:de:0030-drops-173764},
doi = {10.4230/LIPIcs.IPEC.2022.20},
annote = {Keywords: Local Search, Structural parameterization, Fixed-parameter tractability}
}
Document
Parameterized Complexity of a Parallel Machine Scheduling Problem
Abstract
In this paper we consider the parameterized complexity of two versions of a parallel machine scheduling problem with precedence delays, unit processing times and time windows. In the first version - with exact delays - we assume that the delay between two jobs must be exactly respected, whereas in the second version - with minimum delays - the delay between two jobs is a lower bound on the time between them. Two parameters are considered for this analysis: the pathwidth of the interval graph induced by the time windows and the maximum precedence delay value. We prove that our problems are para-NP-complete with respect to any of the two parameters and fixed-parameter tractable parameterized by the pair of parameters.
Cite as
Maher Mallem, Claire Hanen, and Alix Munier-Kordon. Parameterized Complexity of a Parallel Machine Scheduling Problem. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{mallem_et_al:LIPIcs.IPEC.2022.21,
author = {Mallem, Maher and Hanen, Claire and Munier-Kordon, Alix},
title = {{Parameterized Complexity of a Parallel Machine Scheduling Problem}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {21:1--21:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.21},
URN = {urn:nbn:de:0030-drops-173774},
doi = {10.4230/LIPIcs.IPEC.2022.21},
annote = {Keywords: parameterized complexity, scheduling, precedence delays, pathwidth, chains, parallel processors}
}
Document
Anti-Factor Is FPT Parameterized by Treewidth and List Size (But Counting Is Hard)
Abstract
In the general AntiFactor problem, a graph G and, for every vertex v of G, a set X_v ⊆ ℕ of forbidden degrees is given. The task is to find a set S of edges such that the degree of v in S is not in the set X_v. Standard techniques (dynamic programming plus fast convolution) can be used to show that if M is the largest forbidden degree, then the problem can be solved in time (M+2)^{tw}⋅n^{O(1)} if a tree decomposition of width tw is given. However, significantly faster algorithms are possible if the sets X_v are sparse: our main algorithmic result shows that if every vertex has at most x forbidden degrees (we call this special case AntiFactor_x), then the problem can be solved in time (x+1)^{O(tw)}⋅n^{O(1)}. That is, AntiFactor_x is fixed-parameter tractable parameterized by treewidth tw and the maximum number x of excluded degrees.
Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor₁ is already #W[1]-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set X, we denote by X-AntiFactor the special case where every vertex v has the same set X_v = X of forbidden degrees. We show the following lower bound for every fixed set X: if there is an ε > 0 such that #X-AntiFactor can be solved in time (max X+2-ε)^{tw}⋅n^{O(1)} given a tree decomposition of width tw, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails.
Cite as
Dániel Marx, Govind S. Sankar, and Philipp Schepper. Anti-Factor Is FPT Parameterized by Treewidth and List Size (But Counting Is Hard). In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{marx_et_al:LIPIcs.IPEC.2022.22,
author = {Marx, D\'{a}niel and Sankar, Govind S. and Schepper, Philipp},
title = {{Anti-Factor Is FPT Parameterized by Treewidth and List Size (But Counting Is Hard)}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {22:1--22:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.22},
URN = {urn:nbn:de:0030-drops-173780},
doi = {10.4230/LIPIcs.IPEC.2022.22},
annote = {Keywords: Anti-Factor, General Factor, Treewidth, Representative Sets, SETH}
}
Document
Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph
Abstract
We present fixed-parameter tractable (FPT) algorithms for two problems, Maximum Happy Set (MaxHS) and Densest k-Subgraph (DkS) - also known as Maximum Edge Happy Set. Given a graph G and an integer k, MaxHS asks for a set S of k vertices such that the number of happy vertices with respect to S is maximized, where a vertex v is happy if v and all its neighbors are in S. We show that MaxHS can be solved in time 𝒪(2^mw ⋅ mw ⋅ k² ⋅ |V(G)|) and 𝒪(8^cw ⋅ k² ⋅ |V(G)|), where mw and cw denote the modular-width and the clique-width of G, respectively. This answers the open questions on fixed-parameter tractability posed in [Asahiro et al., 2021].
The DkS problem asks for a subgraph with k vertices maximizing the number of edges. If we define happy edges as the edges whose endpoints are in S, then DkS can be seen as an edge-variant of MaxHS. In this paper we show that DkS can be solved in time f(nd)⋅|V(G)|^𝒪(1) and 𝒪(2^{cd}⋅ k² ⋅ |V(G)|), where nd and cd denote the neighborhood diversity and the cluster deletion number of G, respectively, and f is some computable function. This result implies that DkS is also fixed-parameter tractable by twin cover number.
Cite as
Yosuke Mizutani and Blair D. Sullivan. Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{mizutani_et_al:LIPIcs.IPEC.2022.23,
author = {Mizutani, Yosuke and Sullivan, Blair D.},
title = {{Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {23:1--23:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.23},
URN = {urn:nbn:de:0030-drops-173795},
doi = {10.4230/LIPIcs.IPEC.2022.23},
annote = {Keywords: parameterized algorithms, maximum happy set, densest k-subgraph, modular-width, clique-width, neighborhood diversity, cluster deletion number, twin cover}
}
Document
Parameterized Complexity of Streaming Diameter and Connectivity Problems
Abstract
We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is 𝒪(log n) for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in 𝒪(k) passes and 𝒪(k log n) bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where Ω(n/p) bits of memory is needed for any p-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph H, for most H. For some cases, we can also show one-pass, Ω(n log n) bits of memory lower bounds. We also prove a much stronger Ω(n²/p) lower bound for Diameter on bipartite graphs.
Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size k. This yields a kernel of 2k vertices (with 𝒪(k²) edges) produced as a stream in poly(k) passes and only 𝒪(k log n) bits of memory.
Cite as
Jelle J. Oostveen and Erik Jan van Leeuwen. Parameterized Complexity of Streaming Diameter and Connectivity Problems. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{oostveen_et_al:LIPIcs.IPEC.2022.24,
author = {Oostveen, Jelle J. and van Leeuwen, Erik Jan},
title = {{Parameterized Complexity of Streaming Diameter and Connectivity Problems}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {24:1--24:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.24},
URN = {urn:nbn:de:0030-drops-173808},
doi = {10.4230/LIPIcs.IPEC.2022.24},
annote = {Keywords: Stream, Streaming, Graphs, Parameter, Complexity, Diameter, Connectivity, Vertex Cover, Disjointness, Permutation}
}
Document
Applying a Cut-Based Data Reduction Rule for Weighted Cluster Editing in Polynomial Time
Abstract
Given an undirected graph, the task in Cluster Editing is to insert and delete a minimum number of edges to obtain a cluster graph, that is, a disjoint union of cliques. In the weighted variant each vertex pair comes with a weight and the edge modifications have to be of minimum overall weight. In this work, we provide the first polynomial-time algorithm to apply the following data reduction rule of Böcker et al. [Algorithmica, 2011] for Weighted Cluster Editing: For a graph G = (V,E), merge a vertex set S ⊆ V into a single vertex if the minimum cut of G[S] is at least the combined cost of inserting all missing edges within G[S] plus the cost of cutting all edges from S to the rest of the graph. Complementing our theoretical findings, we experimentally demonstrate the effectiveness of the data reduction rule, shrinking real-world test instances from the PACE Challenge 2021 by around 24% while previous heuristic implementations of the data reduction rule only achieve 8%.
Cite as
Hjalmar Schulz, André Nichterlein, Rolf Niedermeier, and Christopher Weyand. Applying a Cut-Based Data Reduction Rule for Weighted Cluster Editing in Polynomial Time. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{schulz_et_al:LIPIcs.IPEC.2022.25,
author = {Schulz, Hjalmar and Nichterlein, Andr\'{e} and Niedermeier, Rolf and Weyand, Christopher},
title = {{Applying a Cut-Based Data Reduction Rule for Weighted Cluster Editing in Polynomial Time}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {25:1--25:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.25},
URN = {urn:nbn:de:0030-drops-173816},
doi = {10.4230/LIPIcs.IPEC.2022.25},
annote = {Keywords: Correlation Clustering, Minimum Cut, Maximum s-t-Flow}
}
Document
The PACE 2022 Parameterized Algorithms and Computational Experiments Challenge: Directed Feedback Vertex Set
Abstract
The Parameterized Algorithms and Computational Experiments challenge (PACE) 2022 was devoted to engineer algorithms solving the NP-hard Directed Feedback Vertex Set (DFVS) problem. The DFVS problem is to find a minimum subset X ⊆ V in a given directed graph G = (V,E) such that, when all vertices of X and their adjacent edges are deleted from G, the remainder is acyclic.
Overall, the challenge had 90 participants from 26 teams, 12 countries, and 3 continents that submitted their implementations to this year’s competition. In this report, we briefly describe the setup of the challenge, the selection of benchmark instances, as well as the ranking of the participating teams. We also briefly outline the approaches used in the submitted solvers.
Cite as
Ernestine Großmann, Tobias Heuer, Christian Schulz, and Darren Strash. The PACE 2022 Parameterized Algorithms and Computational Experiments Challenge: Directed Feedback Vertex Set. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gromann_et_al:LIPIcs.IPEC.2022.26,
author = {Gro{\ss}mann, Ernestine and Heuer, Tobias and Schulz, Christian and Strash, Darren},
title = {{The PACE 2022 Parameterized Algorithms and Computational Experiments Challenge: Directed Feedback Vertex Set}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {26:1--26:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.26},
URN = {urn:nbn:de:0030-drops-173826},
doi = {10.4230/LIPIcs.IPEC.2022.26},
annote = {Keywords: Feedback Vertex Set, Algorithm Engineering, FPT, Kernelization, Heuristics}
}
Document
PACE Solver Description
PACE Solver Description: DiVerSeS - A Heuristic Solver for the Directed Feedback Vertex Set Problem
Abstract
This article briefly describes the most important algorithms and techniques used in the directed feedback vertex set heuristic solver called "DiVerSeS", submitted to the 7th Parameterized Algorithms and Computational Experiments Challenge (PACE 2022).
Cite as
Sylwester Swat. PACE Solver Description: DiVerSeS - A Heuristic Solver for the Directed Feedback Vertex Set Problem. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 27:1-27:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{swat:LIPIcs.IPEC.2022.27,
author = {Swat, Sylwester},
title = {{PACE Solver Description: DiVerSeS - A Heuristic Solver for the Directed Feedback Vertex Set Problem}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {27:1--27:3},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.27},
URN = {urn:nbn:de:0030-drops-173835},
doi = {10.4230/LIPIcs.IPEC.2022.27},
annote = {Keywords: Directed feedback vertex set, heuristic solver, graph algorithms, PACE 2022}
}
Document
PACE Solver Description
PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set
Abstract
In this document we describe the techniques we used and implemented for our submission to the Parameterized Algorithms and Computational Experiments Challenge (PACE) 2022. The given problem is Directed Feedback Vertex Set (DFVS), where one is given a directed graph G = (V,E) and wants to find a minimum S ⊆ V such that G-S is acyclic. We approach this problem by first exhaustively applying a set of reduction rules. In order to find a minimum DFVS on the remaining instance, we create and solve a series of Vertex Cover instances.
Cite as
Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt. PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 28:1-28:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{angrick_et_al:LIPIcs.IPEC.2022.28,
author = {Angrick, Sebastian and Bals, Ben and Casel, Katrin and Cohen, Sarel and Friedrich, Tobias and Hastrich, Niko and Hradilak, Theresa and Issac, Davis and Ki{\ss}ig, Otto and Schmidt, Jonas and Wendt, Leo},
title = {{PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {28:1--28:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.28},
URN = {urn:nbn:de:0030-drops-173847},
doi = {10.4230/LIPIcs.IPEC.2022.28},
annote = {Keywords: directed feedback vertex set, vertex cover, reduction rules}
}
Document
PACE Solver Description
PACE Solver Description: Hust-Solver - A Heuristic Algorithm of Directed Feedback Vertex Set Problem
Abstract
A directed graph is formed by vertices and arcs from one vertex to another. The feedback vertex set problem (FVSP) consists in making a given directed graph acyclic by removing as few vertices as possible. In this write-up, we outline the core techniques used in the heuristic feedback vertex set algorithm, submitted to the heuristic track of the 2022 PACE challenge.
Cite as
YuMing Du, QingYun Zhang, JunZhou Xu, ShunGen Zhang, Chao Liao, ZhiHuai Chen, ZhiBo Sun, ZhouXing Su, JunWen Ding, Chen Wu, PinYan Lu, and ZhiPeng Lv. PACE Solver Description: Hust-Solver - A Heuristic Algorithm of Directed Feedback Vertex Set Problem. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 29:1-29:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{du_et_al:LIPIcs.IPEC.2022.29,
author = {Du, YuMing and Zhang, QingYun and Xu, JunZhou and Zhang, ShunGen and Liao, Chao and Chen, ZhiHuai and Sun, ZhiBo and Su, ZhouXing and Ding, JunWen and Wu, Chen and Lu, PinYan and Lv, ZhiPeng},
title = {{PACE Solver Description: Hust-Solver - A Heuristic Algorithm of Directed Feedback Vertex Set Problem}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {29:1--29:3},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.29},
URN = {urn:nbn:de:0030-drops-173855},
doi = {10.4230/LIPIcs.IPEC.2022.29},
annote = {Keywords: directed feedback vertex set, local search, simulated annealing, set covering}
}
Document
PACE Solver Description
PACE Solver Description: GraPA-JAVA
Abstract
We present an exact solver for the DFVS, submitted for the exact track of the Parameterized Algorithms and Computational Experiments challenge (PACE) in 2022. The solver heavily relies on data reduction (known from the literature and new reduction rules). The instances are then further processed by integer linear programming approaches. We implemented the algorithm in the scope of a student project at the University of Bremen.
Cite as
Moritz Bergenthal, Jona Dirks, Thorben Freese, Jakob Gahde, Enna Gerhard, Mario Grobler, and Sebastian Siebertz. PACE Solver Description: GraPA-JAVA. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 30:1-30:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bergenthal_et_al:LIPIcs.IPEC.2022.30,
author = {Bergenthal, Moritz and Dirks, Jona and Freese, Thorben and Gahde, Jakob and Gerhard, Enna and Grobler, Mario and Siebertz, Sebastian},
title = {{PACE Solver Description: GraPA-JAVA}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {30:1--30:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.30},
URN = {urn:nbn:de:0030-drops-173861},
doi = {10.4230/LIPIcs.IPEC.2022.30},
annote = {Keywords: complexity theory, parameterized complexity, linear programming, java, directed feedback vertex set, PACE 2022}
}
Document
PACE Solver Description
PACE Solver Description: DreyFVS
Abstract
We describe DreyFVS, a heuristic for Directed Feedback Vertex Set submitted to the 2022 edition of Parameterized Algorithms and Computational Experiments Challenge. The Directed Feedback Vertex Set problem asks to remove a minimal number of vertices from a digraph such that the resulting digraph is acyclic. Our algorithm first performs a guess on a reduced instance by leveraging the Sinkhorn-Knopp algorithm, to then improve this solution by pipelining two local search methods.
Cite as
Gabriel Bathie, Gaétan Berthe, Yoann Coudert-Osmont, David Desobry, Amadeus Reinald, and Mathis Rocton. PACE Solver Description: DreyFVS. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 31:1-31:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bathie_et_al:LIPIcs.IPEC.2022.31,
author = {Bathie, Gabriel and Berthe, Ga\'{e}tan and Coudert-Osmont, Yoann and Desobry, David and Reinald, Amadeus and Rocton, Mathis},
title = {{PACE Solver Description: DreyFVS}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {31:1--31:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.31},
URN = {urn:nbn:de:0030-drops-173870},
doi = {10.4230/LIPIcs.IPEC.2022.31},
annote = {Keywords: Directed Feedback Vertex Set, Heuristic, Sinkhorn algorithm, Local search}
}
Document
PACE Solver Description
PACE Solver Description: DAGer - Cutting out Cycles with MaxSAT
Abstract
We describe the solver DAGer for the Directed Feedback Vertex Set (DFVS) problem, as it was submitted to the exact track of the 2022 PACE Challenge. Our approach first applies a wide range of preprocessing techniques involving both well-known data reductions for DFVS as well as non-trivial adaptations from the vertex cover problem. For the actual solving, we found that using a MaxSAT solver with incremental constraints achieves a good performance.
Cite as
Rafael Kiesel and André Schidler. PACE Solver Description: DAGer - Cutting out Cycles with MaxSAT. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 32:1-32:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kiesel_et_al:LIPIcs.IPEC.2022.32,
author = {Kiesel, Rafael and Schidler, Andr\'{e}},
title = {{PACE Solver Description: DAGer - Cutting out Cycles with MaxSAT}},
booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
pages = {32:1--32:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-260-0},
ISSN = {1868-8969},
year = {2022},
volume = {249},
editor = {Dell, Holger and Nederlof, Jesper},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.32},
URN = {urn:nbn:de:0030-drops-173885},
doi = {10.4230/LIPIcs.IPEC.2022.32},
annote = {Keywords: Directed Feeback Vertex Set, Data Reductions, Incremental MaxSAT}
}