Volume
LIPIcs, Volume 148
14th International Symposium on Parameterized and Exact Computation (IPEC 2019)
Event
IPEC 2019, September 11-13, 2019, Munich, GermanyEditors
Publication Details
- published at: 2019-12-04
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-129-0
- DBLP: db/conf/iwpec/ipec2019
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Complete Volume
LIPIcs, Volume 148, IPEC'19, Complete Volume
Abstract
LIPIcs, Volume 148, IPEC'19, Complete Volume
Cite as
14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@Proceedings{jansen_et_al:LIPIcs.IPEC.2019,
title = {{LIPIcs, Volume 148, IPEC'19, Complete Volume}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019},
URN = {urn:nbn:de:0030-drops-116409},
doi = {10.4230/LIPIcs.IPEC.2019},
annote = {Keywords: Theory of computation, Parameterized complexity and exact algorithms}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{jansen_et_al:LIPIcs.IPEC.2019.0,
author = {Jansen, Bart M. P. and Telle, Jan Arne},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {0:i--0:xvi},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.0},
URN = {urn:nbn:de:0030-drops-114618},
doi = {10.4230/LIPIcs.IPEC.2019.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Finding and Counting Permutations via CSPs
Abstract
Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k.
In this work we give two new algorithms for this well-studied problem, one whose running time is n^{k/4 + o(k)}, and a polynomial-space algorithm whose running time is the better of O(1.6181^n) and O(n^{k/2 + 1}). These results improve the earlier best bounds of n^{0.47k + o(k)} and O(1.79^n) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when k in Omega(log{n}). We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction.
Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time f(k) * n^{o(k/log{k})} would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing and 3-decreasing permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that an algorithm with sub-exponential running time is unlikely, even for patterns from these restricted classes.
Cite as
Benjamin Aram Berendsohn, László Kozma, and Dániel Marx. Finding and Counting Permutations via CSPs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{berendsohn_et_al:LIPIcs.IPEC.2019.1,
author = {Berendsohn, Benjamin Aram and Kozma, L\'{a}szl\'{o} and Marx, D\'{a}niel},
title = {{Finding and Counting Permutations via CSPs}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {1:1--1:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.1},
URN = {urn:nbn:de:0030-drops-114627},
doi = {10.4230/LIPIcs.IPEC.2019.1},
annote = {Keywords: permutations, pattern matching, constraint satisfaction, exponential time}
}
Document
Width Parameterizations for Knot-Free Vertex Deletion on Digraphs
Abstract
A knot in a directed graph G is a strongly connected subgraph Q of G with at least two vertices, such that no vertex in V(Q) is an in-neighbor of a vertex in V(G)\V(Q). Knots are important graph structures, because they characterize the existence of deadlocks in a classical distributed computation model, the so-called OR-model. Deadlock detection is correlated with the recognition of knot-free graphs as well as deadlock resolution is closely related to the Knot-Free Vertex Deletion (KFVD) problem, which consists of determining whether an input graph G has a subset S subseteq V(G) of size at most k such that G[V\S] contains no knot. Because of natural applications in deadlock resolution, KFVD is closely related to Directed Feedback Vertex Set. In this paper we focus on graph width measure parameterizations for KFVD. First, we show that: (i) KFVD parameterized by the size of the solution k is W[1]-hard even when p, the length of a longest directed path of the input graph, as well as kappa, its Kenny-width, are bounded by constants, and we remark that KFVD is para-NP-hard even considering many directed width measures as parameters, but in FPT when parameterized by clique-width; (ii) KFVD can be solved in time 2^{O(tw)} x n, but assuming ETH it cannot be solved in 2^{o(tw)} x n^{O(1)}, where tw is the treewidth of the underlying undirected graph. Finally, since the size of a minimum directed feedback vertex set (dfv) is an upper bound for the size of a minimum knot-free vertex deletion set, we investigate parameterization by dfv and we show that (iii) KFVD can be solved in FPT-time parameterized by either dfv+kappa or dfv+p. Results of (iii) cannot be improved when replacing dfv by k due to (i).
Cite as
Stéphane Bessy, Marin Bougeret, Alan D. A. Carneiro, Fábio Protti, and Uéverton S. Souza. Width Parameterizations for Knot-Free Vertex Deletion on Digraphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bessy_et_al:LIPIcs.IPEC.2019.2,
author = {Bessy, St\'{e}phane and Bougeret, Marin and Carneiro, Alan D. A. and Protti, F\'{a}bio and Souza, U\'{e}verton S.},
title = {{Width Parameterizations for Knot-Free Vertex Deletion on Digraphs}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {2:1--2:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.2},
URN = {urn:nbn:de:0030-drops-114631},
doi = {10.4230/LIPIcs.IPEC.2019.2},
annote = {Keywords: Knot, deadlock, width measure, FPT, W\lbrack1\rbrack-hard, directed feedback vertex set}
}
Document
Parameterized Valiant’s Classes
Abstract
We define a theory of parameterized algebraic complexity classes in analogy to parameterized Boolean counting classes. We define the classes VFPT and VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define appropriate reductions and completeness notions. Our main contribution is the VW[1]-completeness proof of the parameterized clique family. This proof is far more complicated than in the Boolean world. It requires some new concepts like composition theorems for bounded exponential sums and Boolean-arithmetic formulas. In addition, we also look at two polynomials linked to the permanent with vastly different parameterized complexity.
Cite as
Markus Bläser and Christian Engels. Parameterized Valiant’s Classes. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{blaser_et_al:LIPIcs.IPEC.2019.3,
author = {Bl\"{a}ser, Markus and Engels, Christian},
title = {{Parameterized Valiant’s Classes}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {3:1--3:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.3},
URN = {urn:nbn:de:0030-drops-114648},
doi = {10.4230/LIPIcs.IPEC.2019.3},
annote = {Keywords: Algebraic complexity theory, parameterized complexity theory, Valiant’s classes}
}
Document
Hierarchy of Transportation Network Parameters and Hardness Results
Abstract
The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or k-Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension.
We investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parameters distance to linear forest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentioned problems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeleton dimension is upper bounded by the max leaf number and that for any graph on at least three vertices there are edge weights such that both parameters are equal.
Then we show that computing the highway dimension according to most recent definition is NP-hard, which answers an open question stated by Feldmann et al. [Andreas Emil Feldmann et al., 2015]. Finally we prove that on graphs G=(V,E) of skeleton dimension O(log^2 |V|) it is NP-hard to approximate the k-Center problem within a factor less than 2.
Cite as
Johannes Blum. Hierarchy of Transportation Network Parameters and Hardness Results. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{blum:LIPIcs.IPEC.2019.4,
author = {Blum, Johannes},
title = {{Hierarchy of Transportation Network Parameters and Hardness Results}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {4:1--4:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.4},
URN = {urn:nbn:de:0030-drops-114656},
doi = {10.4230/LIPIcs.IPEC.2019.4},
annote = {Keywords: Graph Parameters, Skeleton Dimension, Highway Dimension, k-Center}
}
Document
Metric Dimension Parameterized by Treewidth
Abstract
A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polytime algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth.
We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time f(pw)n^{o(pw)} on n-vertex graphs of constant degree, with pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. [SIAM J. Discrete Math. '17] with respect to the combined parameter tl+Delta, where tl is the tree-length and Delta the maximum-degree of the input graph.
Cite as
Édouard Bonnet and Nidhi Purohit. Metric Dimension Parameterized by Treewidth. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2019.5,
author = {Bonnet, \'{E}douard and Purohit, Nidhi},
title = {{Metric Dimension Parameterized by Treewidth}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {5:1--5:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.5},
URN = {urn:nbn:de:0030-drops-114668},
doi = {10.4230/LIPIcs.IPEC.2019.5},
annote = {Keywords: Metric Dimension, Treewidth, Parameterized Hardness}
}
Document
Faster Subgraph Counting in Sparse Graphs
Abstract
A fundamental graph problem asks to compute the number of induced copies of a k-node pattern graph H in an n-node graph G. The fastest algorithm to date is still the 35-years-old algorithm by Nešetřil and Poljak [Nešetřil and Poljak, 1985], with running time f(k) * O(n^{omega floor[k/3] + 2}) where omega <=2.373 is the matrix multiplication exponent. In this work we show that, if one takes into account the degeneracy d of G, then the picture becomes substantially richer and leads to faster algorithms when G is sufficiently sparse. More precisely, after introducing a novel notion of graph width, the DAG-treewidth, we prove what follows. If H has DAG-treewidth tau(H) and G has degeneracy d, then the induced copies of H in G can be counted in time f(d,k) * O~(n^{tau(H)}); and, under the Exponential Time Hypothesis, no algorithm can solve the problem in time f(d,k) * n^{o(tau(H)/ln tau(H))} for all H. This result characterises the complexity of counting subgraphs in a d-degenerate graph. Developing bounds on tau(H), then, we obtain natural generalisations of classic results and faster algorithms for sparse graphs. For example, when d=O(poly log(n)) we can count the induced copies of any H in time f(k) * O~(n^{floor[k/4] + 2}), beating the Nešetřil-Poljak algorithm by essentially a cubic factor in n.
Cite as
Marco Bressan. Faster Subgraph Counting in Sparse Graphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bressan:LIPIcs.IPEC.2019.6,
author = {Bressan, Marco},
title = {{Faster Subgraph Counting in Sparse Graphs}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {6:1--6:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.6},
URN = {urn:nbn:de:0030-drops-114673},
doi = {10.4230/LIPIcs.IPEC.2019.6},
annote = {Keywords: subgraph counting, tree decomposition, degeneracy}
}
Document
Towards a Theory of Parameterized Streaming Algorithms
Abstract
Parameterized complexity attempts to give a more fine-grained analysis of the complexity of problems: instead of measuring the running time as a function of only the input size, we analyze the running time with respect to additional parameters. This approach has proven to be highly successful in delineating our understanding of NP-hard problems. Given this success with the TIME resource, it seems but natural to use this approach for dealing with the SPACE resource. First attempts in this direction have considered a few individual problems, with some success: Fafianie and Kratsch [MFCS'14] and Chitnis et al. [SODA'15] introduced the notions of streaming kernels and parameterized streaming algorithms respectively. For example, the latter shows how to refine the Omega(n^2) bit lower bound for finding a minimum Vertex Cover (VC) in the streaming setting by designing an algorithm for the parameterized k-VC problem which uses O(k^{2}log n) bits.
In this paper, we initiate a systematic study of graph problems from the paradigm of parameterized streaming algorithms. We first define a natural hierarchy of space complexity classes of FPS, SubPS, SemiPS, SupPS and BrutePS, and then obtain tight classifications for several well-studied graph problems such as Longest Path, Feedback Vertex Set, Dominating Set, Girth, Treewidth, etc. into this hierarchy (see Figure 1 and Table 1). On the algorithmic side, our parameterized streaming algorithms use techniques from the FPT world such as bidimensionality, iterative compression and bounded-depth search trees. On the hardness side, we obtain lower bounds for the parameterized streaming complexity of various problems via novel reductions from problems in communication complexity. We also show a general (unconditional) lower bound for space complexity of parameterized streaming algorithms for a large class of problems inspired by the recently developed frameworks for showing (conditional) kernelization lower bounds.
Parameterized algorithms and streaming algorithms are approaches to cope with TIME and SPACE intractability respectively. It is our hope that this work on parameterized streaming algorithms leads to two-way flow of ideas between these two previously separated areas of theoretical computer science.
Cite as
Rajesh Chitnis and Graham Cormode. Towards a Theory of Parameterized Streaming Algorithms. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{chitnis_et_al:LIPIcs.IPEC.2019.7,
author = {Chitnis, Rajesh and Cormode, Graham},
title = {{Towards a Theory of Parameterized Streaming Algorithms}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {7:1--7:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.7},
URN = {urn:nbn:de:0030-drops-114682},
doi = {10.4230/LIPIcs.IPEC.2019.7},
annote = {Keywords: Parameterized Algorithms, Streaming Algorithms, Kernels}
}
Document
FPT Inapproximability of Directed Cut and Connectivity Problems
Abstract
Cut problems and connectivity problems on digraphs are two well-studied classes of problems from the viewpoint of parameterized complexity. After a series of papers over the last decade, we now have (almost) tight bounds for the running time of several standard variants of these problems parameterized by two parameters: the number k of terminals and the size p of the solution. When there is evidence of FPT intractability, then the next natural alternative is to consider FPT approximations. In this paper, we show two types of results for directed cut and connectivity problems, building on existing results from the literature: first is to circumvent the hardness results for these problems by designing FPT approximation algorithms, or alternatively strengthen the existing hardness results by creating "gap-instances" under stronger hypotheses such as the (Gap-)Exponential Time Hypothesis (ETH). Formally, we show the following results:
Cutting paths between a set of terminal pairs, i.e., Directed Multicut: Pilipczuk and Wahlstrom [TOCT '18] showed that Directed Multicut is W[1]-hard when parameterized by p if k=4. We complement this by showing the following two results:
- Directed Multicut has a k/2-approximation in 2^{O(p^2)}* n^{O(1)} time (i.e., a 2-approximation if k=4),
- Under Gap-ETH, Directed Multicut does not admit an (59/58-epsilon)-approximation in f(p)* n^{O(1)} time, for any computable function f, even if k=4.
Connecting a set of terminal pairs, i.e., Directed Steiner Network (DSN): The DSN problem on general graphs is known to be W[1]-hard parameterized by p+k due to Guo et al. [SIDMA '11]. Dinur and Manurangsi [ITCS '18] further showed that there is no FPT k^{1/4-o(1)}-approximation algorithm parameterized by k, under Gap-ETH. Chitnis et al. [SODA '14] considered the restriction to special graph classes, but unfortunately this does not lead to FPT algorithms either: DSN on planar graphs is W[1]-hard parameterized by k. In this paper we consider the DSN_Planar problem which is an intermediate version: the graph is general, but we want to find a solution whose cost is at most that of an optimal planar solution (if one exists). We show the following lower bounds for DSN_Planar:
- DSN_Planar has no (2-epsilon)-approximation in FPT time parameterized by k, under Gap-ETH. This answers in the negative a question of Chitnis et al. [ESA '18].
- DSN_Planar is W[1]-hard parameterized by k+p. Moreover, under ETH, there is no (1+epsilon)-approximation for DSN_Planar in f(k,p,epsilon)* n^{o(k+sqrt{p+1/epsilon})} time for any computable function f.
Pairwise connecting a set of terminals, i.e., Strongly Connected Steiner Subgraph (SCSS): Guo et al. [SIDMA '11] showed that SCSS is W[1]-hard parameterized by p+k, while Chitnis et al. [SODA '14] showed that SCSS remains W[1]-hard parameterized by p, even if the input graph is planar. In this paper we consider the SCSS_Planar problem which is an intermediate version: the graph is general, but we want to find a solution whose cost is at most that of an optimal planar solution (if one exists). We show the following lower bounds for SCSS_Planar:
- SCSS_Planar is W[1]-hard parameterized by k+p. Moreover, under ETH, there is no (1+epsilon)-approximation for SCSS_Planar in f(k,p,epsilon)* n^{o(sqrt{k+p+1/epsilon})} time for any computable function f.
Previously, the only known FPT approximation results for SCSS applied to general graphs parameterized by k: a 2-approximation by Chitnis et al. [IPEC '13], and a matching (2-epsilon)-hardness under Gap-ETH by Chitnis et al. [ESA '18].
Cite as
Rajesh Chitnis and Andreas Emil Feldmann. FPT Inapproximability of Directed Cut and Connectivity Problems. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{chitnis_et_al:LIPIcs.IPEC.2019.8,
author = {Chitnis, Rajesh and Feldmann, Andreas Emil},
title = {{FPT Inapproximability of Directed Cut and Connectivity Problems}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {8:1--8:20},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.8},
URN = {urn:nbn:de:0030-drops-114692},
doi = {10.4230/LIPIcs.IPEC.2019.8},
annote = {Keywords: Directed graphs, cuts, connectivity, Steiner problems, FPT inapproximability}
}
Document
C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width
Abstract
For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA'95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA'20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs.
We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.
Cite as
Giordano Da Lozzo, David Eppstein, Michael T. Goodrich, and Siddharth Gupta. C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{dalozzo_et_al:LIPIcs.IPEC.2019.9,
author = {Da Lozzo, Giordano and Eppstein, David and Goodrich, Michael T. and Gupta, Siddharth},
title = {{C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {9:1--9:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.9},
URN = {urn:nbn:de:0030-drops-114705},
doi = {10.4230/LIPIcs.IPEC.2019.9},
annote = {Keywords: Clustered planarity, carving-width, non-crossing partitions, FPT}
}
Document
The Complexity of Packing Edge-Disjoint Paths
Abstract
We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem.
Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard.
Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs.
Cite as
Jan Dreier, Janosch Fuchs, Tim A. Hartmann, Philipp Kuinke, Peter Rossmanith, Bjoern Tauer, and Hung-Lung Wang. The Complexity of Packing Edge-Disjoint Paths. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{dreier_et_al:LIPIcs.IPEC.2019.10,
author = {Dreier, Jan and Fuchs, Janosch and Hartmann, Tim A. and Kuinke, Philipp and Rossmanith, Peter and Tauer, Bjoern and Wang, Hung-Lung},
title = {{The Complexity of Packing Edge-Disjoint Paths}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {10:1--10:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.10},
URN = {urn:nbn:de:0030-drops-114710},
doi = {10.4230/LIPIcs.IPEC.2019.10},
annote = {Keywords: parameterized complexity, embedding, packing, covering, Hamiltonian path, unary binpacking, path-perfect graphs}
}
Document
Hardness of FO Model-Checking on Random Graphs
Abstract
It is known that FO model-checking is fixed-parameter tractable on Erdős - Rényi graphs G(n,p(n)) if the edge-probability p(n) is sufficiently small [Grohe, 2001] (p(n)=O(n^epsilon/n) for every epsilon>0). A natural question to ask is whether this result can be extended to bigger probabilities. We show that for Erdős - Rényi graphs with vertex colors the above stated upper bound by Grohe is the best possible.
More specifically, we show that there is no FO model-checking algorithm with average FPT run time on vertex-colored Erdős - Rényi graphs G(n,n^delta/n) (0 < delta < 1) unless AW[*]subseteq FPT/poly. This might be the first result where parameterized average-case intractability of a natural problem with a natural probability distribution is linked to worst-case complexity assumptions.
We further provide hardness results for FO model-checking on other random graph models, including G(n,1/2) and Chung-Lu graphs, where our intractability results tightly match known tractability results [E. D. Demaine et al., 2014]. We also provide lower bounds on the size of shallow clique minors in certain Erdős - Rényi and Chung - Lu graphs.
Cite as
Jan Dreier and Peter Rossmanith. Hardness of FO Model-Checking on Random Graphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{dreier_et_al:LIPIcs.IPEC.2019.11,
author = {Dreier, Jan and Rossmanith, Peter},
title = {{Hardness of FO Model-Checking on Random Graphs}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {11:1--11:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.11},
URN = {urn:nbn:de:0030-drops-114721},
doi = {10.4230/LIPIcs.IPEC.2019.11},
annote = {Keywords: random graphs, FO model-checking}
}
Document
Computing the Largest Bond of a Graph
Abstract
A bond of a graph G is an inclusion-wise minimal disconnecting set of G, i.e., bonds are cut-sets that determine cuts [S,V\S] of G such that G[S] and G[V\S] are both connected. Given s,t in V(G), an st-bond of G is a bond whose removal disconnects s and t. Contrasting with the large number of studies related to maximum cuts, there are very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond and the largest st-bond of a graph. Although cuts and bonds are similar, we remark that computing the largest bond of a graph tends to be harder than computing its maximum cut. We show that Largest Bond remains NP-hard even for planar bipartite graphs, and it does not admit a constant-factor approximation algorithm, unless P = NP. We also show that Largest Bond and Largest st-Bond on graphs of clique-width w cannot be solved in time f(w) x n^{o(w)} unless the Exponential Time Hypothesis fails, but they can be solved in time f(w) x n^{O(w)}. In addition, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, but they do not admit polynomial kernels unless NP subseteq coNP/poly.
Cite as
Gabriel L. Duarte, Daniel Lokshtanov, Lehilton L. C. Pedrosa, Rafael C. S. Schouery, and Uéverton S. Souza. Computing the Largest Bond of a Graph. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{duarte_et_al:LIPIcs.IPEC.2019.12,
author = {Duarte, Gabriel L. and Lokshtanov, Daniel and Pedrosa, Lehilton L. C. and Schouery, Rafael C. S. and Souza, U\'{e}verton S.},
title = {{Computing the Largest Bond of a Graph}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {12:1--12:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.12},
URN = {urn:nbn:de:0030-drops-114732},
doi = {10.4230/LIPIcs.IPEC.2019.12},
annote = {Keywords: bond, cut, maximum cut, connected cut, FPT, treewidth, clique-width}
}
Document
Parameterized Algorithms for Maximum Cut with Connectivity Constraints
Abstract
We study two variants of Maximum Cut, which we call Connected Maximum Cut and Maximum Minimal Cut, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas Maximum Cut on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size.
Cite as
Hiroshi Eto, Tesshu Hanaka, Yasuaki Kobayashi, and Yusuke Kobayashi. Parameterized Algorithms for Maximum Cut with Connectivity Constraints. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{eto_et_al:LIPIcs.IPEC.2019.13,
author = {Eto, Hiroshi and Hanaka, Tesshu and Kobayashi, Yasuaki and Kobayashi, Yusuke},
title = {{Parameterized Algorithms for Maximum Cut with Connectivity Constraints}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {13:1--13:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.13},
URN = {urn:nbn:de:0030-drops-114747},
doi = {10.4230/LIPIcs.IPEC.2019.13},
annote = {Keywords: Maximum cut, Parameterized algorithm, NP-hardness, Graph parameter}
}
Document
Multistage Vertex Cover
Abstract
Covering all edges of a graph by a small number of vertices, this is the NP-hard Vertex Cover problem, is among the most fundamental algorithmic tasks. Following a recent trend in studying dynamic and temporal graphs, we initiate the study of Multistage Vertex Cover. Herein, having a series of graphs with same vertex set but over time changing edge sets (known as temporal graph consisting of time layers), the goal is to find for each layer of the temporal graph a small vertex cover and to guarantee that the two vertex cover sets between two subsequent layers differ not too much (specified by a given parameter). We show that, different from classic Vertex Cover and some other dynamic or temporal variants of it, Multistage Vertex Cover is computationally hard even in fairly restricted settings. On the positive side, however, we also spot several fixed-parameter tractability results based on some of the most natural parameterizations.
Cite as
Till Fluschnik, Rolf Niedermeier, Valentin Rohm, and Philipp Zschoche. Multistage Vertex Cover. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{fluschnik_et_al:LIPIcs.IPEC.2019.14,
author = {Fluschnik, Till and Niedermeier, Rolf and Rohm, Valentin and Zschoche, Philipp},
title = {{Multistage Vertex Cover}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {14:1--14:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.14},
URN = {urn:nbn:de:0030-drops-114753},
doi = {10.4230/LIPIcs.IPEC.2019.14},
annote = {Keywords: NP-hardness, dynamic graph problems, temporal graphs, time-evolving networks, W\lbrack1\rbrack-hardness, fixed-parameter tractability, kernelization}
}
Document
Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems
Abstract
We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from an edge-coloured graph G to an edge-coloured graph H is a vertex-mapping from G to H that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph H, which generalises the classic vertex-colourability property. The question we are interested in is the following: given an edge-coloured graph G, can we perform k graph operations so that the resulting graph admits a homomorphism to H? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs + and -. We denote the corresponding problems (parameterized by k) by Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring. These problems generalise the extensively studied H-Colouring problem (where one has to decide if an input graph admits a homomorphism to a fixed target H). For 2-edge-coloured H, it is known that H-Colouring already captures the complexity of all fixed-target Constraint Satisfaction Problems.
Our main focus is on the case where H is an edge-coloured graph of order at most 2, a case that is already interesting since it includes standard problems such as Vertex Cover, Odd Cycle Transversal and Edge Bipartization. For such a graph H, we give a PTime/NP-complete complexity dichotomy for all three Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring problems. Then, we address their parameterized complexity. We show that all Vertex Deletion-H-Colouring and Edge Deletion-H-Colouring problems for such H are FPT. This is in contrast with the fact that already for some H of order 3, unless PTime = NP, none of the three considered problems is in XP, since 3-Colouring is NP-complete. We show that the situation is different for Switching-H-Colouring: there are three 2-edge-coloured graphs H of order 2 for which Switching-H-Colouring is W[1]-hard, and assuming the ETH, admits no algorithm in time f(k)n^{o(k)} for inputs of size n and for any computable function f. For the other cases, Switching-H-Colouring is FPT.
Cite as
Florent Foucaud, Hervé Hocquard, Dimitri Lajou, Valia Mitsou, and Théo Pierron. Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{foucaud_et_al:LIPIcs.IPEC.2019.15,
author = {Foucaud, Florent and Hocquard, Herv\'{e} and Lajou, Dimitri and Mitsou, Valia and Pierron, Th\'{e}o},
title = {{Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {15:1--15:16},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.15},
URN = {urn:nbn:de:0030-drops-114765},
doi = {10.4230/LIPIcs.IPEC.2019.15},
annote = {Keywords: Graph homomorphism, Graph modification, Edge-coloured graph, Signed graph}
}
Document
On the Fine-Grained Complexity of Least Weight Subsequence in Multitrees and Bounded Treewidth DAGs
Abstract
This paper introduces a new technique that generalizes previously known fine-grained reductions from linear structures to graphs. Least Weight Subsequence (LWS) [Hirschberg and Larmore, 1987] is a class of highly sequential optimization problems with form F(j) = min_{i < j} [F(i) + c_{i,j}] . They can be solved in quadratic time using dynamic programming, but it is not known whether these problems can be solved faster than n^{2-o(1)} time. Surprisingly, each such problem is subquadratic time reducible to a highly parallel, non-dynamic programming problem [Marvin Künnemann et al., 2017]. In other words, if a "static" problem is faster than quadratic time, so is an LWS problem. For many instances of LWS, the sequential versions are equivalent to their static versions by subquadratic time reductions. The previous result applies to LWS on linear structures, and this paper extends this result to LWS on paths in sparse graphs, the Least Weight Subpath (LWSP) problems. When the graph is a multitree (i.e. a DAG where any pair of vertices can have at most one path) or when the graph is a DAG whose underlying undirected graph has constant treewidth, we show that LWSP on this graph is still subquadratically reducible to their corresponding static problems. For many instances, the graph versions are still equivalent to their static versions.
Moreover, this paper shows that if we can decide a property of form Exists x Exists y P(x,y) in subquadratic time, where P is a quickly checkable property on a pair of elements, then on these classes of graphs, we can also in subquadratic time decide whether there exists a pair x,y in the transitive closure of the graph that also satisfy P(x,y).
Cite as
Jiawei Gao. On the Fine-Grained Complexity of Least Weight Subsequence in Multitrees and Bounded Treewidth DAGs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{gao:LIPIcs.IPEC.2019.16,
author = {Gao, Jiawei},
title = {{On the Fine-Grained Complexity of Least Weight Subsequence in Multitrees and Bounded Treewidth DAGs}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {16:1--16:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.16},
URN = {urn:nbn:de:0030-drops-114778},
doi = {10.4230/LIPIcs.IPEC.2019.16},
annote = {Keywords: fine-grained complexity, dynamic programming, graph reachability}
}
Document
Resolving Infeasibility of Linear Systems: A Parameterized Approach
Abstract
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).
Cite as
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.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}
}
Document
Clustering to Given Connectivities
Abstract
We define a general variant of the graph clustering problem where the criterion of density for the clusters is (high) connectivity. In Clustering to Given Connectivities, we are given an n-vertex graph G, an integer k, and a sequence Lambda=<lambda_{1},...,lambda_{t}> of positive integers and we ask whether it is possible to remove at most k edges from G such that the resulting connected components are exactly t and their corresponding edge connectivities are lower-bounded by the numbers in Lambda. We prove that this problem, parameterized by k, is fixed parameter tractable, i.e., can be solved by an f(k)* n^{O(1)}-step algorithm, for some function f that depends only on the parameter k. Our algorithm uses the recursive understanding technique that is especially adapted so to deal with the fact that we do not impose any restriction to the connectivity demands in Lambda.
Cite as
Petr A. Golovach and Dimitrios M. Thilikos. Clustering to Given Connectivities. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{golovach_et_al:LIPIcs.IPEC.2019.18,
author = {Golovach, Petr A. and Thilikos, Dimitrios M.},
title = {{Clustering to Given Connectivities}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {18:1--18:17},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.18},
URN = {urn:nbn:de:0030-drops-114796},
doi = {10.4230/LIPIcs.IPEC.2019.18},
annote = {Keywords: graph clustering, edge connectivity, parameterized complexity}
}
Document
Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization
Abstract
A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [IPEC 2018], we introduce a natural generalization of this problem, which we call d-Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d-Cut on (2d+2)-regular graphs and a polynomial algorithm for graphs of maximum degree at most d+2. The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [IPEC 2018], is a polynomial kernel for d-Cut for every positive integer d, parameterized by the distance to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time O^*(2^n).
Cite as
Guilherme C. M. Gomes and Ignasi Sau. Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{c.m.gomes_et_al:LIPIcs.IPEC.2019.19,
author = {C. M. Gomes, Guilherme and Sau, Ignasi},
title = {{Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {19:1--19:15},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.19},
URN = {urn:nbn:de:0030-drops-114809},
doi = {10.4230/LIPIcs.IPEC.2019.19},
annote = {Keywords: matching cut, bounded degree cut, parameterized complexity, FPT algorithm, polynomial kernel, distance to cluster}
}
Document
Finding Linear Arrangements of Hypergraphs with Bounded Cutwidth in Linear Time
Abstract
Cutwidth is a fundamental graph layout parameter. It generalises to hypergraphs in a natural way and has been studied in a wide range of contexts. For graphs it is known that for a fixed constant k there is a linear time algorithm that for any given G, decides whether G has cutwidth at most k and, in the case of a positive answer, outputs a corresponding linear arrangement. We show that such an algorithm also exists for hypergraphs.
Cite as
Thekla Hamm. Finding Linear Arrangements of Hypergraphs with Bounded Cutwidth in Linear Time. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{hamm:LIPIcs.IPEC.2019.20,
author = {Hamm, Thekla},
title = {{Finding Linear Arrangements of Hypergraphs with Bounded Cutwidth in Linear Time}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {20:1--20:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.20},
URN = {urn:nbn:de:0030-drops-114812},
doi = {10.4230/LIPIcs.IPEC.2019.20},
annote = {Keywords: Fixed parameter linear, Path decomposition, Hypergraph}
}
Document
The Independent Set Problem Is FPT for Even-Hole-Free Graphs
Abstract
The class of even-hole-free graphs is very similar to the class of perfect graphs, and was indeed a cornerstone in the tools leading to the proof of the Strong Perfect Graph Theorem. However, the complexity of computing a maximum independent set (MIS) is a long-standing open question in even-hole-free graphs. From the hardness point of view, MIS is W[1]-hard in the class of graphs without induced 4-cycle (when parameterized by the solution size). Halfway of these, we show in this paper that MIS is FPT when parameterized by the solution size in the class of even-hole-free graphs. The main idea is to apply twice the well-known technique of augmenting graphs to extend some initial independent set.
Cite as
Edin Husić, Stéphan Thomassé, and Nicolas Trotignon. The Independent Set Problem Is FPT for Even-Hole-Free Graphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 21:1-21:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{husic_et_al:LIPIcs.IPEC.2019.21,
author = {Husi\'{c}, Edin and Thomass\'{e}, St\'{e}phan and Trotignon, Nicolas},
title = {{The Independent Set Problem Is FPT for Even-Hole-Free Graphs}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {21:1--21:12},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.21},
URN = {urn:nbn:de:0030-drops-114826},
doi = {10.4230/LIPIcs.IPEC.2019.21},
annote = {Keywords: independent set, FPT algorithm, even-hole-free graph, augmenting graph}
}
Document
Improved Analysis of Highest-Degree Branching for Feedback Vertex Set
Abstract
Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning heuristic. In this paper, we prove that this empirically fast algorithm runs in O(3.460^k n) time, where k is the solution size. This improves the previous best O(3.619^k n)-time deterministic algorithm obtained by Kociumaka and Pilipczuk (Inf. Process. Lett., 2014).
Cite as
Yoichi Iwata and Yusuke Kobayashi. Improved Analysis of Highest-Degree Branching for Feedback Vertex Set. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 22:1-22:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{iwata_et_al:LIPIcs.IPEC.2019.22,
author = {Iwata, Yoichi and Kobayashi, Yusuke},
title = {{Improved Analysis of Highest-Degree Branching for Feedback Vertex Set}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {22:1--22:11},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.22},
URN = {urn:nbn:de:0030-drops-114833},
doi = {10.4230/LIPIcs.IPEC.2019.22},
annote = {Keywords: Feedback Vertex Set, Branch and bound, Measure and conquer}
}
Document
Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs
Abstract
Let C and D be hereditary graph classes. Consider the following problem: given a graph G in D, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to C. We prove that it can be solved in 2^{o(n)} time, where n is the number of vertices of G, if the following conditions are satisfied:
- the graphs in C are sparse, i.e., they have linearly many edges in terms of the number of vertices;
- the graphs in D admit balanced separators of size governed by their density, e.g., O(Delta) or O(sqrt{m}), where Delta and m denote the maximum degree and the number of edges, respectively; and
- the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph.
This leads, for example, to the following corollaries for specific classes C and D:
- a largest induced forest in a P_t-free graph can be found in 2^{O~(n^{2/3})} time, for every fixed t; and
- a largest induced planar graph in a string graph can be found in 2^{O~(n^{3/4})} time.
Cite as
Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski, Erik Jan van Leeuwen, and Bartosz Walczak. Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 23:1-23:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{novotna_et_al:LIPIcs.IPEC.2019.23,
author = {Novotn\'{a}, Jana and Okrasa, Karolina and Pilipczuk, Micha{\l} and Rz\k{a}\.{z}ewski, Pawe{\l} and van Leeuwen, Erik Jan and Walczak, Bartosz},
title = {{Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {23:1--23:11},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.23},
URN = {urn:nbn:de:0030-drops-114845},
doi = {10.4230/LIPIcs.IPEC.2019.23},
annote = {Keywords: subexponential algorithm, feedback vertex set, P\underlinet-free graphs, string graphs}
}
Document
Beating Treewidth for Average-Case Subgraph Isomorphism
Abstract
For any fixed graph G, the subgraph isomorphism problem asks whether an n-vertex input graph has a subgraph isomorphic to G. A well-known algorithm of Alon, Yuster and Zwick (1995) efficiently reduces this to the "colored" version of the problem, denoted G-SUB, and then solves G-SUB in time O(n^{tw(G)+1}) where tw(G) is the treewidth of G. Marx (2010) conjectured that G-SUB requires time Omega(n^{const * tw(G)}) and, assuming the Exponential Time Hypothesis, proved a lower bound of Omega(n^{const * emb(G)}) for a certain graph parameter emb(G) = Omega(tw(G)/log tw(G)). With respect to the size of AC^0 circuits solving G-SUB, Li, Razborov and Rossman (2017) proved an unconditional average-case lower bound of Omega(n^{kappa(G)}) for a different graph parameter kappa(G) = Omega(tw(G)/log tw(G)).
Our contributions are as follows. First, we show that emb(G) is at most O(kappa(G)) for all graphs G. Next, we show that kappa(G) can be asymptotically less than tw(G); for example, if G is a hypercube then kappa(G) is Theta(tw(G)/sqrt{log tw(G)}). Finally, we construct AC^0 circuits of size O(n^{kappa(G)+const}) that solve G-SUB in the average case, on a variety of product distributions. This improves an O(n^{2 kappa(G)+const}) upper bound of Li et al., and shows that the average-case complexity of G-SUB is n^{o(tw(G))} for certain families of graphs G such as hypercubes.
Cite as
Gregory Rosenthal. Beating Treewidth for Average-Case Subgraph Isomorphism. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{rosenthal:LIPIcs.IPEC.2019.24,
author = {Rosenthal, Gregory},
title = {{Beating Treewidth for Average-Case Subgraph Isomorphism}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {24:1--24:14},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.24},
URN = {urn:nbn:de:0030-drops-114850},
doi = {10.4230/LIPIcs.IPEC.2019.24},
annote = {Keywords: subgraph isomorphism, average-case complexity, AC^0, circuit complexity}
}
Document
Invited Paper
The PACE 2019 Parameterized Algorithms and Computational Experiments Challenge: The Fourth Iteration (Invited Paper)
Abstract
The organizers of the 4th Parameterized Algorithms and Computational Experiments challenge (PACE 2019) report on the 4th iteration of the PACE challenge. This year, the first track featured the MinVertexCover problem, which asks given an undirected graph G=(V,E) to output a set S subseteq V of vertices such that for every edge vw in E at least one endpoint belongs to S. The exact decision version of this problem is one of the most discussed problem if not even the prototypical problem in parameterized complexity theory. Another two tracks were dedicated to computing the hypertree width of a given hypergraph, which is a certain generalization of tree decompositions to hypergraphs that has widely been applied to problems in databases, constraint programming, and artificial intelligence. On one track we asked for submissions that compute hypertree decompositions of minimum width (MinHypertreeWidth) and on the other track we asked to heuristically compute hypertree decompositions of small width quickly (HeurHypertreeWidth). We received 28 implementations from 26 teams. This year we asked participants to submit solver descriptions in order to count as a submission for the challenge. We received those from 16 teams with overall 33 participants from 10 countries. One team submitted successful solutions to all three tracks.
Cite as
M. Ayaz Dzulfikar, Johannes K. Fichte, and Markus Hecher. The PACE 2019 Parameterized Algorithms and Computational Experiments Challenge: The Fourth Iteration (Invited Paper). In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 25:1-25:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{dzulfikar_et_al:LIPIcs.IPEC.2019.25,
author = {Dzulfikar, M. Ayaz and Fichte, Johannes K. and Hecher, Markus},
title = {{The PACE 2019 Parameterized Algorithms and Computational Experiments Challenge: The Fourth Iteration}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {25:1--25:23},
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.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.25},
URN = {urn:nbn:de:0030-drops-114861},
doi = {10.4230/LIPIcs.IPEC.2019.25},
annote = {Keywords: Parameterized Algorithms, Vertex Cover Problem, Hypertree Decompositions, Implementation Challenge, FPT}
}