eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
0
0
10.4230/LIPIcs.IPEC.2018
article
LIPIcs, Volume 115, IPEC'18, Complete Volume
Paul, Christophe
Pilipczuk, Michał
LIPIcs, Volume 115, IPEC'18, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018/LIPIcs.IPEC.2018.pdf
Theory of computation, Complexity classes, Theory of computation, Design and analysis of algorithms, Mathematics of computing, Discrete mathematics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
0:i
0:xiv
10.4230/LIPIcs.IPEC.2018.0
article
Front Matter, Table of Contents, Preface, Conference Organization
Paul, Christophe
Pilipczuk, Michal
Front Matter, Table of Contents, Preface, Conference Organization
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.0/LIPIcs.IPEC.2018.0.pdf
Front Matter
Table of Contents
Preface
Conference Organization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
1:1
1:18
10.4230/LIPIcs.IPEC.2018.1
article
Counting Problems in Parameterized Complexity
Curticapean, Radu
1
https://orcid.org/0000-0001-7201-9905
Basic Algorithms Research Copenhagen (BARC) and IT University of Copenhagen, Copenhagen, Denmark
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs.
While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.1/LIPIcs.IPEC.2018.1.pdf
counting complexity
parameterized complexity
graph motifs
perfect matchings
graph minor theory
Hamiltonian cycles
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
2:1
2:13
10.4230/LIPIcs.IPEC.2018.2
article
A Complexity Dichotomy for Hitting Small Planar Minors Parameterized by Treewidth
Baste, Julien
1
Sau, Ignasi
2
https://orcid.org/0000-0002-8981-9287
Thilikos, Dimitrios M.
3
https://orcid.org/0000-0003-0470-1800
Sorbonne Université, Laboratoire d'Informatique de Paris 6, LIP6, Paris, France
LIRMM, CNRS, Université de Montpellier, Montpellier, France
LIRMM, CNRS, Université de Montpellier, Montpellier, France, and, Department of Mathematics, National and Kapodistrian University of Athens, Greece
For a fixed graph H, we are interested in the parameterized complexity of the following problem, called {H}-M-Deletion, parameterized by the treewidth tw of the input graph: given an n-vertex graph G and an integer k, decide whether there exists S subseteq V(G) with |S| <= k such that G setminus S does not contain H as a minor. In previous work [IPEC, 2017] we proved that if H is planar and connected, then the problem cannot be solved in time 2^{o(tw)} * n^{O(1)} under the ETH, and can be solved in time 2^{O(tw * log tw)} * n^{O(1)}. In this article we manage to classify the optimal asymptotic complexity of {H}-M-Deletion when H is a connected planar graph on at most 5 vertices. Out of the 29 possibilities (discarding the trivial case H = K_1), we prove that 9 of them are solvable in time 2^{Theta (tw)} * n^{O(1)}, and that the other 20 ones are solvable in time 2^{Theta (tw * log tw)} * n^{O(1)}. Namely, we prove that K_4 and the diamond are the only graphs on at most 4 vertices for which the problem is solvable in time 2^{Theta (tw * log tw)} * n^{O(1)}, and that the chair and the banner are the only graphs on 5 vertices for which the problem is solvable in time 2^{Theta (tw)} * n^{O(1)}. For the version of the problem where H is forbidden as a topological minor, the case H = K_{1,4} can be solved in time 2^{Theta (tw)} * n^{O(1)}. This exhibits, to the best of our knowledge, the first difference between the computational complexity of both problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.2/LIPIcs.IPEC.2018.2.pdf
parameterized complexity
graph minors
treewidth
hitting minors
topological minors
dynamic programming
Exponential Time Hypothesis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
3:1
3:14
10.4230/LIPIcs.IPEC.2018.3
article
Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth
van Geffen, Bas A. M.
1
Jansen, Bart M. P.
2
https://orcid.org/0000-0001-8204-1268
de Kroon, Arnoud A. W. M.
1
Morel, Rolf
1
University of Oxford
Eindhoven University of Technology
Many combinatorial problems can be solved in time O^*(c^{tw}) on graphs of treewidth tw, for a problem-specific constant c. In several cases, matching upper and lower bounds on c are known based on the Strong Exponential Time Hypothesis (SETH). In this paper we investigate the complexity of solving problems on graphs of bounded cutwidth, a graph parameter that takes larger values than treewidth. We strengthen earlier treewidth-based lower bounds to show that, assuming SETH, Independent Set cannot be solved in O^*((2-epsilon)^{ctw}) time, and Dominating Set cannot be solved in O^*((3-epsilon)^{ctw}) time. By designing a new crossover gadget, we extend these lower bounds even to planar graphs of bounded cutwidth or treewidth. Hence planarity does not help when solving Independent Set or Dominating Set on graphs of bounded width. This sharply contrasts the fact that in many settings, planarity allows problems to be solved much more efficiently.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.3/LIPIcs.IPEC.2018.3.pdf
planarization
dominating set
cutwidth
lower bounds
strong exponential time hypothesis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
4:1
4:13
10.4230/LIPIcs.IPEC.2018.4
article
Multivariate Analysis of Orthogonal Range Searching and Graph Distances
Bringmann, Karl
1
Husfeldt, Thore
2
https://orcid.org/0000-0001-9078-4512
Magnusson, Måns
3
Max-Planck-Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
BARC, IT University of Copenhagen, Denmark, and Lund University, Sweden
Department of Computer Science, Lund University, Sweden
We show that the eccentricities, diameter, radius, and Wiener index of an undirected n-vertex graph with nonnegative edge lengths can be computed in time O(n * binom{k+ceil[log n]}{k} * 2^k k^2 log n), where k is the treewidth of the graph. For every epsilon>0, this bound is n^{1+epsilon}exp O(k), which matches a hardness result of Abboud, Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form log^d n to binom{d+ceil[log n]}{d}, as originally observed by Monier (J. Alg. 1980).
We also investigate the parameterization by vertex cover number.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.4/LIPIcs.IPEC.2018.4.pdf
Diameter
radius
Wiener index
orthogonal range searching
treewidth
vertex cover number
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
5:1
5:13
10.4230/LIPIcs.IPEC.2018.5
article
A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions
Kangas, Kustaa
1
Koivisto, Mikko
2
Salonen, Sami
2
Department of Computer Science, Aalto University, Espoo, Finland
Department of Computer Science, University of Helsinki, Helsinki, Finland
We consider the problem of counting the linear extensions of an n-element poset whose cover graph has treewidth at most t. We show that the problem can be solved in time O~(n^{t+3}), where O~ suppresses logarithmic factors. Our algorithm is based on fast multiplication of multivariate polynomials, and so differs radically from a previous O~(n^{t+4})-time inclusion - exclusion algorithm. We also investigate the algorithm from a practical point of view. We observe that the running time is not well characterized by the parameters n and t alone, fixing of which leaves large variance in running times due to uncontrolled features of the selected optimal-width tree decomposition. For selecting an efficient tree decomposition we adopt the method of empirical hardness models, and show that it typically enables picking a tree decomposition that is significantly more efficient than a random optimal-width tree decomposition.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.5/LIPIcs.IPEC.2018.5.pdf
Algorithm selection
empirical hardness
linear extension
multiplication of polynomials
tree decomposition
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
6:1
6:14
10.4230/LIPIcs.IPEC.2018.6
article
Generalized Distance Domination Problems and Their Complexity on Graphs of Bounded mim-width
Jaffke, Lars
1
https://orcid.org/0000-0003-4856-5863
Kwon, O-joung
2
https://orcid.org/0000-0003-1820-1962
Strømme, Torstein J. F.
1
https://orcid.org/0000-0002-3896-3166
Telle, Jan Arne
1
Department of Informatics, University of Bergen, Norway
Department of Mathematics, Incheon National University, Incheon, South Korea
We generalize the family of (sigma, rho)-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-r dominating set and distance-r independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as k-trapezoid graphs, Dilworth k-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, k-polygon graphs, circular arc graphs, complements of d-degenerate graphs, and H-graphs if given an H-representation. To supplement these findings, we show that many classes of (distance) (sigma, rho)-problems are W[1]-hard parameterized by mim-width + solution size.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.6/LIPIcs.IPEC.2018.6.pdf
Graph Width Parameters
Graph Classes
Distance Domination Problems
Parameterized Complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
7:1
7:14
10.4230/LIPIcs.IPEC.2018.7
article
A Parameterized Complexity View on Collapsing k-Cores
Luo, Junjie
1
Molter, Hendrik
2
Suchý, Ondrej
3
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China , School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany
Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
We study the NP-hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. [AAAI 2017] and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be NP-hard for all r >=0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r.
We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k <=2 and k >= 3. For the latter case it is known that for all x >= 0 Collapsed k-Core is W[P]-hard when parameterized by b. We show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by (b+x) if k <=2. Furthermore, we show that Collapsed k-Core is in FPT when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.7/LIPIcs.IPEC.2018.7.pdf
r-Degenerate Vertex Deletion
Feedback Vertex Set
Fixed-Parameter Tractability
Kernelization Lower Bounds
Graph Algorithms
Social Network Analysis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
8:1
8:14
10.4230/LIPIcs.IPEC.2018.8
article
Parameterized Complexity of Multi-Node Hubs
Saurabh, Saket
1
Zehavi, Meirav
2
University of Bergen, Norway, and Institute of Mathematical Science, HBNI, India
Ben-Gurion University, Israel
Hubs are high-degree nodes within a network. The examination of the emergence and centrality of hubs lies at the heart of many studies of complex networks such as telecommunication networks, biological networks, social networks and semantic networks. Furthermore, identifying and allocating hubs are routine tasks in applications. In this paper, we do not seek a hub that is a single node, but a hub that consists of k nodes. Formally, given a graph G=(V,E), we a seek a set A subseteq V of size k that induces a connected subgraph from which at least p edges emanate. Thus, we identify k nodes which can act as a unit (due to the connectivity constraint) that is a hub (due to the cut constraint). This problem, which we call Multi-Node Hub (MNH), can also be viewed as a variant of the classic Max Cut problem. While it is easy to see that MNH is W[1]-hard with respect to the parameter k, our main contribution is the first parameterized algorithm that shows that MNH is FPT with respect to the parameter p.
Despite recent breakthrough advances for cut-problems like Multicut and Minimum Bisection, MNH is still very challenging. Not only does a connectivity constraint has to be handled on top of the involved machinery developed for these problems, but also the fact that MNH is a maximization problem seems to prevent the applicability of this machinery in the first place. To deal with the latter issue, we give non-trivial reduction rules that show how MNH can be preprocessed into a problem where it is necessary to delete a bounded-in-parameter number of vertices. Then, to handle the connectivity constraint, we use a novel application of the form of tree decomposition introduced by Cygan et al. [STOC 2014] to solve Minimum Bisection, where we demonstrate how connectivity constraints can be replaced by simpler size constraints. Our approach may be relevant to the design of algorithms for other cut-problems of this nature.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.8/LIPIcs.IPEC.2018.8.pdf
hub
bisection
tree decomposition
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
9:1
9:14
10.4230/LIPIcs.IPEC.2018.9
article
The Parameterised Complexity of Computing the Maximum Modularity of a Graph
Meeks, Kitty
1
Skerman, Fiona
2
School of Computing Science, University of Glasgow, Glasgow, UK
Department of Mathematics, Uppsala University, Uppsala, Sweden
The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be NP-complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.9/LIPIcs.IPEC.2018.9.pdf
modularity
community detection
integer quadratic programming
vertex cover
pathwidth
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
10:1
10:14
10.4230/LIPIcs.IPEC.2018.10
article
On the Distance Identifying Set Meta-Problem and Applications to the Complexity of Identifying Problems on Graphs
Barbero, Florian
1
Isenmann, Lucas
1
Thiebaut, Jocelyn
1
LIRMM, Université de Montpellier, 161 rue Ada, 34095, Montpellier, France
Numerous problems consisting in identifying vertices in graphs using distances are useful in domains such as network verification and graph isomorphism. Unifying them into a meta-problem may be of main interest. We introduce here a promising solution named Distance Identifying Set. The model contains Identifying Code (IC), Locating Dominating Set (LD) and their generalizations r-IC and r-LD where the closed neighborhood is considered up to distance r. It also contains Metric Dimension (MD) and its refinement r-MD in which the distance between two vertices is considered as infinite if the real distance exceeds r. Note that while IC = 1-IC and LD = 1-LD, we have MD = infty-MD; we say that MD is not local.
In this article, we prove computational lower bounds for several problems included in Distance Identifying Set by providing generic reductions from (Planar) Hitting Set to the meta-problem. We focus on two families of problem from the meta-problem: the first one, called bipartite gifted local, contains r-IC, r-LD and r-MD for each positive integer r while the second one, called 1-layered, contains LD, MD and r-MD for each positive integer r. We have:
- the 1-layered problems are NP-hard even in bipartite apex graphs,
- the bipartite gifted local problems are NP-hard even in bipartite planar graphs,
- assuming ETH, all these problems cannot be solved in 2^{o(sqrt{n})} when restricted to bipartite planar or apex graph, respectively, and they cannot be solved in 2^{o(n)} on bipartite graphs,
- even restricted to bipartite graphs, they do not admit parameterized algorithms in 2^{O(k)} * n^{O(1)} except if W[0] = W[2]. Here k is the solution size of a relevant identifying set.
In particular, Metric Dimension cannot be solved in 2^{o(n)} under ETH, answering a question of Hartung in [Sepp Hartung and André Nichterlein, 2013].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.10/LIPIcs.IPEC.2018.10.pdf
identifying code
resolving set
metric dimension
distance identifying set
parameterized complexity
W-hierarchy
meta-problem
hitting set
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
11:1
11:14
10.4230/LIPIcs.IPEC.2018.11
article
The Parameterized Complexity of Finding Point Sets with Hereditary Properties
Eppstein, David
1
Lokshtanov, Daniel
2
Computer Science Department, University of California, Irvine, USA
Department of Informatics, University of Bergen, Norway
We consider problems where the input is a set of points in the plane and an integer k, and the task is to find a subset S of the input points of size k such that S satisfies some property. We focus on properties that depend only on the order type of the points and are monotone under point removals. We exhibit a property defined by three forbidden patterns for which finding a k-point subset with the property is W[1]-complete and (assuming the exponential time hypothesis) cannot be solved in time n^{o(k/log k)}. However, we show that problems of this type are fixed-parameter tractable for all properties that include all collinear point sets, properties that exclude at least one convex polygon, and properties defined by a single forbidden pattern.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.11/LIPIcs.IPEC.2018.11.pdf
parameterized complexity
fixed-parameter tractability
point set pattern matching
largest pattern-avoiding subset
order type
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
12:1
12:14
10.4230/LIPIcs.IPEC.2018.12
article
Dual Parameterization of Weighted Coloring
Araújo, Júlio
1
https://orcid.org/0000-0001-7074-2753
Campos, Victor A.
2
Lima, Carlos Vinícius G. C.
3
https://orcid.org/0000-0002-6666-0533
Fernandes dos Santos, Vinícius
3
https://orcid.org/0000-0002-4608-4559
Sau, Ignasi
4
https://orcid.org/0000-0002-8981-9287
Silva, Ana
1
https://orcid.org/0000-0001-8917-0564
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Brazil
Departamento de Computação, Universidade Federal do Ceará, Fortaleza, Brazil
Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
LIRMM, CNRS, Université de Montpellier, Montpellier, France
Given a graph G, a proper k-coloring of G is a partition c = (S_i)_{i in [1,k]} of V(G) into k stable sets S_1,..., S_k. Given a weight function w: V(G) -> R^+, the weight of a color S_i is defined as w(i) = max_{v in S_i} w(v) and the weight of a coloring c as w(c) = sum_{i=1}^{k} w(i). Guan and Zhu [Inf. Process. Lett., 1997] defined the weighted chromatic number of a pair (G,w), denoted by sigma(G,w), as the minimum weight of a proper coloring of G. The problem of determining sigma(G,w) has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on n-vertex trees in time n^{o(log n)} unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree.
We focus on the so-called dual parameterization of the problem: given a vertex-weighted graph (G,w) and an integer k, is sigma(G,w) <= sum_{v in V(G)} w(v) - k? This parameterization has been recently considered by Escoffier [WG, 2016], who provided an FPT algorithm running in time 2^{O(k log k)} * n^{O(1)}, and asked which kernel size can be achieved for the problem.
We provide an FPT algorithm running in time 9^k * n^{O(1)}, and prove that no algorithm in time 2^{o(k)} * n^{O(1)} exists under the ETH. On the other hand, we present a kernel with at most (2^{k-1}+1) (k-1) vertices, and rule out the existence of polynomial kernels unless NP subseteq coNP/poly, even on split graphs with only two different weights. Finally, we identify some classes of graphs on which the problem admits a polynomial kernel, in particular interval graphs and subclasses of split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.12/LIPIcs.IPEC.2018.12.pdf
weighted coloring
max coloring
parameterized complexity
dual parameterization
FPT algorithms
polynomial kernels
split graphs
interval graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
13:1
13:14
10.4230/LIPIcs.IPEC.2018.13
article
Computing Kernels in Parallel: Lower and Upper Bounds
Bannach, Max
1
https://orcid.org/0000-0002-6475-5512
Tantau, Till
1
Institute for Theoretical Computer Science, Universität zu Lübeck, Lübeck, Germany
Parallel fixed-parameter tractability studies how parameterized problems can be solved in parallel. A surprisingly large number of parameterized problems admit a high level of parallelization, but this does not mean that we can also efficiently compute small problem kernels in parallel: known kernelization algorithms are typically highly sequential. In the present paper, we establish a number of upper and lower bounds concerning the sizes of kernels that can be computed in parallel. An intriguing finding is that there are complex trade-offs between kernel size and the depth of the circuits needed to compute them: For the vertex cover problem, an exponential kernel can be computed by AC^0-circuits, a quadratic kernel by TC^0-circuits, and a linear kernel by randomized NC-circuits with derandomization being possible only if it is also possible for the matching problem. Other natural problems for which similar (but quantitatively different) effects can be observed include tree decomposition problems parameterized by the vertex cover number, the undirected feedback vertex set problem, the matching problem, or the point line cover problem. We also present natural problems for which computing kernels is inherently sequential.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.13/LIPIcs.IPEC.2018.13.pdf
parallel computation
fixed-parameter tractability
kernelization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
14:1
14:14
10.4230/LIPIcs.IPEC.2018.14
article
Exploring the Kernelization Borders for Hitting Cycles
Agrawal, Akanksha
1
Jain, Pallavi
2
Kanesh, Lawqueen
2
Misra, Pranabendu
3
Saurabh, Saket
2
Institute of Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary
Institute of Mathematical Sciences, HBNI, Chennai, India
University of Bergen, Bergen, Norway
A generalization of classical cycle hitting problems, called conflict version of the problem, is defined as follows. An input is undirected graphs G and H on the same vertex set, and a positive integer k, and the objective is to decide whether there exists a vertex subset X subseteq V(G) such that it intersects all desired "cycles" (all cycles or all odd cycles or all even cycles) and X is an independent set in H. In this paper we study the conflict version of classical Feedback Vertex Set, and Odd Cycle Transversal problems, from the view point of kernelization complexity. In particular, we obtain the following results, when the conflict graph H belongs to the family of d-degenerate graphs.
1) CF-FVS admits a O(k^{O(d)}) kernel.
2) CF-OCT does not admit polynomial kernel (even when H is 1-degenerate), unless NP subseteq coNP/poly.
For our kernelization algorithm we exploit ideas developed for designing polynomial kernels for the classical Feedback Vertex Set problem, as well as, devise new reduction rules that exploit degeneracy crucially. Our main conceptual contribution here is the notion of "k-independence preserver". Informally, it is a set of "important" vertices for a given subset X subseteq V(H), that is enough to capture the independent set property in H. We show that for d-degenerate graph independence preserver of size k^{O(d)} exists, and can be used in designing polynomial kernel.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.14/LIPIcs.IPEC.2018.14.pdf
Parameterized Complexity
Kernelization
Conflict-free problems
Feedback Vertex Set
Even Cycle Transversal
Odd Cycle Transversal
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
15:1
15:13
10.4230/LIPIcs.IPEC.2018.15
article
Best-Case and Worst-Case Sparsifiability of Boolean CSPs
Chen, Hubie
1
Jansen, Bart M. P.
2
Pieterse, Astrid
2
Birkbeck, University of London, Malet Street, Bloomsbury, London WC1E 7HX, United Kingdom
Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs). The goal in sparsification is to reduce the number of constraints in a problem instance without changing the answer, such that a bound on the number of resulting constraints can be given in terms of the number of variables n. We investigate how the worst-case sparsification size depends on the types of constraints allowed in the problem formulation (the constraint language). Two algorithmic results are presented. The first result essentially shows that for any arity k, the only constraint type for which no nontrivial sparsification is possible has exactly one falsifying assignment, and corresponds to logical OR (up to negations). Our second result concerns linear sparsification, that is, a reduction to an equivalent instance with O(n) constraints. Using linear algebra over rings of integers modulo prime powers, we give an elegant necessary and sufficient condition for a constraint type to be captured by a degree-1 polynomial over such a ring, which yields linear sparsifications. The combination of these algorithmic results allows us to prove two characterizations that capture the optimal sparsification sizes for a range of Boolean CSPs. For NP-complete Boolean CSPs whose constraints are symmetric (the satisfaction depends only on the number of 1 values in the assignment, not on their positions), we give a complete characterization of which constraint languages allow for a linear sparsification. For Boolean CSPs in which every constraint has arity at most three, we characterize the optimal size of sparsifications in terms of the largest OR that can be expressed by the constraint language.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.15/LIPIcs.IPEC.2018.15.pdf
constraint satisfaction problems
kernelization
sparsification
lower bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
16:1
16:14
10.4230/LIPIcs.IPEC.2018.16
article
Parameterized Leaf Power Recognition via Embedding into Graph Products
Eppstein, David
1
Havvaei, Elham
1
Computer Science Department, University of California, Irvine, USA
The k-leaf power graph G of a tree T is a graph whose vertices are the leaves of T and whose edges connect pairs of leaves at unweighted distance at most k in T. Recognition of the k-leaf power graphs for k >= 6 is still an open problem. In this paper, we provide an algorithm for this problem for sparse leaf power graphs. Our result shows that the problem of recognizing these graphs is fixed-parameter tractable when parameterized both by k and by the degeneracy of the given graph. To prove this, we describe how to embed the leaf root of a leaf power graph into a product of the graph with a cycle graph. We bound the treewidth of the resulting product in terms of k and the degeneracy of G. As a result, we can use methods based on monadic second-order logic (MSO_2) to recognize the existence of a leaf power as a subgraph of the product graph.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.16/LIPIcs.IPEC.2018.16.pdf
leaf power
phylogenetic tree
monadic second-order logic
Courcelle's theorem
strong product of graphs
fixed-parameter tractability
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
17:1
17:13
10.4230/LIPIcs.IPEC.2018.17
article
Parameterized Complexity of Independent Set in H-Free Graphs
Bonnet, Édouard
1
Bousquet, Nicolas
2
Charbit, Pierre
3
Thomassé, Stéphan
4
Watrigant, Rémi
1
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
CNRS, G-SCOP laboratory, Grenoble-INP, France
Université Paris Diderot, IRIF, France
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France, Institut Universitaire de France
In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding simultaneously K_{1, 4}, any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning C_4-free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey's theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.17/LIPIcs.IPEC.2018.17.pdf
Parameterized Algorithms
Independent Set
H-Free Graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
18:1
18:14
10.4230/LIPIcs.IPEC.2018.18
article
Multi-Budgeted Directed Cuts
Kratsch, Stefan
1
Li, Shaohua
2
Marx, Dániel
3
Pilipczuk, Marcin
2
Wahlström, Magnus
4
Institut für Informatik, Humboldt-Universität zu Berlin
Institute of Informatics, University of Warsaw
Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI)
Royal Holloway, University of London
In this paper, we study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors 1,2,...,l to some edges and give separate budgets k_1,k_2,...,k_l for colors 1,2,...,l. For every color i in {1,...,l}, let E_i be the set of edges of color i. The solution C for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements (i.e., be a cut, a skew multicut, or a directed feedback arc set, respectively), but also needs to satisfy that |C cap E_i| <= k_i for every i in {1,...,l}.
Contrary to the classic minimum cut problem, the multi-budgeted variant turns out to be NP-hard even for l = 2. We propose FPT algorithms parameterized by k=k_1 +...+ k_l for all three problems. To this end, we develop a branching procedure for the multi-budgeted minimum cut problem that measures the progress of the algorithm not by reducing k as usual, by but elevating the capacity of some edges and thus increasing the size of maximum source-to-sink flow. Using the fact that a similar strategy is used to enumerate all important separators of a given size, we merge this process with the flow-guided branching and show an FPT bound on the number of (appropriately defined) important multi-budgeted separators. This allows us to extend our algorithm to the Skew Multicut and Directed Feedback Arc Set problems.
Furthermore, we show connections of the multi-budgeted variants with weighted variants of the directed cut problems and the Chain l-SAT problem, whose parameterized complexity remains an open problem. We show that these problems admit a bounded-in-parameter number of "maximally pushed" solutions (in a similar spirit as important separators are maximally pushed), giving somewhat weak evidence towards their tractability.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.18/LIPIcs.IPEC.2018.18.pdf
important separators
multi-budgeted cuts
Directed Feedback Vertex Set
fixed-parameter tractability
minimum cut
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
19:1
19:13
10.4230/LIPIcs.IPEC.2018.19
article
Matching Cut: Kernelization, Single-Exponential Time FPT, and Exact Exponential Algorithms
Komusiewicz, Christian
1
https://orcid.org/0000-0003-0829-7032
Kratsch, Dieter
2
Le, Van Bang
3
Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany
Laboratoire de Génie Informatique, de Production et de Maintenance, Université de Lorraine, Metz, France
Universität Rostock, Institut für Informatik, Rostock, Germany
In a graph, a matching cut is an edge cut that is a matching. Matching Cut, which is known to be NP-complete, is the problem of deciding whether or not a given graph G has a matching cut. In this paper we show that Matching Cut admits a quadratic-vertex kernel for the parameter distance to cluster and a linear-vertex kernel for the parameter distance to clique. We further provide an O^*(2^{dc(G)}) time and an O^*(2^{dc^-}(G)}) time FPT algorithm for Matching Cut, where dc(G) and dc^-(G) are the distance to cluster and distance to co-cluster, respectively. We also improve the running time of the best known branching algorithm to solve Matching Cut from O^*(1.4143^n) to O^*(1.3803^n). Moreover, we point out that, unless NP subseteq coNP/poly, Matching Cut does not admit a polynomial kernel when parameterized by treewidth.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.19/LIPIcs.IPEC.2018.19.pdf
matching cut
decomposable graph
graph algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
20:1
20:14
10.4230/LIPIcs.IPEC.2018.20
article
Subset Feedback Vertex Set on Graphs of Bounded Independent Set Size
Papadopoulos, Charis
1
Tzimas, Spyridon
1
Department of Mathematics, University of Ioannina, Greece
The (Weighted) Subset Feedback Vertex Set problem is a generalization of the classical Feedback Vertex Set problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although the two problems exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of a structural graph parameter. Here we consider graphs of bounded independent set number for which it is known that Weighted Feedback Vertex Set can be solved in polynomial time. We provide a dichotomy result with respect to the size of a maximum independent set. In particular we show that Weighted Subset Feedback Vertex Set can be solved in polynomial time for graphs of independent set number at most three, whereas we prove that the problem remains NP-hard for graphs of independent set number four. Moreover, we show that the (unweighted) Subset Feedback Vertex Set problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time n^{O(d)}, where d is the size of a maximum independent set of the input graph. To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size. Based on our findings for Subset Feedback Vertex Set, we settle the complexity of Node Multiway Cut, a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals, as well as its variants where nodes are weighted and/or the terminals are deletable, for every value of the given independent set number.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.20/LIPIcs.IPEC.2018.20.pdf
Subset Feedback Vertex Set
Node Multiway Cut
Terminal Set problem
polynomial-time algorithm
NP-completeness
W[1]-hardness
graphs of bounded independent set size
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
21:1
21:16
10.4230/LIPIcs.IPEC.2018.21
article
Integer Programming in Parameterized Complexity: Three Miniatures
Gavenciak, Tomás
1
Knop, Dusan
2
Koutecký, Martin
3
Department of Applied Mathematics, Charles University, Prague, Czech Republic
Department of Informatics, University of Bergen, Bergen, Norway and Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
Technion - Israel Institute of Technology, Haifa, Israel and Computer Science Institute, Charles University, Prague, Czech Republic
Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra's algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation.
To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f(k) poly(n). We focus on:
- Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used.
- Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups.
- Minding the poly(n): reducing f(k) often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds.
Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, and Max-q-Cut by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, and indefinite quadratic programs in fixed dimension.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.21/LIPIcs.IPEC.2018.21.pdf
graph coloring
parameterized complexity
integer linear programming
integer convex programming
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
22:1
22:14
10.4230/LIPIcs.IPEC.2018.22
article
Solving Target Set Selection with Bounded Thresholds Faster than 2^n
Bliznets, Ivan
1
https://orcid.org/0000-0003-2291-2556
Sagunov, Danil
1
https://orcid.org/0000-0003-3327-9768
St. Petersburg Department of Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia
In this paper we consider the Target Set Selection problem. The problem naturally arises in many fields like economy, sociology, medicine. In the Target Set Selection problem one is given a graph G with a function thr: V(G) -> N cup {0} and integers k, l. The goal of the problem is to activate at most k vertices initially so that at the end of the activation process there is at least l activated vertices. The activation process occurs in the following way: (i) once activated, a vertex stays activated forever; (ii) vertex v becomes activated if at least thr(v) of its neighbours are activated. The problem and its different special cases were extensively studied from approximation and parameterized points of view. For example, parameterizations by the following parameters were studied: treewidth, feedback vertex set, diameter, size of target set, vertex cover, cluster editing number and others.
Despite the extensive study of the problem it is still unknown whether the problem can be solved in O^*((2-epsilon)^n) time for some epsilon >0. We partially answer this question by presenting several faster-than-trivial algorithms that work in cases of constant thresholds, constant dual thresholds or when the threshold value of each vertex is bounded by one-third of its degree. Also, we show that the problem parameterized by l is W[1]-hard even when all thresholds are constant.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.22/LIPIcs.IPEC.2018.22.pdf
exact exponential algorithms
target set
vertex thresholds
social influence
irreversible conversions of graphs
bootstrap percolation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
23:1
23:14
10.4230/LIPIcs.IPEC.2018.23
article
Resolving Conflicts for Lower-Bounded Clustering
Casel, Katrin
1
Hasso Plattner Institute, University of Potsdam, Potsdam, Germany, Universität Trier, Fachbereich IV - Informatikwissenschaften, Germany
This paper considers the effect of non-metric distances for lower-bounded clustering, i.e., the problem of computing a partition for a given set of objects with pairwise distance, such that each set has a certain minimum cardinality (as required for anonymisation or balanced facility location problems). We discuss lower-bounded clustering with the objective to minimise the maximum radius or diameter of the clusters. For these problems there exists a 2-approximation but only if the pairwise distance on the objects satisfies the triangle inequality, without this property no polynomial-time constant factor approximation is possible, unless P=NP. We try to resolve or at least soften this effect of non-metric distances by devising particular strategies to deal with violations of the triangle inequality (conflicts). With parameterised algorithmics, we find that if the number of such conflicts is not too large, constant factor approximations can still be computed efficiently.
In particular, we introduce parameterised approximations with respect to not just the number of conflicts but also for the vertex cover number of the conflict graph (graph induced by conflicts). Interestingly, we salvage the approximation ratio of 2 for diameter while for radius it is only possible to show a ratio of 3. For the parameter vertex cover number of the conflict graph this worsening in ratio is shown to be unavoidable, unless FPT=W[2]. We further discuss improvements for diameter by choosing the (induced) P_3-cover number of the conflict graph as parameter and complement these by showing that, unless FPT=W[1], there exists no constant factor parameterised approximation with respect to the parameter split vertex deletion set.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.23/LIPIcs.IPEC.2018.23.pdf
clustering
triangle inequality
parameterised approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
24:1
24:14
10.4230/LIPIcs.IPEC.2018.24
article
Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness
Roth, Marc
1
Schmitt, Johannes
2
Saarland University and Cluster of Excellence (MMCI), Saarbrücken, Germany
ETH Zürich, Zürich, Switzerland
We investigate the problem #{IndSub}(Phi) of counting all induced subgraphs of size k in a graph G that satisfy a given property Phi. This continues the work of Jerrum and Meeks who proved the problem to be #{W[1]}-hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties Phi, the problem #{IndSub}(Phi) is hard for #{W[1]} if the reduced Euler characteristic of the associated simplicial (graph) complex of Phi is non-zero. This observation links #{IndSub}(Phi) to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that #{IndSub}(Phi) is #{W[1]}-hard for every monotone property Phi that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not k-edge-connected for k > 2. Moreover, we show that for those properties #{IndSub}(Phi) can not be solved in time f(k)* n^{o(k)} for any computable function f unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that #{IndSub}(Phi) is #{W[1]}-hard if Phi is any non-trivial modularity constraint on the number of edges with respect to some prime q or if Phi enforces the presence of a fixed isolated subgraph.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.24/LIPIcs.IPEC.2018.24.pdf
counting complexity
Euler characteristic
homomorphisms
parameterized complexity
simplicial complexes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
25:1
25:13
10.4230/LIPIcs.IPEC.2018.25
article
A Strongly-Uniform Slicewise Polynomial-Time Algorithm for the Embedded Planar Diameter Improvement Problem
Lokshtanov, Daniel
1
de Oliveira Oliveira, Mateus
1
Saurabh, Saket
1
Department of Informatics - University of Bergen, Bergen, Norway
In the embedded planar diameter improvement problem (EPDI) we are given a graph G embedded in the plane and a positive integer d. The goal is to determine whether one can add edges to the planar embedding of G in such a way that planarity is preserved and in such a way that the resulting graph has diameter at most d. Using non-constructive techniques derived from Robertson and Seymour's graph minor theory, together with the effectivization by self-reduction technique introduced by Fellows and Langston, one can show that EPDI can be solved in time f(d)* |V(G)|^{O(1)} for some function f(d). The caveat is that this algorithm is not strongly uniform in the sense that the function f(d) is not known to be computable. On the other hand, even the problem of determining whether EPDI can be solved in time f_1(d)* |V(G)|^{f_2(d)} for computable functions f_1 and f_2 has been open for more than two decades [Cohen at. al. Journal of Computer and System Sciences, 2017]. In this work we settle this later problem by showing that EPDI can be solved in time f(d)* |V(G)|^{O(d)} for some computable function f. Our techniques can also be used to show that the embedded k-outerplanar diameter improvement problem (k-EOPDI), a variant of EPDI where the resulting graph is required to be k-outerplanar instead of planar, can be solved in time f(d)* |V(G)|^{O(k)} for some computable function f. This shows that for each fixed k, the problem k-EOPDI is strongly uniformly fixed parameter tractable with respect to the diameter parameter d.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.25/LIPIcs.IPEC.2018.25.pdf
Embedded Planar Diameter Improvement
Constructive Algorithms
Nooses
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-02-05
115
26:1
26:15
10.4230/LIPIcs.IPEC.2018.26
article
The PACE 2018 Parameterized Algorithms and Computational Experiments Challenge: The Third Iteration
Bonnet, Édouard
1
Sikora, Florian
2
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
The Program Committee of the Third Parameterized Algorithms and Computational Experiments challenge (PACE 2018) reports on the third iteration of the PACE challenge. This year, all three tracks were dedicated to solve the Steiner Tree problem, in which, given an edge-weighted graph and a subset of its vertices called terminals, one has to find a minimum-weight subgraph which spans all the terminals. In Track A, the number of terminals was limited. In Track B, a tree-decomposition of the graph was provided in the input, and the treewidth was limited. Finally, Track C welcomed heuristics. Over 80 participants on 40 teams from 16 countries submitted their implementations to the competition.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol115-ipec2018/LIPIcs.IPEC.2018.26/LIPIcs.IPEC.2018.26.pdf
Steiner tree problem
contest
implementation challenge
FPT