eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
0
0
10.4230/LIPIcs.IPEC.2015
article
LIPIcs, Volume 43, IPEC'15, Complete Volume
Husfeldt, Thore
Kanj, Iyad
LIPIcs, Volume 43, IPEC'15, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015/LIPIcs.IPEC.2015.pdf
Complexity Measures and Classes, Analysis of Algorithms and Problem Complexity, Discrete Mathematics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
i
xiv
10.4230/LIPIcs.IPEC.2015.i
article
Front Matter, Table of Contents, Preface, Program Committee, External Reviewers, List of Authors
Husfeldt, Thore
Kanj, Iyad
Front Matter, Table of Contents, Preface, Program Committee, External Reviewers, List of Authors
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.i/LIPIcs.IPEC.2015.i.pdf
IPEC
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
1
16
10.4230/LIPIcs.IPEC.2015.1
article
Bidimensionality and Parameterized Algorithms (Invited Talk)
Thilikos, Dimitrios M.
We provide an exposition of the main results of the theory of bidimensionality in parameterized algorithm design. This theory applies to graph problems that are bidimensional in the sense that i) their solution value is not increasing when we take minors or contractions of the input graph and ii) their solution value for the (triangulated) (k x k)-grid graph grows as a quadratic function of k. Under certain additional conditions, mainly of logical and combinatorial nature, such problems admit subexponential parameterized algorithms and linear kernels when their inputs are restricted to certain topologically defined graph classes. We provide all formal definitions and concepts in order to present these results in a rigorous way and in their latest update.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.1/LIPIcs.IPEC.2015.1.pdf
Parameterized algorithms
Subexponential FPT-algorithms
Kernelization
Linear kenrels
Bidimensionality
Graph Minors
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
17
29
10.4230/LIPIcs.IPEC.2015.17
article
Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk)
Vassilevska Williams, Virginia
Algorithmic research strives to develop fast algorithms for fundamental problems. Despite its many successes, however, many problems still do not have very efficient algorithms. For years researchers have explained the hardness for key problems by proving NP-hardness, utilizing polynomial time reductions to base the hardness of key problems on the famous conjecture P != NP. For problems that already have polynomial time algorithms, however, it does not seem that one can show any sort of hardness based on P != NP. Nevertheless, we would like to provide evidence that a problem $A$ with a running time O(n^k) that has not been improved in decades, also requires n^{k-o(1)} time, thus explaining the lack of progress on the problem. Such unconditional time lower bounds seem very difficult to obtain, unfortunately. Recent work has concentrated on an approach mimicking NP-hardness: (1) select a few key problems that are conjectured to require T(n) time to solve, (2) use special, fine-grained reductions to prove time lower bounds for many diverse problems in P based on the conjectured hardness of the key problems. In this abstract we outline the approach, give some examples of hardness results based on the Strong Exponential Time Hypothesis, and present an overview of some of the recent work on the topic.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.17/LIPIcs.IPEC.2015.17.pdf
reductions
satisfiability
strong exponential time hypothesis
shortest paths
3SUM
equivalences
fine-grained complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
30
42
10.4230/LIPIcs.IPEC.2015.30
article
Variants of Plane Diameter Completion
Golovach, Petr A.
Requilé, Clément
Thilikos, Dimitrios M.
The Plane Diameter Completion problem asks, given a plane graph G and a positive integer d, if it is a spanning subgraph of a plane graph H that has diameter at most d. We examine two variants of this problem where the input comes with another parameter k. In the first variant, called BPDC, k upper bounds the total number of edges to be added and in the second, called BFPDC, k upper bounds the number of additional edges per face. We prove that both problems are NP-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when k=1 on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in O(n^{3})+2^{2^{O((kd)^2\log d)}} * n steps.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.30/LIPIcs.IPEC.2015.30.pdf
Planar graphs
graph modification problems
parameterized algorithms
dynamic programming
branchwidth
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
43
54
10.4230/LIPIcs.IPEC.2015.43
article
Parameterized and Approximation Algorithms for the Load Coloring Problem
Barbero, Florian
Gutin, Gregory
Jones, Mark
Sheng, Bin
Let c, k be two positive integers. Given a graph G=(V,E), the c-Load Coloring problem asks whether there is a c-coloring varphi: V => [c] such that for every i in [c], there are at least k edges with both endvertices colored i. Gutin and Jones (IPL 2014) studied this problem with c=2. They showed 2-Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of c=2, we obtain a kernel with less than 4k vertices and less than 8k edges. These results imply that for any fixed c >= 2, c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.43/LIPIcs.IPEC.2015.43.pdf
Load Coloring
fixed-parameter tractability
kernelization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
55
65
10.4230/LIPIcs.IPEC.2015.55
article
Scheduling Two Competing Agents When One Agent Has Significantly Fewer Jobs
Hermelin, Danny
Kubitza, Judith-Madeleine
Shabtay, Dvir
Talmon, Nimrod
Woeginger, Gerhard
We study a scheduling problem where two agents (each equipped with a private set of jobs) compete to perform their respective jobs on a common single machine. Each agent wants to keep the weighted sum of completion times of his jobs below a given (agent-dependent) bound. This problem is known to be NP-hard, even for quite restrictive settings of the problem parameters.
We consider parameterized versions of the problem where one of the agents has a small number of jobs (and where this small number constitutes the parameter). The problem becomes much more tangible in this case, and we present three positive algorithmic results for it. Our study is complemented by showing that the general problem is NP-complete even when one agent only has a single job.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.55/LIPIcs.IPEC.2015.55.pdf
Parameterized Complexity
Multiagent Scheduling
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
66
77
10.4230/LIPIcs.IPEC.2015.66
article
On the Workflow Satisfiability Problem with Class-independent Constraints
Crampton, Jason
Gagarin, Andrei
Gutin, Gregory
Jones, Mark
A workflow specification defines sets of steps and users. An authorization policy determines for each user a subset of steps the user is allowed to perform. Other security requirements, such as separation-of-duty, impose constraints on which subsets of users may perform certain subsets of steps. The workflow satisfiability problem (WSP) is the problem of determining whether there exists an assignment of users to workflow steps that satisfies all such authorizations and constraints. An algorithm for solving WSP is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. Given the computational difficulty of WSP, it is important, particularly for the second application, that such algorithms are as efficient as possible.
We introduce class-independent constraints, enabling us to model scenarios where the set of users is partitioned into groups, and the identities of the user groups are irrelevant to the satisfaction of the constraint. We prove that solving WSP is fixed-parameter tractable (FPT) for this class of constraints and develop an FPT algorithm that is useful in practice. We compare the performance of the FPT algorithm with that of SAT4J (a pseudo-Boolean SAT solver) in computational experiments, which show that our algorithm significantly outperforms SAT4J for many instances of WSP. User-independent constraints, a large class of constraints including many practical ones, are a special case of class-independent constraints for which WSP was proved to be FPT (Cohen et al., J. Artif. Intel. Res. 2014). Thus our results considerably extend our knowledge of the fixed-parameter tractability of WSP.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.66/LIPIcs.IPEC.2015.66.pdf
Workflow Satisfiability Problem; Constraint Satisfaction Problem; fixed-parameter tractability; user-independent constraints
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
78
89
10.4230/LIPIcs.IPEC.2015.78
article
Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism
Kim, Eun Jung
Paul, Christophe
Sau, Ignasi
Thilikos, Dimitrios M.
In this paper we design FPT-algorithms for two parameterized problems. The first is List Digraph Homomorphism: given two digraphs G and H and a list of allowed vertices of H for every vertex of G, the question is whether there exists a homomorphism from G to H respecting the list constraints. The second problem is a variant of Multiway Cut, namely Min-Max Multiway Cut: given a graph G, a non-negative integer l, and a set T of r terminals, the question is whether we can partition the vertices of G into r parts such that (a) each part contains one terminal and (b) there are at most l edges with only one endpoint in this part. We parameterize List Digraph Homomorphism by the number w of edges of G that are mapped to non-loop edges of H and we give a time 2^{O(l * log(h) + l^{2 * log(l)}} * n^{4} * log(n) algorithm, where h is the order of the host graph H.We also prove that Min-Max Multiway Cut can be solved in time 2^{O((l * r)^2 * log(l *r))} * n^{4} * log(n). Our approach introduces a general problem, called List Allocation, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an FPT-algorithm for the List Allocation problem that is designed using a suitable adaptation of the randomized contractions technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.78/LIPIcs.IPEC.2015.78.pdf
Parameterized complexity
Fixed-Parameter Tractable algorithm
Multiway Cut
Digraph homomorphism
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
90
101
10.4230/LIPIcs.IPEC.2015.90
article
Improved Exact Algorithms for Mildly Sparse Instances of Max SAT
Sakai, Takayuki
Seto, Kazuhisa
Tamaki, Suguru
Teruyama, Junichi
We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2^{(1-mu(c))n}) for instances with n variables and m=cn clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with mu(c)=1/O(c * log(c)) due to Dantsin and Wolpert [SAT 2006], a randomized polynomial space algorithm with mu(c)=1/O(c * log^3(c)) and a deterministic polynomial space algorithm with mu(c)=1/O(c^2 * log^2(c)) due to Sakai, Seto and Tamaki [Theory Comput. Syst., 2015]. Our first result is a deterministic polynomial space algorithm with mu(c)=1/O(c * log(c)) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2^n only if m=O(n^2).
Our second results are deterministic exponential space algorithms for Max SAT with mu(c)=1/O((c * log(c))^{2/3}) and for Max 3-SAT with mu(c)=1/O(c^{1/2}) that run super-polynomially faster than 2^n when m=o(n^{5/2}/log^{5/2}(n)) and m=o(n^3/log^2(n)) respectively.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.90/LIPIcs.IPEC.2015.90.pdf
maximum satisfiability
weighted
polynomial space
exponential space
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
102
113
10.4230/LIPIcs.IPEC.2015.102
article
Polynomial Fixed-parameter Algorithms: A Case Study for Longest Path on Interval Graphs
Giannopoulou, Archontia C.
Mertzios, George B.
Niedermeier, Rolf
We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time.
The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path; it is NP-hard in general but known to be solvable in O(n^4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k^9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis for polynomial-time solvable problems offers a very fertile ground for future studies for all sorts of algorithmic problems. It may enable a refined understanding of efficiency aspects for polynomial-time solvable problems, similarly to what classical parameterized complexity analysis does for NP-hard problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.102/LIPIcs.IPEC.2015.102.pdf
fixed-parameter algorithm
preprocessing
data reduction
polynomial-time algorithm
longest path problem
interval graphs
proper interval vertex del
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
114
126
10.4230/LIPIcs.IPEC.2015.114
article
Meta-kernelization using Well-structured Modulators
Eiben, Eduard
Ganian, Robert
Szeider, Stefan
Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.114/LIPIcs.IPEC.2015.114.pdf
Kernelization
Parameterized complexity
Structural parameters
Rank-width
Split decompositions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
127
137
10.4230/LIPIcs.IPEC.2015.127
article
Parameter Compilation
Chen, Hubie
In resolving instances of a computational problem, if multiple instances of interest share a feature in common,it may be fruitful to compile this feature into a format that allows for more efficient resolution, even if the compilation is relatively expensive. In this article, we introduce a formal framework for classifying problems according to their compilability. The basic object in our framework is that of a parameterized problem, which here is a language along with a parameterization—a map which provides, for each instance, a so-called parameter on which compilation may be performed. Our framework is positioned within the paradigm of parameterized complexity, and our notions are relatable to established concepts in the theory of parameterized complexity. Indeed, we view our framework as playing a unifying role, integrating together parameterized complexity and compilability theory.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.127/LIPIcs.IPEC.2015.127.pdf
compilation
parameterized complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
138
150
10.4230/LIPIcs.IPEC.2015.138
article
An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion
Kanté, Mamadou Moustapha
Kim, Eun Jung
Kwon, O-joung
Paul, Christophe
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514-528, 2006.], and it is similar to pathwidth, which is the linearized variant of treewidth. Motivated from the results on graph modification problems into graphs of bounded treewidth or pathwidth, we investigate a graph modification problem into the class of graphs having linear rankwidth at most one, called the Linear Rankwidth-1 Vertex Deletion (shortly, LRW1-Vertex Deletion). In this problem, given an n-vertex graph G and a positive integer k, we want to decide whether there is a set of at most k vertices whose removal turns G into a graph of linear rankwidth at most one and if one exists, find such a vertex set. While the meta-theorem of Courcelle, Makowsky, and Rotics implies thatLRW1-Vertex Deletion can be solved in time f(k) * n^3 for some function f, it is not clear whether this problem allows a runtime with a modest exponential function. We establish that LRW1-Vertex Deletion can be solved in time 8^k * n^{O(1)}. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define the necklace graphs and investigate their structural properties.
We also show that the LRW1-Vertex Deletion has a polynomial kernel.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.138/LIPIcs.IPEC.2015.138.pdf
(linear) rankwidth
distance-hereditary graphs
thread graphs
parameterized complexity
kernelization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
151
162
10.4230/LIPIcs.IPEC.2015.151
article
Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices
Suchý, Ondrj
In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk, Pilipczuk, Sankowski, and van Leeuwen [FOCS 2014] gave a polynomial kernel for Steiner Tree in planar graphs, when parameterized by |T|+k, the total number of vertices in the constructed subgraph.
In this paper we present several polynomial time applicable reduction rules for Planar Steiner Tree. In an instance reduced with respect to the presented reduction rules, the number of terminals |T| is at most quadratic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in planar graphs for the parameterization by the number k of Steiner vertices in the solution.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.151/LIPIcs.IPEC.2015.151.pdf
Steiner Tree
polynomial kernel
planar graphs
polynomial-time preprocessing
network sparsification
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
163
174
10.4230/LIPIcs.IPEC.2015.163
article
Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT
Jansen, Bart M. P.
Pieterse, Astrid
We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.163/LIPIcs.IPEC.2015.163.pdf
sparsification
graph coloring
Hamiltonian cycle
satisfiability
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
175
186
10.4230/LIPIcs.IPEC.2015.175
article
Definability Equals Recognizability for k-Outerplanar Graphs
Jaffke, Lars
Bodlaender, Hans L.
One of the most famous algorithmic meta-theorems states that every graph property that can be defined by a sentence in counting monadic second order logic (CMSOL) can be checked in linear time for graphs of bounded treewidth, which is known as Courcelle's Theorem. These algorithms are constructed as finite state tree automata, and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e., every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. We prove this conjecture for k-outerplanar graphs, which are known to have treewidth at most 3k-1.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.175/LIPIcs.IPEC.2015.175.pdf
treewidth
monadic second order logic of graphs
finite state tree automata
$k$-outerplanar graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
187
198
10.4230/LIPIcs.IPEC.2015.187
article
Practical Algorithms for Linear Boolean-width
ten Brinke, Chiel B.
van Houten, Frank J. P.
Bodlaender, Hans L.
In this paper, we give a number of new exact algorithms and heuristics to compute linear boolean decompositions, and experimentally evaluate these algorithms. The experimental evaluation shows that significant improvements can be made with respect to running time without increasing the width of the generated decompositions. We also evaluated dynamic programming algorithms on linear boolean decompositions for several vertex subset problems. This evaluation shows that such algorithms are often much faster (up to several orders of magnitude) compared to theoretical worst case bounds.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.187/LIPIcs.IPEC.2015.187.pdf
graph decomposition
boolean-width
heuristics
exact algorithms
vertex subset problems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
199
211
10.4230/LIPIcs.IPEC.2015.199
article
Linear Kernels for Outbranching Problems in Sparse Digraphs
Bonamy, Marthe
Kowalik, Lukasz
Pilipczuk, Michal
Socala, Arkadiusz
In the k-Leaf Out-Branching and k-Internal Out-Branching problems we are given a directed graph D with a designated root r and a nonnegative integer k. The question is to determine the existence of an outbranching rooted at r that has at least k leaves, or at least k internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k^2) vertices are known on general graphs. In this work we show that k-Leaf Out-Branching admits a kernel with O(k) vertices on H-minor-free graphs, for any fixed H, whereas k-Internal Out-Branching admits a kernel with O(k) vertices on any graph class of bounded expansion.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.199/LIPIcs.IPEC.2015.199.pdf
FPT algorithm
kernelization
outbranching
sparse graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
212
223
10.4230/LIPIcs.IPEC.2015.212
article
Maximum Matching Width: New Characterizations and a Fast Algorithm for Dominating Set
Jeong, Jisu
Sæther, Sigve Hortemo
Telle, Jan Arne
We give alternative definitions for maximum matching width, e.g., a graph G has mmw(G) <= k if and only if it is a subgraph of a chordal graph H and for every maximal clique X of H there exists A,B,C \subseteq X with A \cup B \cup C=X and |A|,|B|,|C| <= k such that any subset of X that is a minimal separator of H is a subset of either A, B or C. Treewidth and branchwidth have alternative definitions through intersections of subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We show that mm-width combines both aspects, focusing on nodes and on edges. Based on this we prove that given a graph G and a branch decomposition of mm-width k we can solve Dominating Set in time O^*(8^k), thereby beating O^*(3^{tw(G)}) whenever tw(G) > log_3(8) * k ~ 1.893 k. Note that mmw(G) <= tw(G)+1 <= 3 mmw(G) and these inequalities are tight. Given only the graph G and using the best known algorithms to find decompositions, maximum matching width will be better for solving Dominating Set whenever tw(G) > 1.549 * mmw(G).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.212/LIPIcs.IPEC.2015.212.pdf
FPT algorithms
treewidth
dominating set
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
224
235
10.4230/LIPIcs.IPEC.2015.224
article
Fast Parallel Fixed-parameter Algorithms via Color Coding
Bannach, Max
Stockhusen, Christoph
Tantau, Till
Fixed-parameter algorithms have been successfully applied to solve numerous difficult problems within acceptable time bounds on large inputs. However, most fixed-parameter algorithms are inherently sequential and, thus, make no use of the parallel hardware present in modern computers. We show that parallel fixed-parameter algorithms do not only exist for numerous parameterized problems from the literature - including vertex cover, packing problems, cluster editing, cutting vertices, finding embeddings, or finding matchings - but that there are parallel algorithms working in constant time or at least in time depending only on the parameter (and not on the size of the input) for these problems. Phrased in terms of complexity classes, we place numerous natural parameterized problems in parameterized versions of AC^0. On a more technical level, we show how the color coding method can be implemented in constant time and apply it to embedding problems for graphs of bounded tree-width or tree-depth and to model checking first-order formulas in graphs of bounded degree.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.224/LIPIcs.IPEC.2015.224.pdf
color coding
parallel computation
fixed-parameter tractability
graph packing
cutting $ell$ vertices
cluster editing
tree-width
tree-depth,
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
236
247
10.4230/LIPIcs.IPEC.2015.236
article
Fixed-parameter Tractable Distances to Sparse Graph Classes
Bulian, Jannis
Dawar, Anuj
We show that for various classes C of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph G to C is fixed-parameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT. The second shows that determining the elimination distance of a graph G to a minor-closed class C is FPT.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.236/LIPIcs.IPEC.2015.236.pdf
parameterized complexity
fixed-parameter tractable
distance
graph theory
sparse graphs
graph minor
nowhere dense
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
248
257
10.4230/LIPIcs.IPEC.2015.248
article
Strong ETH and Resolution via Games and the Multiplicity of Strategies
Bonacina, Ilario
Talebanfard, Navid
We consider a restriction of the Resolution proof system in which at most a fixed number of variables can be resolved more than once along each refutation path. This system lies between regular Resolution, in which no variable can be resolved more than once along any path, and general Resolution where there is no restriction on the number of such variables. We show that when the number of re-resolved variables is not too large, this proof system is consistent with the Strong Exponential Time Hypothesis (SETH). More precisely for large n and k we show that there are unsatisfiable k-CNF formulas which require Resolution refutations of size 2^{(1 - epsilon_k)n}, where n is the number of variables and epsilon_k=~O(k^{-1/5}), whenever in each refutation path we only allow at most ~O(k^{-1/5})n variables to be resolved multiple times. However, these re-resolved variables along different paths do not need to be the same. Prior to this work, the strongest proof system shown to be consistent with SETH was regular Resolution [Beck and Impagliazzo, STOC'13]. This work strengthens that result and gives a different and conceptually simpler game-theoretic proof for the case of regular Resolution.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.248/LIPIcs.IPEC.2015.248.pdf
Strong Exponential Time Hypothesis
resolution
proof systems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
258
269
10.4230/LIPIcs.IPEC.2015.258
article
Quick but Odd Growth of Cacti
Kolay, Sudeshna
Lokshtanov, Daniel
Panolan, Fahad
Saurabh, Saket
Let F be a family of graphs. Given an input graph G and a positive integer k, testing whether G has a k-sized subset of vertices S, such that G\S belongs to F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when F is either a family of cactus graphs or a family of odd-cactus graphs. A graph H is called a cactus graph if every pair of cycles in H intersect on at most one vertex. Furthermore, a cactus graph H is called an odd cactus, if every cycle of H is of odd length. Let us denote by C and C_{odd}, families of cactus and odd cactus, respectively. The vertex deletion problems corresponding to C and C_{odd} are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with running time 12^{k}*n^{O(1)} for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.258/LIPIcs.IPEC.2015.258.pdf
Even Cycle Transversal
Diamond Hitting Set
Randomized Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
270
281
10.4230/LIPIcs.IPEC.2015.270
article
A Polynomial Kernel for Block Graph Deletion
Kim, Eun Jung
Kwon, O-joung
In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with O(k^{6}) vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of 'complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time 10^{k} * n^{O(1)}.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.270/LIPIcs.IPEC.2015.270.pdf
block graph
polynomial kernel
single-exponential FPT algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
282
293
10.4230/LIPIcs.IPEC.2015.282
article
Parameterized Complexity of Graph Constraint Logic
van der Zanden, Tom C.
Graph constraint logic is a framework introduced by Hearn and Demaine, which provides several problems that are often a convenient starting point for reductions. We study the parameterized complexity of Constraint Graph Satisfiability and both bounded and unbounded versions of Nondeterministic Constraint Logic (NCL) with respect to solution length, treewidth and maximum degree of the underlying constraint graph as parameters. As a main result we show that restricted NCL remains PSPACE-complete on graphs of bounded bandwidth, strengthening Hearn and Demaine's framework. This allows us to improve upon existing results obtained by reduction from NCL. We show that reconfiguration versions of several classical graph problems (including independent set, feedback vertex set and dominating set) are PSPACE-complete on planar graphs of bounded bandwidth and that Rush Hour, generalized to k*n boards, is PSPACE-complete even when k is at most a constant.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.282/LIPIcs.IPEC.2015.282.pdf
Nondeterministic Constraint Logic
Reconfiguration Problems
Parameterized Complexity
Treewidth
Bandwidth
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
294
306
10.4230/LIPIcs.IPEC.2015.294
article
Complexity and Approximability of Parameterized MAX-CSPs
Dell, Holger
Kim, Eun Jung
Lampis, Michael
Mitsou, Valia
Mömke, Tobias
We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable–constraint incidence graph of the CSP instance.
We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k. We attempt to fully classify them into the following three cases:
1. The exact optimum can be computed in FPT-time.
2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPT-AS), which computes a (1-epsilon)-approximation in time f(k,epsilon) * poly(n).
3. There is no FPT-AS unless FPT=W[1].
For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.294/LIPIcs.IPEC.2015.294.pdf
Approximation
Structural Parameters
Constraint Satisfaction
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
307
318
10.4230/LIPIcs.IPEC.2015.307
article
Enumerating Minimal Connected Dominating Sets in Graphs of Bounded Chordality
Golovach, Petr A.
Heggernes, Pinar
Kratsch, Dieter
Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.307/LIPIcs.IPEC.2015.307.pdf
Minimal connected dominating set
exact algorithms
enumeration
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
319
330
10.4230/LIPIcs.IPEC.2015.319
article
The Graph Motif Problem Parameterized by the Structure of the Input Graph
Bonnet, Édouard
Sikora, Florian
The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. Though, in the many applications of Graph Motif, the input graph originates from real-life and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. Interestingly, we establish that Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.319/LIPIcs.IPEC.2015.319.pdf
Parameterized Complexity
Structural Parameters
Graph Motif
Computational Biology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
331
342
10.4230/LIPIcs.IPEC.2015.331
article
Kernels for Structural Parameterizations of Vertex Cover - Case of Small Degree Modulators
Majumdar, Diptapriyo
Raman, Venkatesh
Saurabh, Saket
Vertex Cover is one of the most well studied problems in the realm of parameterized algorithms and admits a kernel with O(l^2) edges and 2*l vertices. Here, l denotes the size of a vertex cover we are seeking for. A natural question is whether Vertex Cover admits a polynomial kernel (or a parameterized algorithm) with respect to a parameter k, that is, provably smaller than the size of the vertex cover. Jansen and Bodlaender [STACS 2011, TOCS 2013] raised this question and gave a kernel for Vertex Cover of size O(f^3), where f is the size of a feedback vertex set of the input graph. We continue this line of work and study Vertex Cover with respect to a parameter that is always smaller than the solution size and incomparable to the size of the feedback vertex set of the input graph. Our parameter is the number of vertices whose removal results in a graph of maximum degree two. While vertex cover with this parameterization can easily be shown to be fixed-parameter tractable (FPT), we show that it has a polynomial sized kernel.
The input to our problem consists of an undirected graph G, S \subseteq V(G) such that |S| = k and G[V(G)\S] has maximum degree at most 2 and a positive integer l. Given (G,S,l), in polynomial time we output an instance (G',S',l') such that |V(G')|<= O(k^5), |E(G')|<= O(k^6) and G has a vertex cover of size at most l if and only if G' has a vertex cover of size at most l'.
When G[V(G)\S] has maximum degree at most 1, we improve the known kernel bound from O(k^3) vertices to O(k^2) vertices (and O(k^3) edges). In general, if G[V(G)\S] is simply a collection of cliques of size at most d, then we transform the graph in polynomial time to an equivalent hypergraph with O(k^d) vertices and show that, for d >= 3, a kernel with O(k^{d-epsilon}) vertices is unlikely to exist for any epsilon >0 unless NP is a subset of coNO/poly.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.331/LIPIcs.IPEC.2015.331.pdf
Parameterized Complexity
Kernelization
expansion lemma
vertex cover
structural parameterization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
343
354
10.4230/LIPIcs.IPEC.2015.343
article
Parameterized Complexity of Critical Node Cuts
Hermelin, Danny
Kaspi, Moshe
Komusiewicz, Christian
Navon, Barak
We consider the following graph cut problem called Critical Node Cut (CNC): Given a graph G on n vertices, and two positive integers k and x, determine whether G has a set of k vertices whose removal leaves G with at most x connected pairs of vertices. We analyze this problem in the framework of parameterized complexity. That is, we are interested in whether or not this problem is solvable in f(kappa) * n^{O(1)} time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters kappa. We consider four such parameters:
- The size k of the required cut.
- The upper bound x on the number of remaining connected pairs.
- The lower bound y on the number of connected pairs to be removed.
- The treewidth w of G.
We determine whether or not CNC is fixed-parameter tractable for each of these parameters. We determine this also for all possible aggregations of these four parameters, apart from w+k. Moreover, we also determine whether or not CNC admits a polynomial kernel for all these parameterizations. That is, whether or not there is an algorithm that reduces each instance of CNC in polynomial time to an equivalent instance of size kappa^{O(1)}, where kappa is the given parameter.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.343/LIPIcs.IPEC.2015.343.pdf
graph cut problem
NP-hard problem
treewidth
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
355
364
10.4230/LIPIcs.IPEC.2015.355
article
Parameterized Complexity of Sparse Linear Complementarity Problems
Sumita, Hanna
Kakimura, Naonori
Makino, Kazuhisa
In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with all the parameters. We also show that if we drop any of the three parameters, then the LCP is fixed-parameter intractable.
In addition, we discuss the nonexistence of a polynomial kernel for the LCP.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.355/LIPIcs.IPEC.2015.355.pdf
linear complementarity problem
sparsity
parameterized complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
365
376
10.4230/LIPIcs.IPEC.2015.365
article
Parameterized Lower Bound and Improved Kernel for Diamond-free Edge Deletion
Sandeep, R. B.
Sivadasan, Naveen
A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks to find whether there exist at most k edges in the input graph whose deletion results in a diamond-free graph. The problem was proved to be NP-complete and a polynomial kernel of O(k^4) vertices was found by Fellows et. al. (Discrete Optimization, 2011).
In this paper, we give an improved kernel of O(k^3) vertices for Diamond-free Edge Deletion. We give an alternative proof of the NP-completeness of the problem and observe that it cannot be solved in time 2^{o(k)} * n^{O(1)}, unless the Exponential Time Hypothesis fails.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.365/LIPIcs.IPEC.2015.365.pdf
edge deletion problems
polynomial kernelization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
377
388
10.4230/LIPIcs.IPEC.2015.377
article
On Kernelization and Approximation for the Vector Connectivity Problem
Kratsch, Stefan
Sorge, Manuel
In the Vector Connectivity problem we are given an undirected graph G=(V,E), a demand function phi: V => {0,...,d}, and an integer k. The question is whether there exists a set S of at most k vertices such that every vertex v in V\S has at least phi(v) vertex-disjoint paths to S; this abstractly captures questions about placing servers in a network, or warehouses on a map, relative to demands. The problem is NP-hard already for instances with d=4 (Cicalese et al., Theor. Comput. Sci. 2015), admits a log-factor approximation (Boros et al., Networks 2014), and is fixed-parameter tractable in terms of k (Lokshtanov, unpublished 2014).
We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector d-Connectivity where the upper bound d on demands is a constant. For Vector d-Connectivity we give a factor d-approximation algorithm and construct a vertex-linear kernelization, i.e., an efficient reduction to an equivalent instance with f(d)k=O(k) vertices. For Vector Connectivity we get a factor opt-approximation and we show that it has no kernelization to size polynomial in k+d unless NP \subseteq coNP/poly, making f(d)\poly(k) optimal for Vector d-Connectivity. Finally, we provide a write-up for fixed-parameter tractability of Vector Connectivity(k) by giving a different algorithm based on matroid intersection.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.377/LIPIcs.IPEC.2015.377.pdf
parameterized complexity
kernelization
approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
389
401
10.4230/LIPIcs.IPEC.2015.389
article
B-Chromatic Number: Beyond NP-Hardness
Panolan, Fahad
Philip, Geevarghese
Saurabh, Saket
The b-chromatic number of a graph G, chi_b(G), is the largest integer k such that G has a k-vertex coloring with the property that each color class has a vertex which is adjacent to at least one vertex in each of the other color classes. In the B-Chromatic Number problem, the objective is to decide whether chi_b(G) >= k. Testing whether chi_b(G)=Delta(G)+1, where Delta(G) is the maximum degree of a graph, itself is NP-complete even for connected bipartite graphs (Kratochvil, Tuza and Voigt, WG 2002). In this paper we study B-Chromatic Number in the realm of parameterized complexity and exact exponential time algorithms. We show that B-Chromatic Number is W[1]-hard when parameterized by k, resolving the open question posed by Havet and Sampaio (Algorithmica 2013). When k=Delta(G)+1, we design an algorithm for B-Chromatic Number running in time 2^{O(k^2 * log(k))}*n^{O(1)}. Finally, we show that B-Chromatic Number for an n-vertex graph can be solved in time O(3^n * n^{4} * log(n)).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.389/LIPIcs.IPEC.2015.389.pdf
b-chromatic number
exact algorithm
parameterized complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-11-19
43
402
413
10.4230/LIPIcs.IPEC.2015.402
article
Fast Biclustering by Dual Parameterization
Drange, Pål Grønås
Reidl, Felix
Sánchez Villaamil, Fernando
Sikdar, Somnath
We study two clustering problems, Starforest Editing, the problem of adding and deleting edges to obtain a disjoint union of stars, and the generalization Bicluster Editing. We show that, in addition to being NP-hard, none of the problems can be solved in subexponential time unless the exponential time hypothesis fails.
Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the number of connected components in the solution should not make the problem easier: In particular, they argue that the subexponential time algorithm for editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J. Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p is a secondary parameter, bounding the number of components in the solution.
However, upon bounding the number of stars or bicliques in the solution, we obtain algorithms which run in time O(2^{3*sqrt(pk)} + n + m) for p-Starforest Editing and O(2^{O(p * sqrt(k) * log(pk))} + n + m) for p-Bicluster Editing. We obtain a similar result for the more general case of t-Partite p-Cluster Editing. This is subexponential in k for a fixed number of clusters, since p is then considered a constant.
Our results even out the number of multivariate subexponential time algorithms and give reasons to believe that this area warrants further study.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol043-ipec2015/LIPIcs.IPEC.2015.402/LIPIcs.IPEC.2015.402.pdf
graph editing
subexponential algorithms
parameterized complexity