Results 1  10
of
20
Parameterized Complexity
, 1998
"... the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs ..."
Abstract

Cited by 1213 (77 self)
 Add to MetaCart
the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs into the toolkit of every algorithm designer. The purpose of the seminar was to bring together leading experts from all over the world, and from the diverse areas of computer science that have been attracted to this new framework. The seminar was intended as the rst larger international meeting with a specic focus on parameterized complexity, and it hopefully serves as a driving force in the development of the eld. 1 We had 49 participants from Australia, Canada, India, Israel, New Zealand, USA, and various European countries. During the workshop 25 lectures were given. Moreover, one night session was devoted to open problems and Thursday was basically used for problem discussion
Scheduling Split Intervals
, 2002
"... We consider the problem of scheduling jobs that are given as groups of nonintersecting segments on the real line. Each job Jj is associated with an interval, Ij, which consists of up to t segments, for some t _) 1, a of their segments intersect. Such jobs show up in a I.I Problem Statement and Mo ..."
Abstract

Cited by 63 (5 self)
 Add to MetaCart
We consider the problem of scheduling jobs that are given as groups of nonintersecting segments on the real line. Each job Jj is associated with an interval, Ij, which consists of up to t segments, for some t _) 1, a of their segments intersect. Such jobs show up in a I.I Problem Statement and Motivation. We wide range of applications, including the transmission consider the problem of scheduling jobs that are given of continuousmedia data, allocation of linear resources as groups of nonintersecting segments on the real line. (e.g. bandwidth in linear processor arrays), and in Each job Jj is associated with a tinterval, Ij, which
On the parameterized complexity of multipleinterval graph problems
 Theor. Comput. Sci
"... Abstract. Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multipleinterval graphs, often allowing for more robustness in the modeling of the specifi ..."
Abstract

Cited by 50 (8 self)
 Add to MetaCart
(Show Context)
Abstract. Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multipleinterval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multipleinterval graphs was initiated. In this sequel, we study multipleinterval graph problems from the perspective of parameterized complexity. The problems under consideration are kIndependent Set, kDominating Set, and kClique, which are all known to be W[1]hard for general graphs, and NPcomplete for multipleinterval graphs. We prove that kClique is in FPT, while kIndependent Set and kDominating Set are both W[1]hard. We also prove that kIndependent Dominating Set, a hybrid of the two above problems, is also W[1]hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]hardness via a reduction from the kMulticolored Clique problem, a variant of kClique. We believe this technique has interest in its own right, as it should help in simplifying W[1]hardness results which are notoriously hard to construct and technically tedious.
Optimization problems in multipleinterval graphs
 In Proceedings of the 18th annual Symposium On Discrete Algorithms (SODA
, 2007
"... Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multipleinterval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
(Show Context)
Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multipleinterval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multipleinterval graph. For Minimum Vertex Cover, we give a (2 − 1/t)approximation algorithm which also works when a tinterval representation of our given graph is absent. Following this, we give a t 2approximation algorithm for Minimum Dominating Set which adapts well to more general variants of the problem. We then proceed to prove that Maximum Clique is NPhard already for 3interval graphs, and provide a (t 2 −t+ 1)/2approximation algorithm for general values of t ≥ 2, using bounds proven for the socalled transversal number of tinterval families.
Localized and compact datastructure for comparability graphs
, 2009
"... We show that every comparability graph of any twodimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O(n) time, if two linear extensions of the poset are given, to produce an O(n) space datastructure supporting distance queries in constant time. The datastructure i ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
We show that every comparability graph of any twodimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O(n) time, if two linear extensions of the poset are given, to produce an O(n) space datastructure supporting distance queries in constant time. The datastructure is localized and given as a distance labeling, that is each vertex receives a label of O(log n) bits so that distance queries between any two vertices are answered by inspecting their labels only. This result improves the previous scheme due to Katz, Katz and Peleg [M. Katz, N.A. Katz, D. Peleg, Distance labeling schemes for wellseparated graph classes, Discrete Applied Mathematics 145 (2005) 384–402] by a log n factor. As a byproduct, our datastructure supports allpair shortestpath queries in O(d) time for distanced pairs, and so identifies in constant time the first edge along a shortest path between any source and destination. More fundamentally, we show that this optimal space and time datastructure cannot be extended for higher dimension posets. More precisely, we prove that for comparability graphs of threedimensional posets, every distance labeling scheme requires Ω(n 1/3) bit labels.
Multitrack Interval Graphs
, 1995
"... . A dtrack interval is a union of d intervals, one each from d parallel lines. The intersection graphs of dtrack intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The trac ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
. A dtrack interval is a union of d intervals, one each from d parallel lines. The intersection graphs of dtrack intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The track number for Km;n is determined by proving that the arboricity of Km;n equals its "caterpillar arboricity". Recognition of graphs with track number 2 is shown to be NPcomplete. 1. Introduction Combinatorial properties of interval systems were investigated as early as the 1930's by Tibor Gallai. His two beautiful unpublished remarks are now phrased as saying that interval graphs and their complements are perfect graphs. In 1968, Gallai suggested considering more general set systems consisting of unions of d intervals, one each from d parallel lines. Such set systems have been called separated dintervals, since the d parallel lines can be viewed as d disjoint host intervals on a single li...
A NOTE ON THE INTERVAL NUMBER OF A GRAPH
, 1985
"... Three results on the interval number of a graph on n vertices are presented. (1) The interval number of almost every graph is between n/4lg n and n/4 (this also holds for almost every bipartite graph). (2) There exist Km,+free bipartite graphs with interval number at least c(m)n’*I ( m+lj/lg n, wh ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Three results on the interval number of a graph on n vertices are presented. (1) The interval number of almost every graph is between n/4lg n and n/4 (this also holds for almost every bipartite graph). (2) There exist Km,+free bipartite graphs with interval number at least c(m)n’*I ( m+lj/lg n, which can be improved to &$4+0(&) for m = 2 and (n/2):/lg n for m = 3. (3) There exists a regular graph of girth at least g with interval number at least &I 1)/2)1'(92).
A Sharp Edge Bound on the Interval Number of a Graph
, 2003
"... The interval number of a graph G, denoted by i(G), is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here we settle a conjecture of Griggs and West about bounding i(G) in terms of e, that is the number of edges in G. Nam ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
The interval number of a graph G, denoted by i(G), is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here we settle a conjecture of Griggs and West about bounding i(G) in terms of e, that is the number of edges in G. Namely, it will be shown that i(G) ^ d 1=2 p e e + 1. It is also observed that the edge bound induces i(G) ^ p 3=2fl(G) + o(1), where fl(G) is the genus of G.
kGap interval graphs
 IN: PROC. OF THE 10TH LATIN AMERICAN THEORETICAL INFORMATICS SYMPOSIUM (LATIN) (2012). AVAILABLE AT: ARXIV:1112.3244
, 2012
"... We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersectio ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a kgap interval graph if it has a multiple interval representation with at most n + k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k = 0), we parameterize graph problems by k, and find FPT algorithms for several