Results 1  10
of
11
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
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
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.
Approximating the kMulticut Problem
"... We study the kmulticut problem: Given an edgeweighted undirected graph, a set of l pairs of vertices, and a target k ≤ l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the kmulticut problem ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
We study the kmulticut problem: Given an edgeweighted undirected graph, a set of l pairs of vertices, and a target k ≤ l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the kmulticut problem on trees can be approximated within a factor of 8 3 + ɛ, for any fixed ɛ> 0, and within O(log 2 n log log n) on general graphs, where n is the number of vertices in the graph. For any fixed ɛ> 0, we also obtain a polynomial time algorithm for kmulticut on trees which returns a solution of cost at most (2 + ɛ) · OP T, that separates at least (1 − ɛ) · k pairs, where OP T is the cost of the optimal solution separating k pairs. Our techniques also give a simple 2approximation algorithm for the multicut problem on trees using total unimodularity, matching the best known algorithm [8].
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.
Improved Approximations for Guarding 1.5Dimensional Terrains
"... We present a 4approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 (see [14]). Unlike most of the previous techniques, our method is base ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
We present a 4approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 (see [14]). Unlike most of the previous techniques, our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.
LAGRANGIAN RELAXATION AND PARTIAL COVER (EXTENDED ABSTRACT)
, 2008
"... Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving ( ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LMP) αapproximation as a black box can yield an approximation factor better than 4 3 α. This matches the upper bound given by Könemann et al. (ESA 2006, pages 468–479). Faced with this limitation we study a specific, yet broad class of covering problems: Partial Totally Balanced Cover. By carefully analyzing the inner workings of the LMP algorithm we are able to give an almost tight characterization of the integrality gap of the standard linear relaxation of the problem. As a consequence we obtain improved approximations for the Partial version of Multicut and Path Hitting on Trees, Rectangle Stabbing, and Set Cover with ρBlocks.
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
LAGRANGIAN RELAXATION AND PARTIAL COVER
"... Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LM ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LMP) αapproximation as a black box can yield an approximation factor better than 4/3 α. This matches the upper bound given by Könemann et al. (ESA 2006, pages 468–479). Faced with this limitation we study a specific, yet broad class of covering problems: Partial Totally Balanced Cover. By carefully analyzing the inner workings of the LMP algorithm we are able to give an almost tight characterization of the integrality gap of the standard linear relaxation of the problem. As a consequence we obtain improved approximations for the Partial version of Multicut and Path Hitting on Trees, Rectangle Stabbing, and Set Cover with ρBlocks.
Primaldual algorithms for combinatorial optimization problems
, 2007
"... Combinatorial optimization problems such as routing, scheduling, covering and packing problems abound in everyday life. At a very high level, a combinatorial optimization problem amounts to finding a solution with minimum or maximum cost among a large number of feasible solutions. An algorithm for a ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Combinatorial optimization problems such as routing, scheduling, covering and packing problems abound in everyday life. At a very high level, a combinatorial optimization problem amounts to finding a solution with minimum or maximum cost among a large number of feasible solutions. An algorithm for a given optimization problem is said to be exact if it always returns an optimal solution and is said to be efficient if it runs in time polynomial on the size of its input. The theory of NPcompleteness suggests that exact and efficient algorithms are unlikely to exist for the class of NPhard problems. Unfortunately, a large number of natural and interesting combinatorial optimization problems are NPhard. One way to cope with NPhardness is to relax the optimality requirement and instead look for solutions that are provably close to the optimum. This is the main idea behind approximation algorithms. An algorithm is said to be a ρapproximation if it always returns a solution whose cost is at most a ρ factor away from the optimal cost. Arguably, one of the most important techniques in the design of combinatorial algorithms is the primaldual schema in which the cost of the primal solution is compared to the cost of a dual solution. In this dissertation we study the primaldual schema in the design of approximation algorithms for a number of covering and scheduling problems.
On the MultiRadius Cover Problem
"... An instance of the multiradius cover problem consists of a graph G = (V, E) with edge lengths l: E → R +. Each vertex u ∈ V represents a transmission station for which a transmission radius ru must be picked. Edges represent a continuum of demand points to be satisfied, that is, for every edge (u, ..."
Abstract
 Add to MetaCart
(Show Context)
An instance of the multiradius cover problem consists of a graph G = (V, E) with edge lengths l: E → R +. Each vertex u ∈ V represents a transmission station for which a transmission radius ru must be picked. Edges represent a continuum of demand points to be satisfied, that is, for every edge (u, v) ∈ E we ask that ru + rv ≥ luv. The cost of transmitting at radius r from vertex u is given by an arbitrary nondecreasing cost function cu(r). Our goal is to find a cover with minimum total cost P u cu(ru). The multiradius cover problem is NPhard as it generalizes the wellknown vertex cover problem. In this paper we present a 2approximation algorithm for it.