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25
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 ..."
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Cited by 1213 (77 self)
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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 ..."
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Cited by 50 (8 self)
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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.
A unified approach to approximating partial covering problems
 IN PROCEEDINGS OF THE 14TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS
, 2006
"... An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U. Each element e ∈ U is associated with a profit p(e), whereas each subset S ∈ S has a cost c(S). The objective is to find a minimum cost subcollection S ′ ⊆ S such that the combined prof ..."
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Cited by 24 (2 self)
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An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U. Each element e ∈ U is associated with a profit p(e), whereas each subset S ∈ S has a cost c(S). The objective is to find a minimum cost subcollection S ′ ⊆ S such that the combined profit of the elements covered by S ′ is at least P, a specified profit bound. In the prizecollecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e ∈ U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection S ′ ⊆ S that minimizes the cost of S ′ plus the penalties of uncovered elements. Although problemspecific connections between the partial cover and the prizecollecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first nontrivial approximability results.
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 ..."
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Cited by 18 (5 self)
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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.
Algorithms for capacitated rectangle stabbing and lotsizing with joint setup costs
, 2007
"... In the rectangle stabbing problem we are given a set of axis parallel rectangles and a set of horizontal and vertical lines, and our goal is to find a minimum size subset of lines that intersect all the rectangles. In this paper we study the capacitated version of this problem in which the input inc ..."
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Cited by 14 (2 self)
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In the rectangle stabbing problem we are given a set of axis parallel rectangles and a set of horizontal and vertical lines, and our goal is to find a minimum size subset of lines that intersect all the rectangles. In this paper we study the capacitated version of this problem in which the input includes an integral capacity for each line. The capacity of a line bounds the number of rectangles that the line can cover. We consider two versions of this problem. In the first, one is allowed to use only a single copy of each line (hard capacities), and in the second, one is allowed to use multiple copies of every line, but the multiplicities are counted in the size (or weight) of the solution (soft capacities). We present an exact polynomialtime algorithm for the weighted one dimensional case with hard capacities that can be extended to the one dimensional weighted case with soft capacities. This algorithm is also extended to solve a certain capacitated multiitem lot sizing inventory problem with joint setup costs. For the case of ddimensional rectangle stabbing with soft capacities, we present a 3dapproximation algorithm for the unweighted case. For ddimensional rectangle stabbing problem with hard capacities, we present a bicriteria algorithm that computes 4dapproximate solutions that use at most two copies of every line. Finally, we present hardness results for rectangle stabbing when the dimension is part of the input and for a twodimensional weighted version with hard capacities.
Rounding to an integral program
 In Proceedings of the 4th International Workshop on Efficient and Experimental Algorithms (WEA’05
, 2005
"... Abstract. We present a general framework for approximating several NPhard problems that have two underlying properties in common. First, the problems we consider can be formulated as integer covering programs, possibly with additional side constraints. Second, the number of covering options is rest ..."
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Cited by 11 (0 self)
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Abstract. We present a general framework for approximating several NPhard problems that have two underlying properties in common. First, the problems we consider can be formulated as integer covering programs, possibly with additional side constraints. Second, the number of covering options is restricted in some sense, although this property may be well hidden. Our method is a natural extension of the threshold rounding technique. 1
Algorithmic Aspects of the ConsecutiveOnes Property
, 2009
"... We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition ..."
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Cited by 10 (1 self)
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We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition algorithms, and applications such as integer linear programming and solving Set Cover.
Partial multicuts in trees
 IN PROCEEDINGS OF THE 3RD INTERNATIONAL WORKSHOP ON APPROXIMATION AND ONLINE ALGORITHMS
, 2005
"... Let T = (V, E) be an undirected tree, in which each edge is associated with a nonnegative cost, and let {s1, t1},..., {sk, tk} be a collection of k distinct pairs of vertices. Given a requirement parameter t ≤ k, the partial multicut on a tree problem asks to find a minimum cost set of edges whos ..."
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Cited by 8 (4 self)
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Let T = (V, E) be an undirected tree, in which each edge is associated with a nonnegative cost, and let {s1, t1},..., {sk, tk} be a collection of k distinct pairs of vertices. Given a requirement parameter t ≤ k, the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from T disconnects at least t out of these k pairs. This problem generalizes the wellknown multicut on a tree problem, in which we are required to disconnect all given pairs. The main contribution of this paper is an ( 8 + ɛ)approximation algo3 rithm for partial multicut on a tree, whose run time is strongly polynomial for any fixed ɛ> 0. This result is achieved by introducing problemspecific insight to the general framework of using the Lagrangian relaxation technique in approximation algorithms. Our algorithm utilizes a heuristic for the closely related prizecollecting variant, in which we are not required to disconnect all pairs, but rather incur penalties for failing to do so. We provide a Lagrangian multiplier preserving algorithm for the latter problem, with an approximation factor of 2. Finally, we present a new 2approximation algorithm for multicut on a tree, based on LProunding.
P.J.: Separating points by axisparallel lines
 International Journal of Computational Geometry & Applications
, 2005
"... PengJun Wan ¢ We study the problem of separating £ points in the plane, no two of which have the same ¤ or ¥coordinate using a minimum number of vertical and horizontal lines avoiding the points, so that each cell of the subdivision contains at most one point. We prove that this problem and some v ..."
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Cited by 6 (0 self)
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PengJun Wan ¢ We study the problem of separating £ points in the plane, no two of which have the same ¤ or ¥coordinate using a minimum number of vertical and horizontal lines avoiding the points, so that each cell of the subdivision contains at most one point. We prove that this problem and some variants of it are NPcomplete. We give an approximation algorithm with ratio ¦ for the planar problem, and a ratio § approximation algorithm for the §dimensional variant, in which the points are to be separated using axisparallel hyperplanes. We reduce the problem to the rectangle stabbing problem studied by Gaur et al [5]. Their approximation algorithm uses LProunding. Our algorithm presents an alternative LProunding procedure which also works for the rectangle stabbing problem. We also discuss some dual problems suggested by the linear programs used to solve the separation problem. 1
Parameterized Complexity of Stabbing Rectangles and Squares in the Plane
, 2009
"... The NPcomplete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the ..."
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Cited by 6 (1 self)
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The NPcomplete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines. While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far—only its generalization to three or more dimensions was known to be W[1]hard. Giving two fixedparameter reductions, one from the W[1]complete problem Multicolored Clique and one to the W[1]complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]complete with respect to the parameter k, which in particular means that there is no hope for fixedparameter tractability with respect to the parameter k. Our reductions show also the W[1]completeness of the more general problem Set Cover on instances that “almost have the consecutiveones property”, that is, on instances whose matrix representation has at most two blocks of 1s per row. For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k · n O(1) time, we show that Rectangle Stabbing is fixedparameter tractable in the still NPhard case where both these restrictions apply.