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21
On the RedBlue Set Cover Problem
 In Proceedings of the 11th Annual ACMSIAM Symposium on Discrete Algorithms
, 2000
"... Given a finite set of "red" elements R, a finite set of "blue" elements B and a family S ` 2 R[B , the redblue set cover problem is to find a subfamily C ` S which covers all blue elements, but which covers the minimum possible number of red elements. We note that RedBlue Set Cover is closely r ..."
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Cited by 33 (0 self)
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Given a finite set of "red" elements R, a finite set of "blue" elements B and a family S ` 2 R[B , the redblue set cover problem is to find a subfamily C ` S which covers all blue elements, but which covers the minimum possible number of red elements. We note that RedBlue Set Cover is closely related to several combinatorial optimization problems studied earlier. These include the group Steiner problem, directed Steiner problem, minimum label path, minimum monotone satisfying assignment and symmetric label cover. From the equivalence of RedBlue Set Cover and MMSA3 it follows that, unless P=NP, even the restriction of RedBlue Set Cover where every set contains only one blue and two red elements cannot be approximated to within O(2 log 1\Gammaffi n ) , where ffi = 1= log log c n, for any constant c ! 1=2 (where n = S). We give integer programming formulations of the problem and use them to obtain a 2 p n approximation algorithm for the restricted case of RedBlue Set Cove...
Local search for the minimum label spanning tree problem with bounded color classes
, 2003
"... In the Minimum Label Spanning Tree problem ..."
Approximation algorithms and hardness results for labeled connectivity problems
 In 31st MFCS
, 2006
"... Abstract. Let G = (V, E) be a connected multigraph, whose edges are associated with labels specified by an integervalued function L: E → N. In addition, each label ℓ ∈ N to which at least one edge is mapped has a nonnegative cost c(ℓ). The minimum label spanning tree problem (MinLST) asks to find ..."
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Cited by 12 (4 self)
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Abstract. Let G = (V, E) be a connected multigraph, whose edges are associated with labels specified by an integervalued function L: E → N. In addition, each label ℓ ∈ N to which at least one edge is mapped has a nonnegative cost c(ℓ). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I ⊆ N such that the edge set {e ∈ E: L(e) ∈ I} forms a connected subgraph spanning all vertices. Similarly, in the minimum label st path problem (MinLP) the goal is to identify an st path minimizing the combined cost of its labels, where s and t are provided as part of the input. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP. As a secondary objective, we make a concentrated effort to relate the algorithmic methods utilized in approximating these problems to a number of wellknown techniques, originally studied in the context of integer covering. 1
Hyperrectanglebased discriminative data generalization and applications in data mining
, 2007
"... The ultimate goal of data mining is to extract knowledge from massive data. Knowledge is ideally represented as humancomprehensible patterns from which endusers can gain intuitions and insights. Axisparallel hyperrectangles provide interpretable generalizations for multidimensional data points ..."
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The ultimate goal of data mining is to extract knowledge from massive data. Knowledge is ideally represented as humancomprehensible patterns from which endusers can gain intuitions and insights. Axisparallel hyperrectangles provide interpretable generalizations for multidimensional data points with numerical attributes. In this dissertation, we study the fundamental problem of rectanglebased discriminative data generalization in the context of several useful data mining applications: cluster description, rule learning, and Nearest Rectangle classification. Clustering is one of the most important data mining tasks. However, most clustering methods output sets of points as clusters and do not generalize them into interpretable patterns. We perform a systematic study of cluster description, where we propose novel description formats leading to enhanced expressive power and introduce novel description problems specifying different tradeoffs between interpretability and accuracy. We also present efficient heuristic algorithms for the introduced problems in the proposed formats. Ifthen rules are
Approximation and Hardness Results for Label Cut and Related Problems
"... We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels ..."
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We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels of minimum cardinality (or minimum total weight), such that the removal of all edges with these labels disconnects s and t. We give the first nontrivial approximation and hardness results for the Label Cut problem. Firstly, we present an O ( √ m)approximation algorithm for the Label Cut problem, where m is the number of edges in the input graph. Secondly, we show that it is NPhard to approximate Label Cut within 2 log1−1 / log logc n n for any constant c < 1/2, where n is the input length of the problem. Thirdly, our techniques can be applied to other previously considered optimization problems. In particular we show that the Minimum Label Path problem has the same approximation hardness as that of Label Cut, simultaneously improving and unifying two known hardness results for this problem which were previously the best (but incomparable due to different complexity assumptions). 1
An Effective Genetic Algorithm for the MinimumLabel Spanning Tree Problem
"... Given a connected, undirected graph G with labeled edges, the minimumlabel spanning tree problem seeks a spanning tree on G to whose edges are attached the smallest possible number of labels. A greedy heuristic for this NPhard problem greedily chooses labels so as to reduce the number of component ..."
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Given a connected, undirected graph G with labeled edges, the minimumlabel spanning tree problem seeks a spanning tree on G to whose edges are attached the smallest possible number of labels. A greedy heuristic for this NPhard problem greedily chooses labels so as to reduce the number of components in the subgraphs they induce as quickly as possible. A genetic algorithm for the problem encodes candidate solutions as permutations of the labels; an initial segment of such a chromosome lists the labels that appear on the edges in the chromosome’s tree. Three versions of the GA apply generic or heuristic crossover and mutation operators and a local search step. In tests on 27 randomlygenerated instances of the minimumlabel spanning tree problem, versions of the GA that apply generic operators, with and without the local search step, perform less well than the greedy heuristic, but a version that applies the local search step and operators tailored to the problem returns solutions that require on average 10 % fewer labels than the heuristic’s.
The labeled maximum matching problem
 Computers and Operations Research
"... Given a graph G where a label is associated with each edge, we address the problem of looking for a maximum matching of G using the minimum number of different labels, namely the Labeled Maximum Matching Problem. It is a relatively new problem whose application is related to the timetabling problem ..."
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Cited by 3 (1 self)
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Given a graph G where a label is associated with each edge, we address the problem of looking for a maximum matching of G using the minimum number of different labels, namely the Labeled Maximum Matching Problem. It is a relatively new problem whose application is related to the timetabling problem [15]. We prove it is NPcomplete and present four different mathematical formulations. Moreover, we propose an exact algorithm based on a branchandbound approach to solve it. We evaluate the performance of our algorithm on a wide set of instances and compare our computational times with the ones required by CPLEX to solve the proposed mathematical formulations. Test results show the effectiveness of our procedure, that hugely outperforms the solver. Key words: PACS:
On Labeled Traveling Salesman Problems
"... Abstract. We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum (or minimum) number of colors. We derive results regarding hardness of approximation, and analyze approximation algorithms for both ve ..."
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Abstract. We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum (or minimum) number of colors. We derive results regarding hardness of approximation, and analyze approximation algorithms for both versions of the problem. For the maximization version we give a 1approximation algorithm and show that it is APX2 hard. For the minimization version, we show that it is not approximable within n 1−ǫ for every ǫ> 0. When every color appears in the graph at most r times and r is an increasing function of n the problem is not O(r 1−ǫ)approximable. For fixed constant r we analyze a polynomialtime (r+Hr)/2approximation algorithm (Hr is the rth harmonic number), and prove APXhardness. Analysis of the studied algorithms is shown to be tight. 1
The Parameterized Complexity of Some Minimum Label Problems
"... We study the parameterized complexity of several minimum label graph problems, in which we are given an undirected graph whose edges are labeled, and a property Π, and we are asked to find a subset of edges satisfying property Π that uses the minimum number of labels. These problems have a lot of ap ..."
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We study the parameterized complexity of several minimum label graph problems, in which we are given an undirected graph whose edges are labeled, and a property Π, and we are asked to find a subset of edges satisfying property Π that uses the minimum number of labels. These problems have a lot of applications in networking. We show that all the problems under consideration are W[2]hard when parameterized by the number of used labels, and that they remain W[2]hard even on graphs whose pathwidth is bounded above by a small constant. On the positive side, we prove that most of these problems are FPT when parameterized by the solution size, that is, the size of the sought edge set. For example, we show that computing a maximum matching or an edge dominating set that uses the minimum number of labels, is FPT when parameterized by the solution size. Proving that some of these problems are FPT is nontrivial, and requires interesting and elegant algorithmic methods that we develop in this paper. 1