## On the Red-Blue Set Cover Problem (2000)

Venue: | In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms |

Citations: | 34 - 0 self |

### BibTeX

@INPROCEEDINGS{Carr00onthe,

author = {Robert D. Carr and Srinivas Doddi and Goran Konjevod and Madhav Marathe},

title = {On the Red-Blue Set Cover Problem},

booktitle = {In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms},

year = {2000},

pages = {345--353}

}

### Years of Citing Articles

### OpenURL

### Abstract

Given a finite set of "red" elements R, a finite set of "blue" elements B and a family S ` 2 R[B , the red-blue 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 Red-Blue 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 Red-Blue Set Cover and MMSA3 it follows that, unless P=NP, even the restriction of Red-Blue 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 Red-Blue Set Cove...

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Citation Context ...r program. The only general way to solve a linear program with more than polynomially many constraints uses the ellipsoid method and requires a separation oracle (see Grotschel, Lov'asz and Schrijver =-=[15]-=-). In our case, the separation oracle needs to be able to solve weighted Hitting Set. The question that the oracle must answer is: Does there exist a blue element b 2 B and a hitting set for S b whose... |

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Citation Context ...e sets from S. 2 Related problems Red-Blue Set Cover contains the classical set cover and several known generalizations of this problem as its special cases. Set cover. The Set Cover problem (Johnson =-=[17]-=-) can be viewed as the Red-Blue Set Cover problem where each set S 2 S contains exactly one red element, and no red element is contained in more than one set. The goal is to cover all blue elements us... |

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Citation Context ...roblem of classifying (or predicting) by combining the classification results (or predictions) of subordinate classification algorithms also referred to as "experts" in the learning theory l=-=iterature [12, 5]-=-. 7 Acknowledgments and notes on related work During the time between submission and the current version several researchers have provided us pointers to recent results they have obtained on some of t... |

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Citation Context ...he resulting Red-Blue Set Cover instance is equivalent to the original Group Steiner Tree instance. A similar reduction can be used to transform any instance of Directed Steiner Tree (Charikar et al. =-=[6]-=-) into an instance of Red-Blue Set Cover. However, in general, the Red-Blue Set Cover instance will have exponentially many sets (since there may be exponentially many paths from the root to the termi... |

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Citation Context ...t Cover problem. Group Steiner and directed Steiner problems. Another special case of Red-Blue Set Cover is the (rooted, unweighted) group Steiner problem for trees studied by Garg, Konjevod and Ravi =-=[13]-=-. Let a tree G = (V; E) be given together with a family of subsets of vertices G = fg 1 ; : : :; gm g, called groups. Let one vertex v 2 V be marked as the root. We call all vertices belonging to some... |

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Citation Context ...tent (i.e. misclassifies the minimum number of red elements), using a conjunction of these given classifiers. 6.2 Information Retrieval. This example is a slight variation of the example discussed in =-=[12]-=-. A typical problem in this area consists of classifying a large corpus of documents. Such a classification might be related to grouping articles found during a web search (e.g. classifying news artic... |

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Citation Context ...problems and as corollaries obtain stronger non-approximability results for Red-Blue Set Cover and related problems. 3 Symmetric label cover. The label cover problem (see, for example, Arora and Lund =-=[4]-=-) was originally defined in order to make easier the use of the PCP theorem in proving approximation hardness of Set Cover and related problems. Two commonly used variants are called Label Covermax an... |

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Citation Context ...ffi = 1= log log c n, for any constant c ! 1=2. We also obtain a slightly weaker result via a polynomial-time approximation-preserving reduction from Symmetric Label Cover, for which Dodis and Khanna =-=[9]-=- claim nonapproximability to within 2 log 1\Gammaffl n for any ffl ? 0, unless NP`DTIME(n polylog(n) ). On the other hand, we provide a polynomial-time 2 p n approximation algorithm for the restrictio... |

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Citation Context ...on jSjsn 1=4 , where n = maxfjAj; jBj; jUjg. This implies the theorem. Minimum monotone satisfying assignment (MMSA). This problem was independently introduced by Alekhnovich, Buss, Moran and Pitassi =-=[3]-=- and by Goldwasser and Motwani [14] (under the name AND-OR scheduling). The problem MMSA k is specified by a monotone formula (that is, a formula that uses no negations) of depth k (the formula has k ... |

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Citation Context ... that each element of the ground set is covered at least k times by the family. Efficient approximation algorithms for multicover problems were described by Dobson [8] and by Rajagopalan and Vazirani =-=[20]-=-. The approximation guarantees achieved are log kn and log n, respectively, where n is the number of elements in the ground set. In order to write the constraints for this IP, we denote the family of ... |

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Citation Context ... a directed graph). Thus our reduction is only polynomial in the case where there are only polynomially many root-terminal paths in the directed Steiner instance. Minimum color path. Krumke and Wirth =-=[18]-=- discuss the approximability of the following problem: given a graph G = (V; E) with a function c : E ! C, where C is a finite set of colors, find a spanning tree of G whose edges are labeled with the... |

18 |
private communication
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Citation Context ...alent to Red-Blue Set Cover, the same holds for Red-Blue Set Cover. Another way of showing inapproximability of RedBlue Set Cover is to use two-prover proof systems and was pointed out to us by Feige =-=[11]-=-. 4 Linear programming relaxations Consider the following natural formulation of the RedBlue Set Cover problem. min X r y r y rsxS 8(r; S) : r 2 S X S3b xSs1 8b 2 B x 2 f0; 1g jSj : (4.1) This formula... |

11 | Intractability of assembly sequencing: Unit disks in the plane
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Citation Context ...jBj; jUjg. This implies the theorem. Minimum monotone satisfying assignment (MMSA). This problem was independently introduced by Alekhnovich, Buss, Moran and Pitassi [3] and by Goldwasser and Motwani =-=[14]-=- (under the name AND-OR scheduling). The problem MMSA k is specified by a monotone formula (that is, a formula that uses no negations) of depth k (the formula has k levels of alternating AND and OR ga... |

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(Show Context)
Citation Context ...p Steiner problem, is equivalent to the third level of the minimum monotone satisfying assignment hierarchy, and is a special case of the minimum color path problem. Using a result of Dinur and Safra =-=[7]-=- on the hardness of approximation of MMSA 3 , we find that unless P=NP, the special case of Red-Blue Set Cover where every set contains only one blue and two red elements cannot be approximated to wit... |

2 |
The minimum label path problem. Unpublished manuscript, Los Alamos National Laboratory
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Citation Context ...ase where only one pair of vertices fs; tg ` V is given and the goal is to find an s-t path that uses fewest colors contains Red-Blue Set Cover as a special case. This problem was first considered in =-=[16]-=-. The reduction is as follows: given an instance of Red-Blue Set Cover we construct an equivalent instance of Minimum Color Path. First we create one color for each red element. Then we define the ver... |

1 |
The hardness of approximating spanner problems. Unpublished manuscript
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Citation Context ...rk During the time between submission and the current version several researchers have provided us pointers to recent results they have obtained on some of the problems in this paper. Elkin and Peleg =-=[10]-=- have independently shown the hardness of Red-Blue Set Cover by a reduction from Symmetric Label Cover, obtaining a similar hardness result as claimed here. Their main focus was on the complexity of a... |

1 |
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Citation Context ...over by a reduction from Symmetric Label Cover, obtaining a similar hardness result as claimed here. Their main focus was on the complexity of approximating spanner problems. Panconesi and Srinivasan =-=[19]-=- have also obtained the hardness of the minimum-label path problem via a reduction from the label cover problem. We thank them for communicating these results to us. Venkatesh Guruswami pointed out wh... |

1 |
Private communication
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Citation Context ... a combined classifier. 6.3 Relationship to General Classification and Learning. Red-Blue Set Cover can be thought of as a restricted form of a more general machine learning or classification problem =-=[21]-=-. In this setting sets R and B can be regarded as sets of elements, each of which has an m-dimensional vector of Boolean values. A component i can be considered to be a Boolean variable x i . A compon... |