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## A unified approach to approximating partial covering problems (2006)

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Venue: | IN PROCEEDINGS OF THE 14TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS |

Citations: | 23 - 2 self |

### Citations

1365 | Integer and Combinatorial Optimization - Nemhauser, Wolsey - 1988 |

821 |
Approximation algorithms for combinatorial problems
- Johnson
- 1974
(Show Context)
Citation Context ...version of this paper [23]. i=1 i=1 ⊓⊔s3.1 Set cover, in terms of ∆ Kearns [21, Thm. 5.15] seems to have been the first to study the partial cover problem, showing that the greedy set cover algorithm =-=[20, 25]-=- can be adapted to provide an approximation factor of 2H(|U|) + 3. A slightly different algorithm was suggested by Slavík [32], who obtained a factor of H(min{∆, k}), where ∆ is the maximum size of a ... |

767 | A threshold of ln n for approximating set cover
- Feige
- 1998
(Show Context)
Citation Context ... remark that the partial cover problem contains set cover as a special case, implying that it cannot be approximated within a factor of (1 − ɛ) ln |U| for any ɛ > 0, unless NP ⊂ TIME(n O(log log n) ) =-=[10]-=-. To the best of our knowledge, the greedy heuristic has not been studied in the context of generalized partial cover, and in fact no algorithm is currently known for this problem. In the full version... |

365 | Primal-Dual Approximation Algorithms for Metric Facility Location and k-Median Problems
- Jain, Vazirani
(Show Context)
Citation Context ...lify the presentation, it is convenient to state our approach in terms of the Lagrangian relaxation technique, which was first utilized in the context of approximation algorithms by Jain and Vazirani =-=[24]-=-. Here is a rough outline of how the proof of Theorem 1 will proceed. We begin by formulating the generalized partial cover problem as an integer program. Next, we dualize the complicating constraint ... |

317 |
On the ratio of optimal integral and fractional covers
- Lovász
- 1975
(Show Context)
Citation Context ...version of this paper [23]. i=1 i=1 ⊓⊔s3.1 Set cover, in terms of ∆ Kearns [21, Thm. 5.15] seems to have been the first to study the partial cover problem, showing that the greedy set cover algorithm =-=[20, 25]-=- can be adapted to provide an approximation factor of 2H(|U|) + 3. A slightly different algorithm was suggested by Slavík [32], who obtained a factor of H(min{∆, k}), where ∆ is the maximum size of a ... |

311 | On the power of unique 2-prover 1-round games
- Khot
- 2002
(Show Context)
Citation Context ...mation using the region growing method, a hardness result of Ω(log log n) was given by Chawla, Krauthgamer, Kumar, Rabani and Sivakumar [5], assuming a stronger version of the Unique Games Conjecture =-=[22]-=-. Based on Räcke’s hierarchical decomposition method [31], Alon, Awerbuch, Azar, Buchbinder and Naor [1] have shown how to simulate multicuts in general graphs by multicuts in the corresponding decomp... |

268 | Combinatorial Optimization - Cook - 1998 |

156 | Approximating max-flow min-(multi)cut theorems and their applications
- Garg, Yannakakis, et al.
- 1996
(Show Context)
Citation Context ... factor of 8 3 + ɛ, for any fixed ɛ > 0. General graphs. When the input graph is no longer restricted to be a tree, the multicut problem becomes significantly harder to approximate. While Garg et al. =-=[14]-=- devised an O(log k)-approximation using the region growing method, a hardness result of Ω(log log n) was given by Chawla, Krauthgamer, Kumar, Rabani and Sivakumar [5], assuming a stronger version of ... |

146 |
Primal-dual approximation algorithms for integral flow and multicuts in trees
- Garg, Yannakakis, et al.
- 1997
(Show Context)
Citation Context ...ing a special case of set cover: The elements to cover are the input pairs, and an edge e ∈ E covers {si, ti} if it resides on the unique path in T connecting si and ti. Garg, Vazirani and Yannakakis =-=[15]-=- presented a primal-dual 2-approximation for this problem, which was also shown to be at least as hard to approximate as vertex cover. The corresponding partial cover problem, in which we are required... |

108 | Minimizing congestion in general networks
- Räcke
- 2002
(Show Context)
Citation Context ... of Ω(log log n) was given by Chawla, Krauthgamer, Kumar, Rabani and Sivakumar [5], assuming a stronger version of the Unique Games Conjecture [22]. Based on Räcke’s hierarchical decomposition method =-=[31]-=-, Alon, Awerbuch, Azar, Buchbinder and Naor [1] have shown how to simulate multicuts in general graphs by multicuts in the corresponding decomposition tree. As observed by Golovin et al. [17], this me... |

104 |
Set partitioning: a survey
- Balas, Padberg
- 1976
(Show Context)
Citation Context ...ngs may appear to be very simple at first glance, they still capture computational tasks of great theoretical and practical importance, as the reader may verify by consulting directly related surveys =-=[2, 13, 18, 28]-=- and the references therein. In the present paper we focus our attention on the generalized partial cover problem, whose input consists of a ground set of elements U and a family S of subsets of U. In... |

99 | D.: On the hardness of approximating multicut and sparsest-cut
- Chawla, Krauthgamer, et al.
- 2006
(Show Context)
Citation Context ... approximate. While Garg et al. [14] devised an O(log k)-approximation using the region growing method, a hardness result of Ω(log log n) was given by Chawla, Krauthgamer, Kumar, Rabani and Sivakumar =-=[5]-=-, assuming a stronger version of the Unique Games Conjecture [22]. Based on Räcke’s hierarchical decomposition method [31], Alon, Awerbuch, Azar, Buchbinder and Naor [1] have shown how to simulate mul... |

87 | Algorithms for facility location problems with outliers - Charikar, Khuller, et al. - 1985 |

81 | Matching: A well-solved class of integer linear programs
- Edmonds, Johnson
- 1970
(Show Context)
Citation Context ...nt to the special case of set cover in which each subset consists of exactly two elements. We note that edge cover is actually a matching problem in disguise, implying its polynomial time solvability =-=[8, 26]-=-. Plesník [30] proved that unit-profit partial edge cover, which is also known as the k-edge cover problem, can be solved to optimality by reducing it to standard edge cover. However, when arbitrary p... |

71 | The Computational Complexity of Machine Learning - Kearns - 1990 |

70 |
Approximating covering and packing problems: set cover, vertex cover, independent set, and related problems
- Hochbaum
- 1997
(Show Context)
Citation Context ...ngs may appear to be very simple at first glance, they still capture computational tasks of great theoretical and practical importance, as the reader may verify by consulting directly related surveys =-=[2, 13, 18, 28]-=- and the references therein. In the present paper we focus our attention on the generalized partial cover problem, whose input consists of a ground set of elements U and a family S of subsets of U. In... |

67 | Saving an epsilon: a 2-approximation for the k-mst problem in graphs - Garg |

58 | Approximation Algorithms for Partial Covering Problems.JournalofAlgorithms 53(1
- Gandhi, Khuller, et al.
- 2004
(Show Context)
Citation Context ...sed an algorithm for generalized partial cover whose approximation guarantee is f , a result that was independently obtained by Fujito [11] using a primadual algorithm. Gandhi, Khuller and Srinivasan =-=[12]-=- achieved a similar ratio for partial cover. In the full version of this paper [23], we present a combinatorial f -LMP algorithm for the prize-collecting set cover problem, showing that every weighted... |

57 |
A linear-time approximation algorithm for the weighted vertex cover problem
- Bar-Yehuda, Even
- 1981
(Show Context)
Citation Context ...≥ 1 ∀ e ∈ U S∈Se xS, ze ≥ 0 ∀ S ∈ S, e ∈ U maximize � e∈U ye subject to � ye ≤ c(S) ∀ S ∈ S e∈S 0 ≤ ye ≤ π(e) ∀ e ∈ U Our method is based on modifying the primal-dual algorithm of Bar-Yehuda and Even =-=[4]-=-. It is not difficult to verify that although the latter algorithm constructs a solution whose cost is at most (f + 1)OPT(P), it does not satisfy the LMP property. For this purpose, we employ an addit... |

51 | Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation
- Chudak, Roughgarden, et al.
(Show Context)
Citation Context ...s dependence appears to be the primary factor limiting the employment of Lagrangian relaxations in most problems of interest. The latter drawback was pointed out by Chudak, Roughgarden and Williamson =-=[6]-=-, who asked whether it is possible to devise more general variants of the Lagrangian relaxation framework that apply to a broader class of problems. We answer this question in the affirmative, by deve... |

45 | The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema - Hajiaghayi, Jain - 2006 |

42 | A general approach to online network optimization problems
- Alon, Awerbuch, et al.
(Show Context)
Citation Context ...r, Kumar, Rabani and Sivakumar [5], assuming a stronger version of the Unique Games Conjecture [22]. Based on Räcke’s hierarchical decomposition method [31], Alon, Awerbuch, Azar, Buchbinder and Naor =-=[1]-=- have shown how to simulate multicuts in general graphs by multicuts in the corresponding decomposition tree. As observed by Golovin et al. [17], this method extends to approximate the partial multicu... |

26 |
Using homogeneous weights for approximating the partial cover problem
- Bar-Yehuda
(Show Context)
Citation Context ...urroughs [4] and Hochbaum [19] in the context of partial vertex cover, is approximating partial cover in terms of f , the maximum frequency of any element. Based on the local-ratio method, Bar-Yehuda =-=[3]-=- devised an algorithm for generalized partial cover whose approximation guarantee is f , a result that was independently obtained by Fujito [11] using a primadual algorithm. Gandhi, Khuller and Sriniv... |

25 |
Improved performance of the greedy algorithm for partial
- Slavík
- 1997
(Show Context)
Citation Context ...he partial cover problem, showing that the greedy set cover algorithm [20, 25] can be adapted to provide an approximation factor of 2H(|U|) + 3. A slightly different algorithm was suggested by Slavík =-=[32]-=-, who obtained a factor of H(min{∆, k}), where ∆ is the maximum size of a set in S and k is the coverage requirement. We remark that the partial cover problem contains set cover as a special case, imp... |

24 |
Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem
- Gaur, Ibaraki, et al.
(Show Context)
Citation Context ... instance, the first algorithm that approximates partial rectangle stabbing in Rd , noting that the resulting factor of ( 4 3 + ɛ)d nearly matches the dapproximation of Gaur, Ibaraki and Krishnamurti =-=[16]-=- for the full coverage version of this problem. In addition, we obtain an alternative, albeit non-combinatorial, ( 4 3 + ɛ) f - approximation for partial set cover with maximum element frequency f .s3... |

23 | Massaging a linear programming solution to give a 2approximation for a generalization of the vertex cover problem
- Bshouty, Burroughs
- 1998
(Show Context)
Citation Context ...2 Set cover, in terms of f Let fe be the number of sets in S that contain the element e ∈ U; fe is also known as the frequency of e. A recent line of work, that was initiated by Bshouty and Burroughs =-=[4]-=- and Hochbaum [19] in the context of partial vertex cover, is approximating partial cover in terms of f , the maximum frequency of any element. Based on the local-ratio method, Bar-Yehuda [3] devised ... |

22 |
Covering, packing and knapsack problems
- Padberg
- 1982
(Show Context)
Citation Context ...ngs may appear to be very simple at first glance, they still capture computational tasks of great theoretical and practical importance, as the reader may verify by consulting directly related surveys =-=[2, 13, 18, 28]-=- and the references therein. In the present paper we focus our attention on the generalized partial cover problem, whose input consists of a ground set of elements U and a family S of subsets of U. In... |

19 | M.: Approximating the k-multicut problem
- Golovin, Nagarajan, et al.
- 2006
(Show Context)
Citation Context ... S3 by augmenting S2 with a carefully chosen subset of S1. The cost of this subset is bounded by extending the arguments used by Levin and Segev [24] and independently by Golovin, Nagarajan and Singh =-=[17]-=- for approximating the k-multicut problem. Finally, we establish Theorem 1 by proving that the cost of the cheaper of S1 and S3 is at most ( 4 3 + ɛ)r times the cost of an optimal solution. Designing ... |

15 |
The t-vertex cover problem: Extending the half integrality framework with budget constraints
- Hochbaum
- 1998
(Show Context)
Citation Context ...erms of f Let fe be the number of sets in S that contain the element e ∈ U; fe is also known as the frequency of e. A recent line of work, that was initiated by Bshouty and Burroughs [4] and Hochbaum =-=[19]-=- in the context of partial vertex cover, is approximating partial cover in terms of f , the maximum frequency of any element. Based on the local-ratio method, Bar-Yehuda [3] devised an algorithm for g... |

8 | Partial multicuts in trees
- Levin, Segev
- 2006
(Show Context)
Citation Context ...least as hard to approximate as vertex cover. The corresponding partial cover problem, in which we are required to disconnect a specified number of pairs, has recently been studied by Levin and Segev =-=[24]-=- and independently by Golovin et al. [17], who achieved an approximation guarantee of 8 3 +ɛ, for any fixed ɛ > 0. Since the former authors provide a 2-LMP algorithm for the prizecollecting multicut p... |

7 | An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree
- Nagamochi
(Show Context)
Citation Context ...neral tree augmentation problem, and produce solutions whose cost is within factor 2 of optimum. We refer the reader to a short survey of these results [9, Sec. 1]. For the unweighted case, Nagamochi =-=[27]-=- proposed a (1.875 + ɛ)-approximation for any fixed ɛ > 0, a ratio that was later improved to 3 2 by Even, Feldman, Kortsarz and Nutov [9].sIn the generalized partial laminar cover problem, each Vi ∈ ... |

6 | A 3/2-approximation algorithm for augmenting the edge-connectivity of a graph from 1 to 2 using a subset of a given edge set
- Even, Feldman, et al.
- 2001
(Show Context)
Citation Context ...hese results [9, Sec. 1]. For the unweighted case, Nagamochi [27] proposed a (1.875 + ɛ)-approximation for any fixed ɛ > 0, a ratio that was later improved to 3 2 by Even, Feldman, Kortsarz and Nutov =-=[9]-=-.sIn the generalized partial laminar cover problem, each Vi ∈ F is associated with a profit p(Vi). The goal is to identify a minimum cost set of edges E ′ ⊆ E such that the overall profit of the sets ... |

6 |
Optimal Set Covering: A Survey
- Garfinkel, Nemhauser
- 1972
(Show Context)
Citation Context |

6 | Polyhedral techniques for graphic covering problems - Parekh - 2002 |

6 |
Constrained weighted matchings and edge coverings in graphs
- Plesńık
- 1999
(Show Context)
Citation Context ... case of set cover in which each subset consists of exactly two elements. We note that edge cover is actually a matching problem in disguise, implying its polynomial time solvability [8, 26]. Plesník =-=[30]-=- proved that unit-profit partial edge cover, which is also known as the k-edge cover problem, can be solved to optimality by reducing it to standard edge cover. However, when arbitrary profits are all... |

6 | Path hitting in acyclic graphs
- Parekh, Segev
(Show Context)
Citation Context ...d is π(Vi). A slightly different view. We begin by demonstrating that laminar cover can be transformed into an instance of the path hitting problem, that has recently been studied by Parekh and Segev =-=[35]-=-. In the latter, we are given a family of demand paths D and a family of hitting paths H in a common undirected tree, where each path p ∈ H has a non-negative cost. When p ∈ H and q ∈ D share at least... |

5 |
On approximation of the submodular set cover problem
- Fujito
- 1999
(Show Context)
Citation Context ...any element. Based on the local-ratio method, Bar-Yehuda [3] devised an algorithm for generalized partial cover whose approximation guarantee is f , a result that was independently obtained by Fujito =-=[11]-=- using a primadual algorithm. Gandhi, Khuller and Srinivasan [12] achieved a similar ratio for partial cover. In the full version of this paper [23], we present a combinatorial f -LMP algorithm for th... |

2 |
A 1-matching blossom type algorithm for edge covering problems
- Murty, Perin
- 1982
(Show Context)
Citation Context ...nt to the special case of set cover in which each subset consists of exactly two elements. We note that edge cover is actually a matching problem in disguise, implying its polynomial time solvability =-=[8, 26]-=-. Plesník [30] proved that unit-profit partial edge cover, which is also known as the k-edge cover problem, can be solved to optimality by reducing it to standard edge cover. However, when arbitrary p... |