## Greedy in Approximation Algorithms (2006)

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Venue: | PROC. OF ESA |

Citations: | 16 - 1 self |

### BibTeX

@INPROCEEDINGS{Mestre06greedyin,

author = {Julián Mestre},

title = {Greedy in Approximation Algorithms},

booktitle = {PROC. OF ESA},

year = {2006},

publisher = {}

}

### OpenURL

### Abstract

The objective of this paper is to characterize classes of problems for which a greedy algorithm finds solutions provably close to optimum. To that end, we introduce the notion of k-extendible systems, a natural generalization of matroids, and show that a greedy algorithm is a 1-factor approximation for these systems. Many seemly unrelated k problems fit in our framework, e.g.: b-matching, maximum profit scheduling and maximum asymmetric TSP. In the second half of the paper we focus on the maximum weight b-matching problem. The problem forms a 2-extendible system, so greedy gives us a 1-factor solution which runs in 2 O(m log n) time. We improve this by providing two linear time approximation algorithms for the problem: a 1 2-factor algorithm that runs in O(bm) time, and a `2 3 − ǫ ´-factor algorithm which runs in expected O ` bm log 1 ´ time.

### Citations

502 |
Matroid Theory
- Oxley
- 1992
(Show Context)
Citation Context ...on of the implication, i.e., if Greedy finds an optimal solution for any weight function defined on the elements then the problem must have the matroid property. A rich theory of matroids exists, see =-=[24, 20]-=- for a thorough treatment of the subject. Many generalizations along two main directions have been proposed. One approach is to define a more general class of problems. Greedy no longer works, therefo... |

149 | A course in combinatorial optimization
- Schrijver
- 2004
(Show Context)
Citation Context ...on of the implication, i.e., if Greedy finds an optimal solution for any weight function defined on the elements then the problem must have the matroid property. A rich theory of matroids exists, see =-=[24, 20]-=- for a thorough treatment of the subject. Many generalizations along two main directions have been proposed. One approach is to define a more general class of problems. Greedy no longer works, therefo... |

125 | A unified approach to approximating resource allocation and schedualing
- Bar-Noy, Bar-Yehuda, et al.
(Show Context)
Citation Context ...a single machine. Each job i has release time ri, deadline di, and profit wi, all positive integers. Every job takes the same amount of time L ∈ Z + to process. (See [4, 9] for an exact algorithm and =-=[5]-=- for a 2-approximation algorithm when the job lengths are arbitrary.) Our objective is to find a non-preemptive schedule that maximizes the weight of the jobs done on time. A job i is done on time if ... |

105 |
Matroids and the greedy algorithm
- Edmonds
- 1971
(Show Context)
Citation Context ...on of the notion of independence from linear algebra and graph theory. Rado [23] showed that if a given problem has the matroid property then Greedy always finds an optimal solution. In turn, Edmonds =-=[13]-=- proved the other direction of the implication, i.e., if Greedy finds an optimal solution for any weight function defined on the elements then the problem must have the matroid property. A rich theory... |

97 |
On the abstract properties of linear dependence
- Whitney
- 1935
(Show Context)
Citation Context ...eserves independence. A matroid is a subset system (E,L) for which the following property holds: ∀A,B ∈ L and |A| < |B| then ∃z ∈ B \ A such that A + z 1 ∈ L Matroids were first introduced by Whitney =-=[26]-=- as an abstraction of the notion of independence from linear algebra and graph theory. Rado [23] showed that if a given problem has the matroid property then Greedy always finds an optimal solution. I... |

69 |
Minimum partition of a matroid into independent subsets
- Edmonds
- 1965
(Show Context)
Citation Context ...the University of Maryland Dean’s Dissertation Fellowship. 1 The notation A + z means A ∪ {z}, likewise A − z means A \ {z}.sbe designed; examples of this are greedoids [17], two-matroid intersection =-=[12]-=-, and matroid matching [19]. Another approach is to study structures where Greedy finds optimal solutions for some, but not all weight functions; symmetric matroids [8], sympletic matroids [7] and the... |

54 |
Greedy algorithm and symmetric matroids
- Bouchet
- 1987
(Show Context)
Citation Context ...[17], two-matroid intersection [12], and matroid matching [19]. Another approach is to study structures where Greedy finds optimal solutions for some, but not all weight functions; symmetric matroids =-=[8]-=-, sympletic matroids [7] and the work of Vince [25] are along these lines. Although different in nature, both approaches have the same objective in mind: exact solutions. In this paper we study Greedy... |

43 |
An ecient reduction technique for degree-constrained subgraph and bidirected network problems
- Gabow
- 1983
(Show Context)
Citation Context ...l time algorithms exist to solve the problem optimally: A maximum size b-matching can be found in O(nm log n) time and maximum weight in O �� b(v)min(mlog n,n2 ) � time; both results are due to Gabow =-=[14]-=-. Greedy on the other hand produces approximate solutions but has the advantage of being simple and much faster, running in just O(mlog n) time. This time savings can be further improved. For instance... |

38 | Incremental) priority algorithms
- Borodin, Nielsen, et al.
(Show Context)
Citation Context ...s to characterize those problems for which a simple greedy strategy produces nearly optimal solutions and to better understand its shortcomings. Along these lines is the recent work by Borodin et al. =-=[6]-=-, who introduced the paradigm of priority algorithms, a formal class of algorithms that captures most greedy-like algorithms. Lower bounds on the approximation ratio any priority algorithm can achieve... |

38 | Linear time 1/2-approximation algorithm for maximum weighted matching in general graphs,” ser
- Preis
- 2001
(Show Context)
Citation Context ...vantage of being simple and much faster, running in just O(mlog n) time. This time savings can be further improved. For instance, for maximum weight matching (the case where b(v) = 1 for all v) Preis =-=[22]-=- proposed a 1 2-approximation algorithm which runs in linear time. Drake et al. [11] designed an alternative simpler algorithm that greedily finds disjoint heavy paths and keeps the best of the two ma... |

35 | Polynomial time algorithms for minimizing the weighted number of late jobs on a single machine when processing times are equal
- Baptiste
- 1999
(Show Context)
Citation Context ...ing: We are to schedule n jobs on a single machine. Each job i has release time ri, deadline di, and profit wi, all positive integers. Every job takes the same amount of time L ∈ Z + to process. (See =-=[4, 9]-=- for an exact algorithm and [5] for a 2-approximation algorithm when the job lengths are arbitrary.) Our objective is to find a non-preemptive schedule that maximizes the weight of the jobs done on ti... |

35 | A simple approximation algorithm for the weighted matching problem
- DRAKE, HOUGARDY
- 2003
(Show Context)
Citation Context ...avings can be further improved. For instance, for maximum weight matching (the case where b(v) = 1 for all v) Preis [22] proposed a 1 2-approximation algorithm which runs in linear time. Drake et al. =-=[11]-=- designed an alternative simpler algorithm that greedily finds disjoint heavy paths and keeps the best of the two matchings defined on the path; the same authors in later work [10] designed an algorit... |

30 | On local search for weighted k-set packing
- Arkin, Hassin
- 1998
(Show Context)
Citation Context ...e randomized and deterministic algorithms with the same approximation guarantee of 2 1 3 − ǫ which run in O(mlog ǫ ) time. We note that a better approximation ratio can be obtained using local search =-=[2]-=- or the limited-backtrack greedy scheme of Arora et at [3], albeit at a very high running time. The challenge here is to get a fast algorithm with a good approximation guarantee. In the second half of... |

24 | Symplectic matroids
- Borovik, Gelfand, et al.
- 1998
(Show Context)
Citation Context ...ection [12], and matroid matching [19]. Another approach is to study structures where Greedy finds optimal solutions for some, but not all weight functions; symmetric matroids [8], sympletic matroids =-=[7]-=- and the work of Vince [25] are along these lines. Although different in nature, both approaches have the same objective in mind: exact solutions. In this paper we study Greedy from the point of view ... |

22 | On the power of priority algorithms for the facility location and set cover
- Angelopoulos, Borodin
- 2002
(Show Context)
Citation Context ...rithms that captures most greedy-like algorithms. Lower bounds on the approximation ratio any priority algorithm can achieve were derived for scheduling [6], set cover, and facility location problems =-=[1]-=-. In particular, our framework explains why Greedy produces 1 2-approximate solutions for b-matching. Given a graph G = (V,E) with n vertices and m edges and degree constraints b : V → N for the verti... |

19 | Improved linear time approximation algorithms for weighted matchings
- Drake, Hougardy
- 2003
(Show Context)
Citation Context ... time. Drake et al. [11] designed an alternative simpler algorithm that greedily finds disjoint heavy paths and keeps the best of the two matchings defined on the path; the same authors in later work =-=[10]-=- designed an algorithm with an approximation factor of 2 3 − ǫ which runs in O � � m ǫ time. Finally, Pettie and Sanders [21] gave randomized and deterministic algorithms with the same approximation g... |

15 |
A Note on Scheduling Equal-Length Jobs to Maximize Throughput
- Chrobak, Dürr, et al.
- 2006
(Show Context)
Citation Context ...ing: We are to schedule n jobs on a single machine. Each job i has release time ri, deadline di, and profit wi, all positive integers. Every job takes the same amount of time L ∈ Z + to process. (See =-=[4, 9]-=- for an exact algorithm and [5] for a 2-approximation algorithm when the job lengths are arbitrary.) Our objective is to find a non-preemptive schedule that maximizes the weight of the jobs done on ti... |

14 |
An analysis of the greedy algorithm for independence systems, in Algorithmic Aspects of Combinatorics
- Korte, Hausmann
- 1978
(Show Context)
Citation Context ... introduction of k-extendible systems, a natural generalization of matroids. We show that Greedy is a 1 k-factor approximation for k-extendible systems. Given a subset system (E,L), Korte and Hausman =-=[16]-=- showed that for the maximization problem defined by (E,L), Greedy achieves its worst approximation ratio on 0-1 weight functions. Consider the 0-1 function wA defined as wA(x) = 1 for x ∈ A and 0 oth... |

10 |
The matroid matching problem
- Lovász
- 1981
(Show Context)
Citation Context ...Dean’s Dissertation Fellowship. 1 The notation A + z means A ∪ {z}, likewise A − z means A \ {z}.sbe designed; examples of this are greedoids [17], two-matroid intersection [12], and matroid matching =-=[19]-=-. Another approach is to study structures where Greedy finds optimal solutions for some, but not all weight functions; symmetric matroids [8], sympletic matroids [7] and the work of Vince [25] are alo... |

8 |
A simpler linear time 2/3-ǫ approximation for maximum weight matching
- Pettie, Sanders
(Show Context)
Citation Context ...t of the two matchings defined on the path; the same authors in later work [10] designed an algorithm with an approximation factor of 2 3 − ǫ which runs in O � � m ǫ time. Finally, Pettie and Sanders =-=[21]-=- gave randomized and deterministic algorithms with the same approximation guarantee of 2 1 3 − ǫ which run in O(mlog ǫ ) time. We note that a better approximation ratio can be obtained using local sea... |

8 |
A theorem on Independence Relations
- Rado
- 1942
(Show Context)
Citation Context ... ∀A,B ∈ L and |A| < |B| then ∃z ∈ B \ A such that A + z 1 ∈ L Matroids were first introduced by Whitney [26] as an abstraction of the notion of independence from linear algebra and graph theory. Rado =-=[23]-=- showed that if a given problem has the matroid property then Greedy always finds an optimal solution. In turn, Edmonds [13] proved the other direction of the implication, i.e., if Greedy finds an opt... |

7 |
Greedoidsa structural framework for the greedy algorithm
- Korte, Lovasz
- 1984
(Show Context)
Citation Context ...01-05413 and CCF-04-30650, and the University of Maryland Dean’s Dissertation Fellowship. 1 The notation A + z means A ∪ {z}, likewise A − z means A \ {z}.sbe designed; examples of this are greedoids =-=[17]-=-, two-matroid intersection [12], and matroid matching [19]. Another approach is to study structures where Greedy finds optimal solutions for some, but not all weight functions; symmetric matroids [8],... |

4 |
A limited-backtrack greedy schema for approximation algorithms
- Arora, Santosh, et al.
- 1994
(Show Context)
Citation Context ...proximation guarantee of 2 1 3 − ǫ which run in O(mlog ǫ ) time. We note that a better approximation ratio can be obtained using local search [2] or the limited-backtrack greedy scheme of Arora et at =-=[3]-=-, albeit at a very high running time. The challenge here is to get a fast algorithm with a good approximation guarantee. In the second half of the paper we explore this tradeoff for b-matching and pro... |

3 |
Approximating asymmetric maximum TSP
- Lewenstein, Sviridenko
- 2003
(Show Context)
Citation Context ...complete directed graph with non-negative weights and we must find a maximum weight tour that visits every city exactly once. The problem is NP-hard; the best known approximation factor for it is 5 8 =-=[18]-=-. The elements of our subset system are the directed edges of the complete graph; a set is independent if its edges form a collection of vertex disjoint paths or a cycle that visits every vertex exact... |

2 |
The greedy travelling salesman's problem
- JENKYNS
- 1979
(Show Context)
Citation Context ...r subset system are the directed edges of the complete graph; a set is independent if its edges form a collection of vertex disjoint paths or a cycle that visits every vertex exactly once. Theorem 6 (=-=[15]-=-). The subset system for maximum ATSP is 3-extendible. Proof. As usual let C + (x,y) be independent, and D be an extension of C. First remove from D the edges (if any) out of x and into y, these are c... |

2 | A framework for the greedy algorithm
- Vince
- 2002
(Show Context)
Citation Context ...atching [19]. Another approach is to study structures where Greedy finds optimal solutions for some, but not all weight functions; symmetric matroids [8], sympletic matroids [7] and the work of Vince =-=[25]-=- are along these lines. Although different in nature, both approaches have the same objective in mind: exact solutions. In this paper we study Greedy from the point of view of approximation algorithms... |