### Citations

481 |
The ellipsoid method and its consequences in combinatorial optimization,”
- Grotschel, Lovasz, et al.
- 1981
(Show Context)
Citation Context ...∩ Vi]. Since Gi has vertex clique cover size 2, the complement of Gi is a bipartite graph. Since finding weighted maximum independent set in a bipartite graph can be determined in polynomial time [27]=-=[33]-=-, weighted maximum clique in Gi can be computed in polynomial time. We compute weighted maximum clique for each Gi, and the largest one is weighted maximum clique for G. This is because for any weight... |

370 |
Approximation algorithms for NP-complete problems on planar graphs
- Baker
- 1994
(Show Context)
Citation Context ...se of geometric intersection graphs (including disk graphs) and give PTASs for weighted maximum independent set and weighted minimum vertex cover based on a sophisticated use of the shifting strategy =-=[4]-=-. The running time of both algorithms is n O(1/ǫ2 ) for achieving approximation ratio 1 + ǫ. However, both algorithms require a geometric representation as the part of the input. In this section, we s... |

366 |
Graph Coloring Problems
- Jensen, Tort
- 1995
(Show Context)
Citation Context ...imination ordering with respect to the property P. In fact, similar ordering-based concepts exist in the literature, especially with regard to graph coloring; for example, graph degeneracy defined in =-=[32]-=-, and equivalently, degree inductiveness in [28]. Most relevant to this paper, Hochbaum [29] and Halldórsson [28] studied approximation algorithms for the maximum independent set problem, and they bou... |

276 |
Incidence matrices and interval graphs.
- Fulkerson, Gross
- 1965
(Show Context)
Citation Context ...e traced back to Hajnal and Surányi [34] in the late 1950s. One of earliest characterizations of chordal graphs was introduced by Dirac [21] in terms of minimal vertex separators. Fulkerson and Gross =-=[28]-=- gave an alternative characterization of chordal graphs in terms of perfect elimination orderings; this was also observed by Rose [55] in 1970. Following this, Rose, Tarjan and Lueker [56] introduced ... |

194 | Zero knowledge and the chromatic number.
- Feige, Kilian
- 1998
(Show Context)
Citation Context ...reased running time of O(m(|E| + |V |)). 5.3 Minimum Vertex Coloring Minimum vertex coloring is a well-studied NP-hard problem, and is not approximable within n 1−ǫ for any fixed ǫ > 0, unless ZPP=NP =-=[26]-=-. For chordal graphs, a greedy algorithm on the reverse ordering of any perfect elimination ordering gives an optimal coloring. For sequentially k-independent graphs, the same greedy algorithm achieve... |

175 |
A local-ratio theorem for approximating the weighted vertex cover problem.
- Bar-Yehuda, Even
- 1985
(Show Context)
Citation Context ... for weighted minimum vertex cover if G is an inductive k-independent graph. Proof. As observed by Zimny [47], this follows directly from the Local Ratio vertex cover algorithm of Bar Yehuda and Even =-=[9]-=-. Namely, for any triangle in the graph, the local ratio algorithm removes the vertex with minimum weight and reduces the weights of the other two vertices by that weight. The algorithm keeps doing th... |

168 | The intersection graphs of subtrees in trees are exactly the chordal graps - Gavril - 1974 |

152 |
Matroids and the greedy algorithm.
- Edmonds, Gjertson, et al.
- 1971
(Show Context)
Citation Context ...think there is a correspondence between algorithm paradigms and problem structures. We have seen multiple evidences for simple algorithms based on local decisions. The most notable one is the matroid =-=[23]-=- and greedoid [46] structures in correspondence to greedy algorithms, which have been studied extensively in the literature. As illustrated in this paper, various problems solved (or approximated) by ... |

150 |
On Rigid Circuit Graphs ,
- Dirac
- 1961
(Show Context)
Citation Context ...aphs and Their Generalizations The study of chordal graphs can be traced back to Hajnal and Surányi [34] in the late 1950s. One of earliest characterizations of chordal graphs was introduced by Dirac =-=[21]-=- in terms of minimal vertex separators. Fulkerson and Gross [28] gave an alternative characterization of chordal graphs in terms of perfect elimination orderings; this was also observed by Rose [55] i... |

150 | Vertex cover might be hard to approximate to within 2−ε,
- Khot, Regev
- 2008
(Show Context)
Citation Context ...uctive k-independent graphs with k > 1, the algorithm is a greedy algorithm (in the sense of [15]). 3 In fact, modulo unique games conjecture, no polynomial-time algorithm can achieve that ratio; see =-=[35]-=-.Proof. If k = 1 then it is a chordal graph, and hence minimum vertex cover can be done optimally. Now assume k > 1; if the inductive k-independent graph G contains triangles, then since finding a tr... |

131 |
Algorithms for minimum coloring, maximum clique, minimum covering by cliques and maximum independent set of a chordal graph
- GAVRIL
- 1972
(Show Context)
Citation Context ...oposition 1, all algorithms discussed here run in polynomial time when k is a fixed constant. It is well known that many NP-complete problems can be solved optimally when restricted to chordal graphs =-=[26]-=-[46]. We show that if we restrict the graph to be an inductive k-independent graph, we get simple approximation algorithms for several NP-complete problems. In fact, the structure of inductive k-indep... |

130 | Farbung von Graphen deren samtliche bzw. deren ungerade Kreise starr sind, - Berge - 1961 |

114 |
Finding a minimum circuit in a graph,
- Itai, Rodeh
- 1978
(Show Context)
Citation Context ...x v to see if α(G[N(v)]) > k in O(k 2 n k+1 ) time by enumerating all subsets of size k + 1. Therefore this kelimination process terminates in O(k 2 n k+3 ) time. By the observation of Itai and Rodeh =-=[41]-=-, and results in [24], we can improve the time complexities of a 2-elimination process to O(n 4.376 ), a 3-elimination process to O(n 5.334 ) and a 4-elimination process to O(n 6.220 ). The above algo... |

103 | Murky graphs
- Hayward
- 1990
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Citation Context ...BFS); and later, Tarjan [60] gave an even simpler algorithm known as the maximum cardinality search (MCS). Many generalizations of chordal graphs have been proposed and studied, for example as in [18]=-=[36]-=-[43][44]. Two important related classes are the weakly chordal graphs introduced by Hayward and the [h, s, t] framework that arises from the study of constant tolerance intersection graphs of subtrees... |

101 | Polynomial-time approximation schemes for geometric graphs
- Erlebach, Jansen, et al.
- 2001
(Show Context)
Citation Context ...ertices are adjacent if two associated translates are overlapping. One special case of such graphs are unit disk graphs, for which a robust PTAS for weighted maximum independent set is known [54]. In =-=[25]-=-, Erlebach, Jansen and Seidel consider a more general case of geometric intersection graphs (including disk graphs) and give PTASs for weighted maximum independent set and weighted minimum vertex cove... |

96 | A characterization of rigid circuit graphs. - Buneman - 1974 |

90 |
E!cient bounds for the stable set, vertex cover and set packing problems.
- Hochbaum
- 1983
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Citation Context ...ence number. This parameter is an inductive property based on local independent set size and it gives a natural extension of chordal graphs. A similar concept is the degree inductiveness. Hochhaum in =-=[39]-=- and separately Halldórsson in [35] studied approximation algorithms for the maximum independent set problem, and they bounded the approximation ratios in terms of the degree inductiveness. In a recen... |

84 | Derandomized graph products. - Alon, Feige, et al. - 1995 |

73 | Approximating the throughput of multiple machines under real-time scheduling
- Bar-Noy, Guha, et al.
(Show Context)
Citation Context ...dering and the underlying intersection graph of ISP is a chordal graph. The job interval scheduling problem is an extension of ISP and has been extensively studied in the literature, for example, see =-=[6]-=-[19][58]. In JISP, each interval belongs to a job. A job can be scheduled onto one and only one of its intervals. The objective is to find the maximum number (or total weight in the weighted case) of ... |

61 | Asteroidal Triple-Free Graphs.
- Corneil, Olariu, et al.
- 1997
(Show Context)
Citation Context ... is known [48] that the intersection of asteroidal triple-free graphs and chordal graphs is exactly interval graphs. One immediate question is what is the intersection of asteroidal triplefree graphs =-=[20]-=- and sequentially k-independent graphs for k ≥ 2? Do they present any interesting properties? Our third question is on the algorithmic aspects of sequentially k-independent graphs. We have studied the... |

53 | Fixed-parameter tractability and completeness
- Downey, Fellows
- 1995
(Show Context)
Citation Context ...dge and we conclude that G is not an inductive k-independent graph. ⊓⊔ Note that the maximum independent set problem is W[1]-complete when parameterized by the size of the independent set size k, see =-=[19]-=-. By a reduction from the maximum independent set problem, finding the inductive independence number is also complete for W[1], hence it is unlikely to have a fixed parameter tractable solution. But t... |

47 |
A d/2 approximation for maximum weight independent set in d-claw free graphs. Algorithm Theory-SWAT 2000
- Berman
- 2000
(Show Context)
Citation Context ... problems solved (or approximated) by the local ratio technique seem to be connected with the graphs having small sequential independence number. One other interesting class is d-claw-free graphs. In =-=[10]-=-, Berman gives a d 2 approximation for maximum weight independent set in d-claw-free graphs. We also note that Edmond’s weighted matching algorithm [22] has been extended to 3-claw-free graphs [50][52... |

44 | Substitution decomposition on chordal graphs and its applications - Hsu, Ma - 1991 |

41 | Incremental) priority algorithms
- Borodin, Nielsen, et al.
(Show Context)
Citation Context ...ithm. For some graph classes such as those defined by the JISP and axis paralell rectangles problems mentioned in section 1, the orderings satisfy the locally defined orderings in priority algorithms =-=[14]-=- and hence we obtain greedy (in the unweighted case) or greedy-like stack algorithms. For completeness, we include a proof of the weighted MIS result in this subsection. We study a natural generalizat... |

35 | Extremal values of the interval number of a graph
- Griggs, West
(Show Context)
Citation Context ...e such graphs in polynomial time for k ≥ 3. Another relevant (but not comparable) class of graphs are the k-interval graphs as defined by Trotter and Harary [62], and independently by Griggs and West =-=[32]-=-. In a recent paper [16], Butman, Hermelin, Lewenstein and Rawitz studied minimumvertex cover, minimum dominating set and maximum clique on k-interval graphs and were able to obtain approximation alg... |

34 | Approximation algorithms for the job interval selection problem and related scheduling problems
- Chuzhoy, Ostrovsky, et al.
(Show Context)
Citation Context ...ing and the underlying intersection graph of ISP is a chordal graph. The job interval scheduling problem is an extension of ISP and has been extensively studied in the literature, for example, see [6]=-=[19]-=-[58]. In JISP, each interval belongs to a job. A job can be scheduled onto one and only one of its intervals. The objective is to find the maximum number (or total weight in the weighted case) of jobs... |

27 |
Mathematical structures underlying greedy algorithms, in: F. G?ecseg (Ed.), Fundamentals of Computation Theory,
- Korte, Lovasz
- 1981
(Show Context)
Citation Context ...orrespondence between algorithm paradigms and problem structures. We have seen multiple evidences for simple algorithms based on local decisions. The most notable one is the matroid [23] and greedoid =-=[46]-=- structures in correspondence to greedy algorithms, which have been studied extensively in the literature. As illustrated in this paper, various problems solved (or approximated) by the local ratio te... |

24 | On the complexity of fixed parameter clique and dominating set.
- Eisenbrand, Grandoni
- 2004
(Show Context)
Citation Context ...)]) > k in O(k 2 n k+1 ) time by enumerating all subsets of size k + 1. Therefore this kelimination process terminates in O(k 2 n k+3 ) time. By the observation of Itai and Rodeh [41], and results in =-=[24]-=-, we can improve the time complexities of a 2-elimination process to O(n 4.376 ), a 3-elimination process to O(n 5.334 ) and a 4-elimination process to O(n 6.220 ). The above algorithms only use linea... |

18 | Optimization problems in multipleinterval graphs
- Butman, Hermelin, et al.
(Show Context)
Citation Context ...ial time for k ≥ 3. Another relevant (but not comparable) class of graphs are the k-interval graphs as defined by Trotter and Harary [62], and independently by Griggs and West [32]. In a recent paper =-=[16]-=-, Butman, Hermelin, Lewenstein and Rawitz studied minimumvertex cover, minimum dominating set and maximum clique on k-interval graphs and were able to obtain approximation algorithms for each of prob... |

18 | Approximation algorithms for the weighted independent set problem
- Kako, Ono, et al.
- 2005
(Show Context)
Citation Context ...arately Halldórsson in [35] studied approximation algorithms for the maximum independent set problem, and they bounded the approximation ratios in terms of the degree inductiveness. In a recent paper =-=[45]-=-, Halldórsson et al introduced a weighted measure of inductiveness, which deals with the weighted case of the maximum independent set problem. Here, we focus on graph class with a fixed local independ... |

16 |
On approximating a vertex cover for planar graphs
- Bar-Yehuda, Even
- 1982
(Show Context)
Citation Context ... In this section, we show a (2 − 1 k )-approximation algorithm for minimum vertex cover on sequentially k-independent graphs. The algorithm shares the same spirit of the result of Bar-Yehuda and Even =-=[8]-=- for the 5 3 approximation of vertex cover for planar graphs, and can be viewed as a generalization of that algorithm. Baker’s PTAS algorithm [4] for minimum vertex cover on planar graphs depends on a... |

15 | Improved algorithms for weakly chordal graphs - Hayward, Spinrad, et al. |

13 | On the chromatic number of intersection graphs of convex sets in the plane
- Kim, Kostochka, et al.
(Show Context)
Citation Context ...nning time of both algorithms is n O(1/ǫ2 ) for achieving approximation ratio 1 + ǫ. However, both algorithms require a geometric representation as the part of the input. Kim, Kostochka and Nakprasit =-=[36]-=- proved that any intersection graph of those translates with clique number k are (3k −3)-degenerate; i.e., ˜ G(|V |3k−3) in our notation. Within that proof, they showed the neighborhood of the topmost... |

12 | How well can primal-dual and local-ratio algorithms perform
- Borodin, Cashman, et al.
- 2005
(Show Context)
Citation Context ...lly k-independent graphs. The local ratio technique is usually described as a recursive algorithm. As in Berman and DasGupta [11], we view it as an iterative algorithm, modeled as stack algorithms in =-=[13]-=-. The Stack Algorithm for packing problems Push Phase:while V is not empty select the next data item v according to some rule 2 decide to either push v onto the stack or discard it end while Pop Phas... |

12 |
Constant tolerance intersection graphs of subtrees of a tree
- Jamison, Mulder
- 2005
(Show Context)
Citation Context ...d later, Tarjan [60] gave an even simpler algorithm known as the maximum cardinality search (MCS). Many generalizations of chordal graphs have been proposed and studied, for example as in [18][36][43]=-=[44]-=-. Two important related classes are the weakly chordal graphs introduced by Hayward and the [h, s, t] framework that arises from the study of constant tolerance intersection graphs of subtrees in a tr... |

9 |
Tolerance intersection graphs on binary trees with constant tolerance 3, Discrete Math
- Jamison, Mulder
- 2000
(Show Context)
Citation Context ...; and later, Tarjan [60] gave an even simpler algorithm known as the maximum cardinality search (MCS). Many generalizations of chordal graphs have been proposed and studied, for example as in [18][36]=-=[43]-=-[44]. Two important related classes are the weakly chordal graphs introduced by Hayward and the [h, s, t] framework that arises from the study of constant tolerance intersection graphs of subtrees in ... |

8 |
Improvements in throughout maximization for real-time scheduling
- Berman, DasGupta
- 2000
(Show Context)
Citation Context ...e ∆ it is NP-hard to approximate within ∆ ǫ for some ǫ > 0 even for the unweighted case [3], but polynomial time solvable for chordal graphs. Akcoglu et al [2] show that the local ratio technique ([7]=-=[11]-=-) achieves a kapproximation for the weighted maximum independent set on sequentially k-independent graphs. The local ratio technique is usually described as a recursive algorithm. As in Berman and Das... |

7 | On the structure of bull-free perfect graphs
- Figueiredo, Maffray, et al.
- 1997
(Show Context)
Citation Context ...i}) ∩ Vi]. Since Gi has vertex clique cover size 2, the complement of Gi is a bipartite graph. Since finding weighted maximum independent set in a bipartite graph can be determined in polynomial time =-=[27]-=-[33], weighted maximum clique in Gi can be computed in polynomial time. We compute weighted maximum clique for each Gi, and the largest one is weighted maximum clique for G. This is because for any we... |

6 |
Equivalences and the complete hierarchy of intersection graphs of paths in a tree
- Golumbic, Lipshteyn, et al.
(Show Context)
Citation Context ...y if the corresponding subtrees have at least t vertices in common in the host tree. A complete hierarchy of intersection graphs of paths in a tree was recently given by Golumbic, Lipshteyn and Stern =-=[31]-=-, and the relationship with weakly chordal graph has also been further investigated. Here we show that sequentially 2-independent graphs cannot be characterized by the [h, s, t] framework with fixed c... |

6 | Constant tolerance representations of graphs in a trees - Jamison, Mulder |

5 | Simple Approximation Algorithm for Nonoverlapping Local Alignments
- Berman, DasGupta, et al.
- 2002
(Show Context)
Citation Context ...me, it is easy to see that the intersection graph for the JISP problem is a sequentially 2-independent graph. – The axis parallel rectangles problem: This problem is studied by Berman and DasGupta in =-=[12]-=- and is motivated by applications to non-overlapping local alignment problems in computational molecular biology. The input is a set of axis parallel rectangles such that, for each axis, the projectio... |

3 | Finding a maximum-weight induced k-partite subgraph of an i-triangulated graph
- Berry, Kennedy, et al.
(Show Context)
Citation Context ... one red edge and we conclude that G is not a k-sequentially independent graph. ⊓⊔ Note that by a reduction to the independent set problem, finding the sequential independence number is complete for W=-=[1]-=-, hence it is unlikely to have a fixed parameter tractable solution. But this does not exclude the possibility to improve the current complexity bound for a small k. It is interesting to note that rec... |

3 |
A generalization of chordal graphs and the maximum clique problem. Information Processing Letters 62:111–120
- Chmeiss, Jégou
- 1997
(Show Context)
Citation Context ...h (LBFS); and later, Tarjan [60] gave an even simpler algorithm known as the maximum cardinality search (MCS). Many generalizations of chordal graphs have been proposed and studied, for example as in =-=[18]-=-[36][43][44]. Two important related classes are the weakly chordal graphs introduced by Hayward and the [h, s, t] framework that arises from the study of constant tolerance intersection graphs of subt... |

2 |
cost algorithms for combinatorial auctions. Applied Optimization
- Opportunity
(Show Context)
Citation Context ...raphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. Akcoglu et al =-=[2]-=- define an extension of chordal graphs whereby the neighbors of v that occur later than v in the elimination order have at most k independent vertices. We refer to such graphs as sequentially k-indepe... |

2 | Approximating maximum weight K-colorable subgraphs
- Chakaravarthy, Roy
- 2009
(Show Context)
Citation Context ...r graph theory literature. For chordal graphs, the unweighted case of the problem is polynomial time solvable for any fixed m, but NP-complete otherwise [66]. In a recent paper, Chakaravarthy and Roy =-=[17]-=- showed that the problem has a simple 2-approximation algorithm for the weighted case. Here we strengthen their approximation result. Theorem 8. For all fixed constant k ≥ 1, there is a polynomial tim... |

1 |
Local ratio: A unied framework for approxmation algorithms in memoriam: Shimon Even 1935-2004
- Bar-Yehuda, Bendel, et al.
- 2004
(Show Context)
Citation Context ...gree ∆ it is NP-hard to approximate within ∆ ǫ for some ǫ > 0 even for the unweighted case [3], but polynomial time solvable for chordal graphs. Akcoglu et al [2] show that the local ratio technique (=-=[7]-=-[11]) achieves a kapproximation for the weighted maximum independent set on sequentially k-independent graphs. The local ratio technique is usually described as a recursive algorithm. As in Berman and... |

1 |
Über die Auflösung von Graphen in Voltständige Teilgraphen
- Hajnal, Surányi
- 1958
(Show Context)
Citation Context ...mber in Section 3. Fig.1. Examples of k-sequentially independent graphs for k = 2, 3,4. 2 Chordal Graphs and Their Generalizations The study of chordal graphs can be traced back to Hajnal and Surányi =-=[34]-=- in the late 1950s. One of earliest characterizations of chordal graphs was introduced by Dirac [21] in terms of minimal vertex separators. Fulkerson and Gross [28] gave an alternative characterizatio... |

1 |
Approximations of weighed independent set and hereditary subset problems
- Halldórsson
- 2000
(Show Context)
Citation Context ...nductive property based on local independent set size and it gives a natural extension of chordal graphs. A similar concept is the degree inductiveness. Hochhaum in [39] and separately Halldórsson in =-=[35]-=- studied approximation algorithms for the maximum independent set problem, and they bounded the approximation ratios in terms of the degree inductiveness. In a recent paper [45], Halldórsson et al int... |

1 |
A unified pproach to approximating resource allocation and scheduling
- Bar-Noy, Bar-Yehuda, et al.
- 2001
(Show Context)
Citation Context .... Both of the JISP and local alignment problems are MAX SNP-hard [41][45], although the current inapproximations are very weak. The existence of local ratio approximation algorithms in Bar-Noy et al. =-=[5]-=- and Berman and DasGupta in [12][13] for the above two problems was our initial motivation for investigating how the intersection graph structure underlies the success of those algorithms. In fact, su... |