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## Algorithms for Maximum Independent Set Applied to Map Labelling (2000)

Citations: | 18 - 0 self |

### Citations

5049 | Optimization by simulated annealing,”
- Kirkpatrick, Gelatt, et al.
- 1983
(Show Context)
Citation Context ...abel. A disadvantage of this method is that it easily gets stuck in local maxima. To escape from local optima in the search space, simulated annealing can be used, see Kirkpatrick, Gelatt, and Vecchi =-=[23]-=-. Moves in the search space that make the solution worse are allowed with a certain probability. Simulated annealing uses the following approach: 1. Compute an initial labelling L = L 0 2. Initialise ... |

2337 | A note on two problems in connexion with graphs.
- Dijkstra
- 1959
(Show Context)
Citation Context ...a long one. In our implementation, we restricted ourselves to shortest path with length at most 1.125. Shortest paths in a graph with non-negative edge lengths can be found using Dijkstra's algorithm =-=[14]-=-. Hence, we can find a closed walk C := (v = v 0 , e 1 , v 1 , e 2 , v 2 , . . . , v k-1 , e k , v k = v) in G with odd k that is minimal with respect to c and |C| by using Dijkstra's algorithm to fin... |

1394 |
Geometric algorithms and combinatorial optimization,”
- Grotschel, Lovasz, et al.
- 1987
(Show Context)
Citation Context ...lities, x(V (H)) # #|V (H)|/2#. Suppose we are given a vector x LP # PE . We first find an odd cycle starting from some node v # V using the construction described by Grotschel, Lovasz, and Schrijver =-=[18]-=-. To find a shortest odd cycle containing node v, Grotschel et al. construct an auxiliary bipartite graph G = ((V 1 , V 2 ), E) and cost vectors c # [0, 1] E andsc # [0, 1]sE as follows. Each node v #... |

416 |
Integer Programming.
- Wolsey
- 1998
(Show Context)
Citation Context ...algorithm in each node of the search tree. Here we give a short description of a basic version of branch and bound and of a cutting plane algorithm. For further details we refer to the book by Wolsey =-=[42]-=-. 20 Consider the problem of determining z OPT = max{z(x) : x # P, x integer}, where z is a linear function in x, and where P is a polyhedron. We refer to this problem as problem #. Branch and bound m... |

198 | A branch-and-cut algorithm for the resolution of largescale symmetric traveling salesman problems
- Padberg, Rinaldi
- 1991
(Show Context)
Citation Context ... cost belongs to the folklore and is used by many authors to improve the formulation of zero-one integer programming problems (for example by Crowder, Johnson, and Padberg [9] and Padberg and Rinaldi =-=[33]-=-). Note that the new bounds are a function of #, z LP , and z # . As a consequence, each time that we find a new best integer feasible solution in the branch-and-bound algorithm we can recompute the b... |

159 | An Empirical Study of Algorithms for PointFeature Label Placement
- Christensen, Marks, et al.
- 1995
(Show Context)
Citation Context ...problem can be approximated within O(log n) of the optimum. For the optimisation problem di#erent heuristic algorithms (including simulated annealing) are discussed by Christensen, Marks, and Shieber =-=[7]-=-. Van Dijk, Thierens, and de Berg [13] 2 considered genetic algorithms, Cromly [8] proposed a semi-automatic LP-based approach for finding feasible solutions to the optimisation variant. Zoraster [44,... |

154 | Vertex packings: structural properties and algorithms. - Nemhauser, Trotter - 1975 |

150 | Treewidth: Algorithmic techniques and results, in: The Twenty
- Bodlaender
- 1997
(Show Context)
Citation Context ...in non-decreasing lexicographic order of the pairs (| 1 2 -x v |, -|N 1 (v) #V (H)|), where ties are broken at random. To compute the coe#cients #, we make use of a path decomposition (see Bodlaender =-=[3]-=-, and de Fluiter [15]) of the graph induced by the nodes in the hole and the nodes we already lifted. Here we describe how we maintain the path decomposition. How to use path decompositions to compute... |

130 |
On the facial structure of set packing polyhedra.
- Padberg
- 1973
(Show Context)
Citation Context ...nt set can have at most one node from C. Hence, the clique inequality x(C) # 1 (4) is a valid inequality for P IS for every clique C. If C is maximal, we call (4) a maximal clique inequality. Padberg =-=[32]-=- showed that the clique inequality on C is facet-defining for P IS if and only if C is a maximal clique. Let C be a collection of not necessarily maximal cliques in G such that for each edge {u, v} # ... |

124 |
Solving Large-Scale Zero-One Linear Programming Problems
- Crowder, Johnson, et al.
- 1983
(Show Context)
Citation Context ...ble setting based on reduced cost belongs to the folklore and is used by many authors to improve the formulation of zero-one integer programming problems (for example by Crowder, Johnson, and Padberg =-=[9]-=- and Padberg and Rinaldi [33]). Note that the new bounds are a function of #, z LP , and z # . As a consequence, each time that we find a new best integer feasible solution in the branch-and-bound alg... |

108 | A packing problem with applications to lettering of maps
- Formann, Wagner
- 1991
(Show Context)
Citation Context ...n be found in which it is only enforced that labels are placed close to the point they refer to and do not overlap. Such formulations are commonly used in the literature (see, e.g., Forman and Wagner =-=[16]-=-). The map labelling community has mainly focused on heuristics. For modern applications, such as embedding automated map labelling in real-time systems for navigation, being able to produce a good ma... |

91 | Label placement by maximum independent set in rectangles.
- Agarwal, Kreveld, et al.
- 1998
(Show Context)
Citation Context ... with a guaranteed approximation factor better than 1 2 provided that P #= NP . Wagner and Wol# [37] proposed a hybrid heuristic using the above 1 2 approximation algorithm. Agarwal, Kreveld and Suri =-=[1]-=- presented a polynomial time approximation scheme for the optimisation problem. They also showed that when the labels are not restricted to have equal height, the problem can be approximated within O(... |

77 |
Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In:
- Beale, Tomlin
- 1970
(Show Context)
Citation Context ...pper bound (GUB) constraints. A GUB constraint models the situation in which we have to choose one option from a set of mutually exclusive options. GUB constraints were introduced by Beale and Tomlin =-=[2]-=-. The advantage of having GUB constraints is that they can be used to define a branching scheme known as branching on GUBs. Let P GUB C = {(x, s) # R V R C | x(C) + s C = 1 for all C # C, x # 0, s # 0... |

68 |
Positioning Names on Maps
- Imhof
- 1975
(Show Context)
Citation Context ...me-consuming task and it is natural to try to automate it. Automating map labelling requires an abstract formulation of the problem. Early formulations, which were based on rules, were given by Imhof =-=[21]-=-, and Yoeli [43]. In this paper we consider a highly abstracted formulation that can be found in which it is only enforced that labels are placed close to the point they refer to and do not overlap. S... |

61 | A computational study of search strategies for mixed integer programming
- Linderoth, Savelsbergh
- 1999
(Show Context)
Citation Context ...sible region. Details of our branching scheme can be found in Verweij [35]. Using pseudo-costs dates back to the nineteen seventies and has most recently been reported on by Linderoth and Savelsbergh =-=[27]-=-. 6 Computational Results on Map Labelling Instances We tested our heuristic and optimisation algorithms on the same class of map labelling instances as used by Christensen et al. [7] and van Dijk et ... |

58 |
Facets for a linear inequality in 0–1 variables
- Wolsey
(Show Context)
Citation Context ...t a lifted odd hole is of the form x(V (H)) + X v#V \V (H) # v x v # #|V (H)|/2# for some suitable vector # # R V + . We compute values # # N V + using sequential lifting (see Padberg [32] and Wolsey =-=[40, 41]-=-), to obtain facet-defining inequalities of P IS . The separation algorithm for lifted odd hole inequalities consists of two parts. The first part derives an odd hole H from x LP that defines a violat... |

52 |
Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane
- Imai, Asano
- 1983
(Show Context)
Citation Context ...{u, v} # C for each {u, v} # E. Then, P IS # P C # PE . 3.2 The Maximal Clique Formulation Suppose we are given a set Q of axis-parallel rectangular regions in R 2 . It was observed by Imai and Asano =-=[20]-=- that there exists a correspondence between the cliques in GQ and maximal non-empty regions that are contained in the intersection of subsets of regions from Q. Lemma 3.2. Let Q be a set of axis-paral... |

50 | Point Labeling with Sliding Labels
- Kreveld, Strijk, et al.
- 1999
(Show Context)
Citation Context ...et of candidate (square) regions. Kakoulis and Tollis [22] exploit this to unify several slightly more general map labelling problems. A di#erent approach reported on by van Kreveld, Strijk, and Wol# =-=[25]-=- and Klau and Mutzel [24] is to allow a label to take any position, as long as its corresponding point is on its boundary. This leads to so-called sliding map labelling models. The advantage of a slid... |

50 |
Properties of vertex packing and independence system polyhedra,
- Nemhauser, Jr
- 1974
(Show Context)
Citation Context ...xample, if x = (1 - #)# S # + ## V \S # for some maximum independent set S # and any # such that 0 # #s1 2 , then the algorithm produces # S # as an answer, an optimal solution. Nemhauser and Trotter =-=[29]-=- have shown that if x is obtained by maximising any objective function over PE , then x is a vector with components 0, 1 2 , and 1 only. In this case the simple LP rounding algorithm degenerates to se... |

44 |
The Solution of Large 0-1 Integer Programming Problems Encountered in Automated Cartography
- Zoraster
- 1990
(Show Context)
Citation Context ... [7]. Van Dijk, Thierens, and de Berg [13] 2 considered genetic algorithms, Cromly [8] proposed a semi-automatic LP-based approach for finding feasible solutions to the optimisation variant. Zoraster =-=[44, 45]-=- used Lagrangian relaxation to make a heuristic algorithm for the optimisation variant. All the results mentioned so far concern the problem in which each point is labelled with one (square) region fr... |

39 |
Routing trains through railway stations : Model formulation and algorithms
- Zwaneveld, Kroon, et al.
- 1996
(Show Context)
Citation Context ...of P v#V O(|#(v)| 2 ) # O(|V ||E|) time for all nodes. The following two propositions, illustrated in Figure 3, are special cases of the dominance criteria reported by Zwaneveld, Kroon and Van Hoesel =-=[46]-=-. Proposition 3.7. (Node-Dominance) Let u, v # V be nodes in G such that N 1 (u) # N 1 (v). Then there exists a maximum independent set in G that does not contain v. Proof. Let I # be a maximum indepe... |

38 |
Facets and Strong Valid Inequalities for Integer Programs
- Wolsey
- 1976
(Show Context)
Citation Context ...t a lifted odd hole is of the form x(V (H)) + X v#V \V (H) # v x v # #|V (H)|/2# for some suitable vector # # R V + . We compute values # # N V + using sequential lifting (see Padberg [32] and Wolsey =-=[40, 41]-=-), to obtain facet-defining inequalities of P IS . The separation algorithm for lifted odd hole inequalities consists of two parts. The first part derives an odd hole H from x LP that defines a violat... |

33 |
The Logic of Automated Map Lettering
- Yoeli
- 1972
(Show Context)
Citation Context ...k and it is natural to try to automate it. Automating map labelling requires an abstract formulation of the problem. Early formulations, which were based on rules, were given by Imhof [21], and Yoeli =-=[43]-=-. In this paper we consider a highly abstracted formulation that can be found in which it is only enforced that labels are placed close to the point they refer to and do not overlap. Such formulations... |

29 |
Integer Programming Applied to the Map Label Placement Problem. Cartographica
- Zoraster
- 1986
(Show Context)
Citation Context ... [7]. Van Dijk, Thierens, and de Berg [13] 2 considered genetic algorithms, Cromly [8] proposed a semi-automatic LP-based approach for finding feasible solutions to the optimisation variant. Zoraster =-=[44, 45]-=- used Lagrangian relaxation to make a heuristic algorithm for the optimisation variant. All the results mentioned so far concern the problem in which each point is labelled with one (square) region fr... |

27 |
A strong cutting plane/branch-and-bound algorithm for node packing
- Nemhauser, Sigismondi
- 1992
(Show Context)
Citation Context ...v 1 , . . . , v i-1 }) \ N 1 (v i )]}. In order to compute the lifting coe#cients, we have to compute several maximum weight independent sets, one for each lifting coe#cient. Nemhauser and Sigismondi =-=[28]-=- observed that # v = 0 for v # V \V (H) if |N 1 (v)#V (H)| # 2. This implies that the independent set problems that have to be solved in order to compute the lifting coe#cients # are relatively small ... |

26 |
Stabulus: a technique for finding stable sets in large graphs with tabu search, Computing 42
- Friden, Hertz, et al.
- 1989
(Show Context)
Citation Context ...number of tabu searches performed by the main loop is bounded by O(|V | 2 ). 4.4.2 Finding an Augmentation using Tabu Search We employ the same neighbourhood as proposed by Friden, Herz, and de Werra =-=[17]-=-. Given a set S 0 with |S 0 | = # 0 , we try to find a set S that satisfies |S| = # 0 and minimises the objective function f(S) = |E(S)|. For this purpose we maintain a partition (S, V \ S) of the nod... |

26 | A unified approach to labeling graphical features
- Kakoulis, Tollis
- 1998
(Show Context)
Citation Context ...ptimisation variant. All the results mentioned so far concern the problem in which each point is labelled with one (square) region from a finite set of candidate (square) regions. Kakoulis and Tollis =-=[22]-=- exploit this to unify several slightly more general map labelling problems. A di#erent approach reported on by van Kreveld, Strijk, and Wol# [25] and Klau and Mutzel [24] is to allow a label to take ... |

18 | Three Rules Suffice for Good Label Placement
- Wagner, Wolff, et al.
- 2001
(Show Context)
Citation Context ...y the algorithm. The simulated annealing method as described by Christensen et al. [7] has been used in several comparisons of label algorithms. Examples of this are Wagner, Wolff, Kapoor, and Strijk =-=[38]-=- using an algorithm that repeatedly applies three simple rules, and van Dijk, Thierens, and de Berg [12] that describes genetic algorithms. The results were that these algorithms had roughly same perf... |

16 | On the separation of maximally violated mod-k cuts
- Caprara, Fischetti, et al.
- 1997
(Show Context)
Citation Context ...ained in Section 5.7. A more general class of valid inequalities that is known in the literature consists of the so-called mod-k inequalities. These are developed by Caprara, Fischetti, and Letchford =-=[4, 5]-=-. 3.4 Pre-processing and Dominance Criteria In this section we consider ways to recognise nodes that belong to a maximum independent set, or that can be removed from the graph without reducing the siz... |

16 | Placing Text Labels on Maps and Diagrams
- J, Marks, et al.
- 1994
(Show Context)
Citation Context ... is the number of point features. If more than 5n moves are accepted at a temperature stage, the temperature is decreased immediately. The calculation of #E is done as described by Christensen et al. =-=[6]-=- by using the conflict graph and a overlap counter for each label position that counts the number of other labels in the current labelling that overlap the label position. So the time for one iteratio... |

16 | Exact algorithms for a geometric packing problem, in
- Kuera, Mehlhorn, et al.
- 1993
(Show Context)
Citation Context ...iven #, whether there exists a set of squares Q 1 , . . . , Q n as described above. Formann and Wagner [16] showed that this latter problem is NP-complete. Kucera, Mehlhorn, Preis, and Schwarzenecker =-=[26]-=- observed that there are only O(n 2 ) possible values that # # can take. Optimising over those can be done by solving the decision variant only O(log n) times, using binary search with di#erent values... |

14 |
B.: Algorithms for Graphs of Small Treewidth.
- Fluiter
- 1997
(Show Context)
Citation Context ...xicographic order of the pairs (| 1 2 -x v |, -|N 1 (v) #V (H)|), where ties are broken at random. To compute the coe#cients #, we make use of a path decomposition (see Bodlaender [3], and de Fluiter =-=[15]-=-) of the graph induced by the nodes in the hole and the nodes we already lifted. Here we describe how we maintain the path decomposition. How to use path decompositions to compute maximum weight indep... |

13 | An efficient and effective approximation algorithm for the map labeling problem
- Wagner, Wolff
- 1995
(Show Context)
Citation Context ... find an optimal labelling. Christensen, Marks, and Shieber [7, p. 219] reported that optimising is impractical for map labelling instances with more than 50 cities. In the same year, Wagner and Wolf =-=[37]-=- 1smentioned that they were able to solve instances with up to 300 cities to optimality. Verweij and Aardal [36] were the first to show that it is computationally feasible to compute provably optimal ... |

12 | Optimal labelling of point features in the slider model, in
- Klau, Mutzel
- 2000
(Show Context)
Citation Context ...regions. Kakoulis and Tollis [22] exploit this to unify several slightly more general map labelling problems. A di#erent approach reported on by van Kreveld, Strijk, and Wol# [25] and Klau and Mutzel =-=[24]-=- is to allow a label to take any position, as long as its corresponding point is on its boundary. This leads to so-called sliding map labelling models. The advantage of a sliding model is that more la... |

11 | An optimisation algorithm for maximum independent set with applications in map labelling
- Verweij, Aardal
- 1999
(Show Context)
Citation Context ...for map labelling instances with more than 50 cities. In the same year, Wagner and Wolf [37] 1 mentioned that they were able to solve instances with up to 300 cities to optimality. Verweij and Aardal =-=[36]-=- were the first to show that it is computationally feasible to compute provably optimal labellings for maps with up to 800 cities. In this paper we address both heuristic algorithms and optimization a... |

10 |
Selected Applications of Integer Programming: A Computational Study
- Verweij
- 2000
(Show Context)
Citation Context ... optimisation variant of the basic map labelling problem. In the remainder of this paper we will refer to it as the map labelling problem. This study is an extension of the work by Verweij and Aardal =-=[36, 35]-=-. 1.3 Outline In Section 2 we show that the map labelling problem reduces to a maximum independent set problem on an associated graph. Section 3 reviews several formulations of the maximum independent... |

9 |
The map-labeling bibliography. http://www.math-inf.unigreifswald.de/map-labeling/bibliography
- Wolff
(Show Context)
Citation Context ...e optimisation variant is NP-hard. 1.2 Related Literature A recent survey on map labelling is given by Neyer [31]. An exhaustive bibliography concerning map labelling is maintained by Wol# and Strijk =-=[39]-=-. Formann and Wagner [16] developed a 1 2 -approximation algorithm for the size optimisation variant of the map labelling problem. They also proved that no algorithm exists with a guaranteed approxima... |

8 |
Intersection Properties of Boxes in R d
- Danzer, Grnbaum
- 1982
(Show Context)
Citation Context ...o the nodes in C. Because C is a clique, all rectangular regions in S intersect pairwise. This implies that the region R = T Q#S Q is non-empty by the following argument (see e.g. Danzer and Grunbaum =-=[11]-=-). The intersection of a set of axis-parallel rectangular regions is non-empty if and only if the intersection of the projections of the 7 rectangular regions on the axis are non-empty. The projection... |

8 | New algorithms for two-label point labeling
- Qin, Wolff, et al.
- 2000
(Show Context)
Citation Context ...odels. The advantage of a sliding model is that more labels can be placed without overlap. In more recent studies also other shapes of the label regions are considered (see e.g. Qin, Wol#, Xu, and Zu =-=[34]-=-). In this paper we study the optimisation variant of the basic map labelling problem. In the remainder of this paper we will refer to it as the map labelling problem. This study is an extension of th... |

7 | On the design of genetic algorithms for geographical applications, in W. Banzhaf et al
- Dijk, Thierens, et al.
- 1999
(Show Context)
Citation Context ...log n) of the optimum. For the optimisation problem di#erent heuristic algorithms (including simulated annealing) are discussed by Christensen, Marks, and Shieber [7]. Van Dijk, Thierens, and de Berg =-=[13]-=- 2 considered genetic algorithms, Cromly [8] proposed a semi-automatic LP-based approach for finding feasible solutions to the optimisation variant. Zoraster [44, 45] used Lagrangian relaxation to mak... |

5 | Robust genetic algorithms for high quality map labeling
- Dijk, Thierens, et al.
- 1998
(Show Context)
Citation Context ... several comparisons of label algorithms. Examples of this are Wagner, Wol#, Kapoor, and Strijk [38] using an algorithm that repeatedly applies three simple rules, and van Dijk, Thierens, and de Berg =-=[12]-=- that describes genetic algorithms. The results were that these algorithms had roughly same performance as simulated annealing although the running times di#ered. Since simulated annealing is present ... |

4 | An LP relaxation procedure for annotating point features using interactive graphics
- Cromley
- 1985
(Show Context)
Citation Context ...roblem di#erent heuristic algorithms (including simulated annealing) are discussed by Christensen, Marks, and Shieber [7]. Van Dijk, Thierens, and de Berg [13] 2 considered genetic algorithms, Cromly =-=[8]-=- proposed a semi-automatic LP-based approach for finding feasible solutions to the optimisation variant. Zoraster [44, 45] used Lagrangian relaxation to make a heuristic algorithm for the optimisation... |

4 |
An ecient and eective approximation algorithm for the map labeling problem
- Wagner, Wol
- 1995
(Show Context)
Citation Context ... find an optimal labelling. Christensen, Marks, and Shieber [7, p. 219] reported that optimising is impractical for map labelling instances with more than 50 cities. In the same year, Wagner and Wolf =-=[37]-=- 1 mentioned that they were able to solve instances with up to 300 cities to optimality. Verweij and Aardal [36] were the first to show that it is computationally feasible to compute provably optimal ... |

1 |
Map labeling with application to graph labeling
- Neyer
- 2000
(Show Context)
Citation Context ...ng problem that is the subject of this paper. Since the decision variant is NP-complete, the optimisation variant is NP-hard. 1.2 Related Literature A recent survey on map labelling is given by Neyer =-=[31]-=-. An exhaustive bibliography concerning map labelling is maintained by Wol# and Strijk [39]. Formann and Wagner [16] developed a 1 2 -approximation algorithm for the size optimisation variant of the m... |

1 |
Three rules su#ce for good label placement
- Wagner, Wol, et al.
(Show Context)
Citation Context ...by the algorithm. The simulated annealing method as described by Christensen et al. [7] has been used in several comparisons of label algorithms. Examples of this are Wagner, Wol#, Kapoor, and Strijk =-=[38]-=- using an algorithm that repeatedly applies three simple rules, and van Dijk, Thierens, and de Berg [12] that describes genetic algorithms. The results were that these algorithms had roughly same perf... |