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Combinatorial Benders’ Cuts for MixedInteger Linear Programming
 Operations Research
"... MixedInteger Programs (MIP’s) involving logical implications modelled through bigM coefficients, are notoriously among the hardest to solve. In this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependenc ..."
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Cited by 11 (1 self)
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MixedInteger Programs (MIP’s) involving logical implications modelled through bigM coefficients, are notoriously among the hardest to solve. In this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the bigM coefficients. Our solution scheme defines a master Integer Linear Problem (ILP) with no continuous variables, which contains combinatorial information on the feasible integer variable combinations that can be “distilled ” from the original MIP model. The master solutions are sent to a slave Linear Program (LP), which validates them and possibly returns combinatorial inequalities to be added to the current master ILP. The inequalities are associated to minimal (or irreducible) infeasible subsystems of a certain linear system, and can be separated efficiently in case the master solution is integer. The overall solution mechanism resembles closely the Benders ’ one, but the cuts we produce are purely combinatorial and do not depend on the bigM values used in the MIP formulation. This produces an LP relaxation of the master problem which can be considerably tighter than the one associated with original MIP formulation. Computational results on two specific classes of hardtosolve MIP’s indicate the new method produces a reformulation which can be solved some orders of magnitude faster than the original MIP model.
Projected ChvátalGomory cuts for Mixed Integer Linear Programs
, 2006
"... Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure Integer Linear Program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the ChvátalGomory (CG) separation problem as a Mixed Int ..."
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Cited by 8 (6 self)
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Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure Integer Linear Program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the ChvátalGomory (CG) separation problem as a Mixed Integer Linear Program (MILP) which is then solved by a generalpurpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory Mixed Integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive ChvátalGomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, whose solution can typically be carried out in a reasonable amount of computing time.
ForceBased Label Number Maximation
, 2003
"... We present a forcebased simulated annealing algorithm to heuristically solve the NPhard label number maximization problem LNM: Given a set of rectangular labels, each of which belongs to a pointfeature in the plane, the task is to find a labeling for a largest subset of the labels. A labeling ..."
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Cited by 8 (0 self)
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We present a forcebased simulated annealing algorithm to heuristically solve the NPhard label number maximization problem LNM: Given a set of rectangular labels, each of which belongs to a pointfeature in the plane, the task is to find a labeling for a largest subset of the labels. A labeling is a placement such that none of the labels overlap and each is placed at its pointfeature. The
A GREEDY RANDOMIZED ADAPTIVE SEARCH PROCEDURE FOR THE POINTFEATURE CARTOGRAPHIC LABEL PLACEMENT
"... The pointfeature cartographic label placement problem (PFCLP) is an NPhard problem which appears during the production of maps. The labels must be placed in predefined places avoiding overlaps and considering cartographic preferences. Due to its high complexity several heuristics have been present ..."
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Cited by 6 (0 self)
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The pointfeature cartographic label placement problem (PFCLP) is an NPhard problem which appears during the production of maps. The labels must be placed in predefined places avoiding overlaps and considering cartographic preferences. Due to its high complexity several heuristics have been presented searching for approximated solutions. This paper proposes a greedy randomized adaptive search procedure (GRASP) for the PFCLP that is based on its associated conflict graph. The computational results show that this metaheuristic is a good strategy for PFCLP, generating better solutions than all those reported in the literature in reasonable computational times.
Label number maximization in the slider model
 in Proc. 12th Internat. Symp. on Graph Drawing (GD'04
, 2004
"... Abstract. We consider the NPhard label number maximization problem lnm: Given a set of rectangular labels, each of which belongs to a point feature in the plane, the task is to find a labeling for a largest subset of the labels. A labeling is a placement such that none of the labels overlap and ea ..."
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Cited by 2 (0 self)
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Abstract. We consider the NPhard label number maximization problem lnm: Given a set of rectangular labels, each of which belongs to a point feature in the plane, the task is to find a labeling for a largest subset of the labels. A labeling is a placement such that none of the labels overlap and each is placed so that its boundary touches the corresponding point feature. The purpose of this paper is twofold: We present a new forcebased simulated annealing algorithm to heuristically solve the problem and we provide the results of a very thorough experimental comparison of the best known labeling methods on widely used benchmark sets. The design of our new method has been guided by the goal to produce labelings that are similar to the results of an experienced human performing the same task. So we are not only looking for a labeling where the number of labels placed is high but also where the distribution of the placed labels is good. Our experimental results show that the new algorithm outperforms the other methods in terms of quality while still being reasonably fast and confirm that the simulated annealing method is wellsuited for map labeling problems. 1
Label updating to avoid pointshaped obstacles in fixed model
, 2006
"... In this paper, we present efficient algorithms for updating the labeling of a set of n points after the presence of a random obstacle that appears on the map repeatedly. We update the labeling so that the given obstacle does not appear in any of the labels, the new labeling is valid, and the labels ..."
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Cited by 2 (2 self)
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In this paper, we present efficient algorithms for updating the labeling of a set of n points after the presence of a random obstacle that appears on the map repeatedly. We update the labeling so that the given obstacle does not appear in any of the labels, the new labeling is valid, and the labels are as large as possible (called the optimal labeling). Each point is assumed to have an axisparallel, squareshaped label of unit size, attached exclusively to that point in the middle of one of its edges. We consider two models: (1) the 2PM model, where each label is attached to its feature only on the middle of one of its horizontal edges, and (2) the r4PM model, where each label is attached to its feature on the middle of either one of its horizontal or vertical edges (known in advance). We assume that a sequence of pointshaped obstacles appear on the map on random locations. Three settings are considered for the behavior of the obstacle: (1) the obstacle is removed afterwards, (2) it remains on the map, and (3) it receives a similar label and remains on the map. Only two operations are permitted on the labels: flipping one or more labels, and/or resizing all labels. In the first setting, we suggest a data structure of O(n) space and O(n lg n) time in the 2PM model, and of O(n 2) time in the r4PM model, so that the updated labeling can be constructed for any obstacle position in O(lg n + k) time, where k is the minimum number of operations needed. For the second and third problems, we suggest an O(n) space and O(n lg n) time data structure that can place each obstacle (possibly with a label) on the map in O(lg n + k) time, if k label flips are sufficient to make room to place the new
Shooting Bricks with Orthogonal Laser Beams: A First Step towards Internal/External Map Labeling
"... We study several variants of a hybrid map labeling problem that combines the following two tasks: (i) a set A of points in a rectangle R needs to be labeled with rectangular labels on the right boundary of R using rectilinear onebend polylines called leaders to connect points and labels; (ii) a max ..."
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We study several variants of a hybrid map labeling problem that combines the following two tasks: (i) a set A of points in a rectangle R needs to be labeled with rectangular labels on the right boundary of R using rectilinear onebend polylines called leaders to connect points and labels; (ii) a maximum subset B ′ of a set B of fixed internal congruent rectangular labels in R needs to be selected such that B ′ is an independent set of labels and no leader intersects any label in B ′. We also call the points in A aliens, the labels of B bricks, and the leaders laser beams. Then the problem translates into every alien shooting a laser beam so that in total as few bricks as possible are destroyed. We provide algorithms and NPhardness results for different variants of the problem. 1
Sliding labels for dynamic point labeling
"... We study a dynamic labeling problem for points on a line that is closely related to labeling of zoomable maps. Typically, labels have a constant size on screen, which means that, as the scale of the map decreases during zooming, the labels grow relatively to the set of points, and conflicts may occu ..."
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We study a dynamic labeling problem for points on a line that is closely related to labeling of zoomable maps. Typically, labels have a constant size on screen, which means that, as the scale of the map decreases during zooming, the labels grow relatively to the set of points, and conflicts may occur due to overlapping labels. Our algorithmic problem is a combined dynamic selection and placement problem in a slidinglabel model: (i) select for each label ℓ a contiguous active range of map scales at which ℓ is displayed, and (ii) place each label at an appropriate position relative to its anchor point by sliding it along the point. The active range optimization (ARO) problem is to select active ranges and slider positions so that no two labels intersect at any scale and the sum of the lengths of active ranges is maximized. We present a dynamic programming algorithm to solve the discrete kposition ARO problem optimally and an FPTAS for the continuous sliding ARO problem. 1