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Resource Constrained Shortest Paths
 PROC. 8TH EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA), SPRINGER LNCS
, 2000
"... The resource constrained shortest path problem (CSP) asks for the computation of a least cost path obeying a set of resource constraints. The problem is NPcomplete. We give theoretical and experimental results for CSP. In the theoretical part we present the hull approach, a combinatorial algorit ..."
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Cited by 27 (1 self)
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The resource constrained shortest path problem (CSP) asks for the computation of a least cost path obeying a set of resource constraints. The problem is NPcomplete. We give theoretical and experimental results for CSP. In the theoretical part we present the hull approach, a combinatorial algorithm for solving a linear programming relaxation and prove that it runs in polynomial time in the case of one resource. In the experimental part we compare the hull approach to previous methods for solving the LP relaxation and give an exact algorithm based on the hull approach. We also compare our exact algorithm to previous exact algorithms and approximation algorithms for the problem.
Towards tighter tables
 In Proceedings of Document Engineering
, 2005
"... Tables are provided in virtually all document formatting systems and are one of the most powerful and useful design elements in current web document standards. Unfortunately, optimal layout of tables which contain text is NPhard for reasonable layout requirements such as minimizing table height for ..."
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Cited by 5 (4 self)
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Tables are provided in virtually all document formatting systems and are one of the most powerful and useful design elements in current web document standards. Unfortunately, optimal layout of tables which contain text is NPhard for reasonable layout requirements such as minimizing table height for a given width [1]. We present two new independentlyapplicable techniques for table layout. The first technique is to solve a continuous approximation to the original layout problem by using a constantarea approximation of the cell content combined with a minimum width and height for the cell. The second technique starts by setting each column to its narrowest possible width and then iteratively reduces the height of the table by judiciously widening its columns. This second technique uses the actual text and linebreak rules rather than the constantarea approximation used by the first technique. We also investigate two hybrid approaches both of which use iterative column widening to improve the quality of an initial solution found using a different technique. In the first hybrid approach we use the continuous approximation technique to compute the initial column widths while in the second hybrid approach a modification of the HTML table layout algorithm is used to compute the initial widths. We found that all four techniques are reasonably fast and give significantly more compact layout than that of HTML layout engines.
Solving the simple continuous table layout problem
 In Proceedings, DocEng ’06
, 2006
"... Automatic table layout is required in web applications. Unfortunately, this is NPhard for reasonable layout requirements such as minimizing table height for a given width. One approach is to solve a continuous relaxation of the layout problem in which each cell must be large enough to contain the a ..."
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Cited by 4 (2 self)
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Automatic table layout is required in web applications. Unfortunately, this is NPhard for reasonable layout requirements such as minimizing table height for a given width. One approach is to solve a continuous relaxation of the layout problem in which each cell must be large enough to contain the area of its content. The solution to this relaxed problem can then guide the solution to the original problem. We give a simple and efficient algorithm for solving this continuous relaxation for the case that cells do not span multiple columns or rows. The algorithm is not only interesting in its own right but also because it provides insight into the geometry of table layout.
CNOP: a package for constrained network optimization
 IN PROC. 3RD INT. WORKSHOP ON ALGORITHM ENGINEERING AND EXPERIMENTS (ALENEX 01), LNCS 2153
, 2001
"... We present a generic package for resource constrained network optimization problems. We illustrate the flexibility and the use of our package by solving four applications: route planning, curve approximation, minimum cost reliability constrained spanning trees and the table layout problem. ..."
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Cited by 3 (0 self)
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We present a generic package for resource constrained network optimization problems. We illustrate the flexibility and the use of our package by solving four applications: route planning, curve approximation, minimum cost reliability constrained spanning trees and the table layout problem.
User Interface Layout with Ordinal and Linear Constraints
, 2006
"... User interfaces as well as documents use tabular layout mechanisms. The HTML table construct and the GridBag layout in Java are typical examples. There are, however, shortcomings of these mechanisms that become obvious with more advanced content like semistructured data or objectoriented models. W ..."
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Cited by 3 (2 self)
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User interfaces as well as documents use tabular layout mechanisms. The HTML table construct and the GridBag layout in Java are typical examples. There are, however, shortcomings of these mechanisms that become obvious with more advanced content like semistructured data or objectoriented models. We present a generalized table construct that solves these shortcomings and generalizes tabular layouts to a foundation of 2D layout. We describe an algorithm for specifying and rendering user interfaces  and 2D documents in general  using simple but expressive mathematical properties. In particular, the new tabular layout is described by ordinal and linear constraints. The ordinal information makes it possible to describe the general structure of a table and merge multiple cells into rectangular areas. The linear constraints allow it to map particular points of the table to particular coordinates or specify the size of areas in an absolute or relative manner. The resulting layout engine is easy to use and can render 2D information in realtime.
Minimum Sized Text Containment Shapes
"... In many textprocessing applications, we would like shapes that expand (or shrink) in size to fit their textual content. We address how to efficiently compute the minimum size for such text shapes. A variant of this problem is to take a fixed shape and determine the maximal size font that will still ..."
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Cited by 2 (2 self)
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In many textprocessing applications, we would like shapes that expand (or shrink) in size to fit their textual content. We address how to efficiently compute the minimum size for such text shapes. A variant of this problem is to take a fixed shape and determine the maximal size font that will still allow the content to fit into it. Our approach is to model the problem as a constrained optimisation problem with a single variable that controls the geometry of the text shape. We use a variant of secant search to determine the minimum area for the shape, guided by the area of the text. We represent the shape by regions that are composed of trapezoids whose coordinates are a linear function of the unknown variable. This allows us to use a novel linear time algorithm (based on computing Minkowski difference) that takes a trapezoid list and text height and determines the region in which a line of text of that height and some minimum width can start and still remain inside the shape.
Chips on Wafers, or packing rectangles into grids
 Preliminary version in Proc. WADS (Workshop on Algorithms and Data Structures
, 2005
"... A set of rectangles S is said to be grid packed if there exists a rectangular grid (not necessarily regular) such that every rectangle lies in the grid and there is at most one rectangle of S in each cell. The area of a grid packing is the area of a minimal bounding box that contains all the rectang ..."
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Cited by 1 (0 self)
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A set of rectangles S is said to be grid packed if there exists a rectangular grid (not necessarily regular) such that every rectangle lies in the grid and there is at most one rectangle of S in each cell. The area of a grid packing is the area of a minimal bounding box that contains all the rectangles in the grid packing. We present an approximation algorithm that given a set S of rectangles and a real constant ε> 0 produces a grid packing of S whose area is at most (1 + ε) times larger than an optimal grid packing in polynomial time. If ε is chosen large enough the running time of the algorithm will be linear. We also study several interesting variants, for example the smallest area grid packing containing at least k ≤ n rectangles, and given a region A grid pack as many rectangles as possible within A. Apart from the approximation algorithms we present several hardness results. 1
Towards Optimal Table Layout
, 2005
"... in current web document standards such as (X)HTML, CSS and XSL. Importantly, designers do not need to precisely specify the width of the table columns, instead the designer may allow these to adapt to the viewing context while still preserving the general design intended by the document author. Unfo ..."
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Cited by 1 (0 self)
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in current web document standards such as (X)HTML, CSS and XSL. Importantly, designers do not need to precisely specify the width of the table columns, instead the designer may allow these to adapt to the viewing context while still preserving the general design intended by the document author. Unfortunately, however, the adaptive layout provided by tables is still not powerful enough. We present a new approach to table layout that extends the current standard by allowing the designer to specify arbitrary linear equality and inequality constraints over the column widths and row heights. These may be strict (required) or preferred with weights indicating their relative importance. We also allow cells to be nonrectangular regions and we allow text to flow between arbitrary cells. We view table layout as a constrained optimisation problem. In addition to the designer's constraints there are implicit constraints reflecting the structure of the table and a constraint that each cell must be large enough to contain its content. Accurately and efficiently handling this containment constraint is the main source of difficulty. We present and evaluate two possible approaches, both of which are based on quadratic programming techniques.
Optimal Automatic Table Layout
"... Automatic layout of tables is useful in word processing applications and is required in online applications because of the need to tailor the layout to the viewport width, choice of font and dynamic content. However, if the table contains text, minimizing the height of the table for a fixed maximum ..."
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Automatic layout of tables is useful in word processing applications and is required in online applications because of the need to tailor the layout to the viewport width, choice of font and dynamic content. However, if the table contains text, minimizing the height of the table for a fixed maximum width is a difficult combinatorial optimization problem. We present three different approaches to finding the minimum height layout based on standard approaches for combinatorial optimization. All are guaranteed to find the optimal solution. The first is an A ⋆based approach that uses an admissible heuristic based on the area of the cell content. The second and third are constraint programming (CP) approaches using the same CP model. The second approach uses traditional CP search, while the third approach uses a hybrid CP/SAT approach, lazy clause generation, that uses learning to reduce the search required. We provide a detailed empirical evaluation of the three approaches and also compare them with two mixed integer programming (MIP) encodings due to Bilauca and Healy.