An algorithm for the maximum weight independent set problem on outerstring graphs

Keil, J. Mark; Mitchell, Joseph S. B.; Pradhan, D.; Vatshelle, Martin

doi:10.1016/j.comgeo.2016.05.001

Cited by 33 publications

(31 citation statements)

References 21 publications

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“…If all points are on the boundary, then it is easy to represent each rectangle as a string (i.e., a Jordan curve) such that all strings have a point on the infinite face and two strings intersect if and only if not both rectangles should be taken; see Figure 2. This class of graphs is known as the outer-string graphs for which it is known that maximum-weight independent set is solvable in O(N 3 ) time, where N denotes the number of segments in a geometric representation of the input graph [8]. As such, BARP is solvable in O(n 9 ) time, but this is rather slow.…”

Section: Boundary-anchored Rectanglesmentioning

confidence: 99%

Packing boundary-anchored rectangles and squares

Biedl

Biniaz

Maheshwari

et al. 2020

Computational Geometry

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Consider a set P of n points on the boundary of an axis-aligned square Q. We study the boundary-anchored packing problem on P in which the goal is to find a set of interior-disjoint axis-aligned rectangles in Q such that each rectangle is anchored (has a corner at some point in P ), each point in P is used to anchor at most one rectangle, and the total area of the rectangles is maximized. Here, a rectangle is anchored at a point p in P if one of its corners coincides with p. In this paper, we show how to solve this problem in time linear in n, provided that the points of P are given in sorted order along the boundary of Q. We also consider the problem for anchoring squares and give an O(n 4 )-time algorithm when the points in P lie on two opposite sides of Q. * A preliminary version of this

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Section: Boundary-anchored Rectanglesmentioning

confidence: 99%

Packing boundary-anchored rectangles and squares

Biedl

Biniaz

Maheshwari

et al. 2020

Computational Geometry

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“…Let R be a geometric representation of Γ(G), where C is represented as a simple polygon P , and each curve is represented as a simple polygonal chain inside P such that one of its endpoints coincides with a distinct vertex of P . Keil et al [8] showed that given a geometric representation R of G, one can compute a maximum weight independent set of G in O(s 3 ) time, where s is the number of line segments in R. [8] observed that any maximum weight independent set of strings can be triangulated to create a triangulation P t of P , as shown in Figure 7(c). They used this idea to design a dynamic programming algorithm, as follows.…”

Section: Computing Subsuming Polygonsmentioning

confidence: 99%

Polygon simplification by minimizing convex corners

Bahoo

Durocher

Keil

et al. 2019

Theoretical Computer Science

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Let P be a polygon with r > 0 reflex vertices and possibly with holes and islands. A subsuming polygon of P is a polygon P such that P ⊆ P , each connected component R of P is a subset of a distinct connected component R of P , and the reflex corners of R coincide with those of R . A subsuming chain of P is a minimal path on the boundary of P whose two end edges coincide with two edges of P . Aichholzer et al. proved that every polygon P has a subsuming polygon with O(r) vertices, and posed an open problem to determine the computational complexity of computing subsuming polygons with the minimum number of convex vertices.We prove that the problem of computing an optimal subsuming polygon is NP-complete, but the complexity remains open for simple polygons (i.e., polygons without holes). Our NPhardness result holds even when the subsuming chains are restricted to have constant length and lie on the arrangement of lines determined by the edges of the input polygon. We show that this restriction makes the problem polynomial-time solvable for simple polygons.

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“…For a fixed triple i, j and k, we first compute OPT(I k ) and store the value in a table T , and will then do one look-up when computing the corresponding table entry of S[i, j]. To this end, we first note that index j is irrelevant for computing OPT(I k ) because for a fixed i and k, the set of L-shapes is the same for all k < j ≤ n. Therefore, for all pairs 1 ≤ i < k < n, we compute OPT(I k ) using the algorithm of Keil et al [23] and store it in T [i, k]. Since their algorithm takes O(n 3 ) and there are O(n 2 ) entries for T , the preprocessing step takes O(n 5 ) overall time.…”

Section: A (4 · Log Opt)-approximation Algorithmmentioning

confidence: 99%

“…To apply our algorithm, we now use the "weighted" median of the L-shapes in OPT[i, j]. Moreover, the algorithm of Keil et al [23] for the MIS problem on outerstring graphs works for weighted outerstring graphs as well. Finally, we can still compute the optimal solution for the weighted MIS problem when OPT[i, j] ≤ 4 for all 1 ≤ i < j ≤ j.…”

Section: A (4 · Log Opt)-approximation Algorithmmentioning

confidence: 99%

Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

Bose

Carmi

Keil

et al. 2019

Lecture Notes in Computer Science

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A graph G with n vertices is called an outerstring graph if it has an intersection representation of a set of n curves inside a disk such that one endpoint of every curve is attached to the boundary of the disk. Given an outerstring graph representation, the Maximum Independent Set (MIS) problem of the underlying graph can be solved in O(s 3 ) time, where s is the number of segments in the representation (Keil et al., Comput. Geom., 60:19-25, 2017). If the strings are of constant size (e.g., line segments, L-shapes, etc.), then the algorithm takes O(n 3 ) time.In this paper, we examine the fine-grained complexity of the MIS problem on some well-known outerstring representations. We show that solving the MIS problem on grounded segment and grounded square-L representations is at least as hard as solving MIS on circle graph representations. Note that no O(n 2−δ )-time algorithm, δ > 0, is known for the MIS problem on circle graphs. For the grounded string representations where the strings are y-monotone simple polygonal paths of constant length with segments at integral coordinates, we solve MIS in O(n 2 ) time and show this to be the best possible under the strong exponential time hypothesis (SETH). For the intersection graph of n L-shapes in the plane, we give a (4 · log OPT)-approximation algorithm for MIS (where OPT denotes the size of an optimal solution), improving the previously best-known (4 · log n)-approximation algorithm of Biedl and Derka (WADS 2017).

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An algorithm for the maximum weight independent set problem on outerstring graphs

Cited by 33 publications

References 21 publications

Packing boundary-anchored rectangles and squares

Packing boundary-anchored rectangles and squares

Polygon simplification by minimizing convex corners

Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

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