Maximum Area Independent Sets in Disk Intersection Graphs

Bereg, Sergey; Dumitrescu, Adrian; Jiang, Minghui

doi:10.1142/s0218195910003220

Cited by 11 publications

(15 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The maximum area independent set (MAIS) problem for rectangles (squares, or disks, etc. ), is that of selecting a maximum area packing from a given set [3]; see classic papers such as [6,32,33,34,35] and also more recent ones [10,11,21] for quantitative bounds and constant approximations. These optimization problems are NP-hard, and there is a rich literature on approximation algorithms.…”

Section: Introductionmentioning

confidence: 99%

Anchored rectangle and square packings

Balas

Dumitrescu

Tóth

2017

Discrete Optimization

Self Cite

View full text Add to dashboard Cite

For points p 1 , . . . , p n in the unit square [0,1] 2 , an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r 1 , . . . , r n ⊆ [0, 1] 2 such that point p i is a corner of the rectangle r i (that is, r i is anchored at p i ) for i = 1, . . . , n. We show that for every set of n points in [0,1] 2 , there is an anchored rectangle packing of area at least 7/12 − O(1/n), and for every n ∈ N, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27.The above constructive lower bounds immediately yield constant-factor approximations, of 7/12 − ε for rectangles and 5/32 for squares, for computing anchored packings of maximum area in O(n log n) time. We prove that a simple greedy strategy achieves a 9/47-approximation for anchored square packings, and 1/3 for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n O(1/ε) and exp(poly(log(n/ε))) time, respectively.

show abstract

Section: Introductionmentioning

confidence: 99%

Anchored rectangle and square packings

Balas

Dumitrescu

Tóth

2017

Discrete Optimization

Self Cite

View full text Add to dashboard Cite

show abstract

“…Conjecture 3 (Bereg, Dumitrescu, and Jiang [3]). For any set S of (not necessary congruent) closed disks in the plane, there exists a subset I of pairwise-disjoint disks such that |I|/|F| ≥ Note that a disk D is centrally symmetric; for any finite family F of congruent disks (i.e., translates of a disk) in the plane, ν(F) ≥ Note.…”

Section: Resultsmentioning

confidence: 99%

Piercing Translates and Homothets of a Convex Body

Dumitrescu

Jiang

2010

Algorithmica

Self Cite

View full text Add to dashboard Cite

According to a classical result of Grünbaum, the transversal number τ (F ) of any family F of pairwise-intersecting translates or homothets of a convex body C in R d is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number τ (F ) to the packing number ν(F ) over all finite families F of translates (resp. homothets) of a convex body C in R d . Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in R d , and gave the first bounds on α(C) for convex bodies C in R d and on β(C) for convex bodies C in the plane. Here we show that β(C) is also bounded by a function of d for any convex body C in R d , and present new or improved bounds on both α(C) and β(C) for various convex bodies C in R d for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.

show abstract

“…4, we obtain an improved lower bound for the special case that the centrally symmetric convex body S is a disk. The previous best lower bound for disks was F (S) > 1/8.4898 [2]. Theorem 3 For a disk S, F (S) ≥ 1/λ disk , where λ disk = 8.3539 .…”

Section: Theoremmentioning

confidence: 98%

“…For the one-dimensional case, it is known that f (S) = F (S) = 1/2 for an interval S [2,15]. The aforementioned results of Zalgaller [22] and Ajtai [1], respectively, give lower and upper bounds of 1/8.6 ≤ F (S) ≤ 1/4 − 1/1728 for a square S. In Sect.…”

Section: Introductionmentioning

confidence: 95%

On Covering Problems of Rado

2009

Self Cite

View full text Add to dashboard Cite

T. Rado conjectured in 1928 that if F is a finite set of axis-parallel squares in the plane, then there exists an independent subset I ⊆ F of pairwise disjoint squares, such that I covers at least 1/4 of the area covered by F . He also showed that the greedy algorithm (repeatedly choose the largest square disjoint from those previously selected) finds an independent set of area at least 1/9 of the area covered by F . The analogous question for other shapes and many similar problems have been considered by R. Rado in his three papers (in Proc. up with a surprising example disproving T. Rado's conjecture. We revisit Rado's problem and present improved upper and lower bounds for squares, disks, convex bodies, centrally symmetric convex bodies, and others, as well as algorithmic solutions to these variants of the problem.

show abstract

Maximum Area Independent Sets in Disk Intersection Graphs

Cited by 11 publications

References 21 publications

Anchored rectangle and square packings

Anchored rectangle and square packings

Piercing Translates and Homothets of a Convex Body

On Covering Problems of Rado

Contact Info

Product

Resources

About