Parameterized complexity of Strip Packing and Minimum Volume Packing

Ashok, Pradeesha; Kolay, Sudeshna; Meesum, Syed Mohammad; Saurabh, Saket

doi:10.1016/j.tcs.2016.11.034

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…In this problem, we are given a set of n rectangles S, and positive integers k, W ∈ N, and the objective is to decide whether all the rectangles in S can be packed in a rectangle (called a strip) of dimensions k × W . In [5], it was shown that if the maximum of the dimensions of the input rectangles, denoted by ℓ, is fixed, then the problem is FPT by k. Thus, the question whether the problem is FPT parameterized by k + ℓ remained open. By a straightforward reduction, we resolve this question as a corollary of our theorem.…”

Section: Our Contribution and Main Proof Ideasmentioning

confidence: 99%

Grid recognition: Classical and parameterized computational perspectives

Gupta

Sa'ar

Zehavi

2023

Journal of Computer and System Sciences

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Section: Our Contribution and Main Proof Ideasmentioning

confidence: 99%

Grid recognition: Classical and parameterized computational perspectives

Gupta

Sa'ar

Zehavi

2023

Journal of Computer and System Sciences

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“…In this problem, we are given a set of n rectangles S, and positive integers k, W ∈ N, and the objective is to decide whether all the rectangles in S can be packed in a rectangle (called a strip) of dimensions k × W . In [5], it was shown that if the maximum of the dimensions of the input rectangles, denoted by , is fixed (i.e., a constant independent of the input), then the problem is FPT parameterized by k. Specifically, running time of 8 k n O( 2 ) W was attained, which is not FPT with respect to k + . Thus, the question whether the problem is FPT parameterized by k + remained open.…”

Section: Our Contribution and Main Proof Ideasmentioning

confidence: 99%

“…Assume that f (N i,j (2)) = (12(i − 1) + 9, 8(j − 1) + 5). Observe that N i,j (2) has four neighbors, N i,j (1), N i,j (3), N i,j (4) and N i,j (5), therefore one of them is embedded at (12(i − 1) + 10, 8(j − 1) + 5). This is a contradiction, since from Lemma 4.12 we get that f ((12(i − 1) + 10, 8(j − 1) + 5)) = (12(i − 1) + 10, 8(j − 1) + 5) and f is an injection.…”

Section: Lemma 411 Let G Be a Grid Graph And Let F Be A Grid Graph Em...mentioning

confidence: 99%

“…Assume that f (N i,j (5)) = (12(i − 1) + 8, 8(j − 1) + 5). Since {N i,j (5), N i,j (6)} ∈ E(G) we get that f (N i,j (6)) = (12(i − 1) + 7, 8(j − 1) + 5) or f (N i,j (6)) = (12(i − 1) + 9, 8(j − 1) + 5) or f (N i,j (6)) = (12(i − 1) + 8, 8(j − 1) + 4). If f (N i,j (6)) = (12(i − 1) + 7, 8(j − 1) + 5) we get that one of N i,j (7), N i,j (8), N i,j (9) is embedded to (12(i − 1) + 7, 8(j − 1) + 6), but one of N i,j (3), N i,j (4) must be embedded there, so we get a contradiction.…”

Section: Lemma 411 Let G Be a Grid Graph And Let F Be A Grid Graph Em...mentioning

confidence: 99%

See 1 more Smart Citation

Grid Recognition: Classical and Parameterized Computational Perspectives

Gupta¹,

Sa'ar²,

Zehavi³

2021

Preprint

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Grid graphs, and, more generally, k × r grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph (given a graph G, decide whether it is a grid graph) is particularly hard-it was shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this paper, we provide several positive results in this regard in the framework of parameterized complexity (additionally, we present new and complementary hardness results). Specifically, our contribution is threefold. First, we show that the problem is fixed-parameter tractable (FPT) parameterized by k + mcc where mcc is the maximum size of a connected component of G. This also implies that the problem is FPT parameterized by td + k where td is the treedepth of G (to be compared with the hardness for pathwidth 2 where k = 3). (We note that when k and r are unrestricted, the problem is trivially FPT parameterized by td.) Further, we derive as a corollary that strip packing is FPT with respect to the height of the strip plus the maximum of the dimensions of the packed rectangles, which was previously only known to be in XP. Second, we present a new parameterization, denoted aG, relating graph distance to geometric distance, which may be of independent interest. We show that the problem is para-NP-hard parameterized by aG, but FPT parameterized by aG on trees, as well as FPT parameterized by k + aG. Third, we show that the recognition of k × r grid graphs is NP-hard on graphs of pathwidth 2 where k = 3. Moreover, when k and r are unrestricted, we show that the problem is NP-hard on trees of pathwidth 2, but trivially solvable in polynomial time on graphs of pathwidth 1.

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(Re)packing Equal Disks into Rectangle

Fomin,

Golovach,

Inamdar

et al. 2024

Discrete Comput Geom

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The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h, we ask whether it is possible by changing positions of at most h disks to pack $$n+k$$ n + k disks. Thus the problem of packing equal disks is the special case of our problem with $$n=h=0$$ n = h = 0 . While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for $$h=0$$ h = 0 . Our main algorithmic contribution is an algorithm that solves the repacking problem in time $$(h+k)^{\mathcal {O}(h+k)}\cdot |I|^{\mathcal {O}(1)}$$ ( h + k ) O ( h + k ) · | I | O ( 1 ) , where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h.

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Parameterized complexity of Strip Packing and Minimum Volume Packing

Cited by 4 publications

References 9 publications

Grid recognition: Classical and parameterized computational perspectives

Grid recognition: Classical and parameterized computational perspectives

Grid Recognition: Classical and Parameterized Computational Perspectives

(Re)packing Equal Disks into Rectangle

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