Connecting the Dots (with Minimum Crossings)

Agrawal, Akanksha; Guśpiel, Grzegorz; Madathil, Jayakrishnan; Saurabh, Saket; Zehavi, Meirav

doi:10.4230/lipics.socg.2019.7

Cited by 5 publications

(9 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Assume towards a contradiction that p f (i, j) = +. Then, we get that f (N i,j (1)) = (12(i − 1) + 11, 8(j − 1) + 6), so, by Lemma 4.13 we get that there exists u ∈ P i,j such that f (u) = (12(i − 1) + 15, 8(j − 1) + 6) = (12(i) + 3, 8(j − 1) + 6) = f (S i,j (2)). This is a contradiction, since f is an injection.…”

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

confidence: 90%

“…We now give a polynomial reduction from the Batteries problem to Grid Embedding where the output graph satisfies that a G is bounded by a fixed constant (if it is a yes-instance). 2 For this purpose, we present the battery gadget (see Figures 8 and 9). The battery gadget is composed of a 13 × 9 rectangle, which corresponds to a cell in the battery holder.…”

Section: Corollary 48 There Exists a Polynomial Reduction From Sat To...mentioning

confidence: 99%

“…) such that all the embeddings of the gadgets "agree". Intuitively, we can think of our goal as having a puzzle where we show that every piece of the puzzle is (2,2) (2,10) (…”

Section: Lemma 418 Letmentioning

confidence: 99%

“…Let G be a connected grid graph. For any k ∈ N, if |V (G)| ≥ k 2 , then there exists a simple path P in G with size k.…”

Section: T Has Exactly One 81amentioning

confidence: 99%

“…Most (in particular, the early efforts) have been directed at variants of the classic Crossing Minimization problem, introduced by Turán in 1940 [56], parameterized by the number of crossings (see, e.g., [34,46,26,39,40,47]). However, in the past few years, there is an increasing interest in the analysis of a variety of other problems in graph drawing from the perspective of parameterized complexity (see, e.g., [10,2,35,14,38,6,21,20,11,23,49,48] and the upcoming Dagstuhl seminar [1]).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Grid Recognition: Classical and Parameterized Computational Perspectives

Gupta¹,

Sa'ar²,

Zehavi³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

confidence: 90%

Section: Corollary 48 There Exists a Polynomial Reduction From Sat To...mentioning

confidence: 99%

“…) such that all the embeddings of the gadgets "agree". Intuitively, we can think of our goal as having a puzzle where we show that every piece of the puzzle is (2,2) (2,10) (…”

Section: Lemma 418 Letmentioning

confidence: 99%

“…Let G be a connected grid graph. For any k ∈ N, if |V (G)| ≥ k 2 , then there exists a simple path P in G with size k.…”

Section: T Has Exactly One 81amentioning

confidence: 99%