TY - GEN
T1 - Grid Recognition
T2 - 32nd International Symposium on Algorithms and Computation, ISAAC 2021
AU - Gupta, Siddharth
AU - Sa’ar, Guy
AU - Zehavi, Meirav
N1 - Funding Information:
Funding Supported in part by the United States – Israel Binational Science Foundation (BSF) grant no. 2018302 and Israel Science Foundation (ISF) individual research grant no. 1176/18. Siddharth Gupta: Supported in part by the Zuckerman STEM Leadership Program. Guy Sa’ar: Supported in part by the Israeli Smart Transportation Research Center (ISTRC).
Publisher Copyright:
© Siddharth Gupta, Guy Sa’ar, and Meirav Zehavi.
PY - 2021/12/1
Y1 - 2021/12/1
N2 - 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 can be embedded into 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, as td ≤ mcc (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.
AB - 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 can be embedded into 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, as td ≤ mcc (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.
KW - Grid graph
KW - Grid recognition
KW - Parameterized complexity
UR - http://www.scopus.com/inward/record.url?scp=85122458524&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ISAAC.2021.37
DO - 10.4230/LIPIcs.ISAAC.2021.37
M3 - Conference contribution
AN - SCOPUS:85122458524
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 32nd International Symposium on Algorithms and Computation, ISAAC 2021
A2 - Ahn, Hee-Kap
A2 - Sadakane, Kunihiko
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Y2 - 6 December 2021 through 8 December 2021
ER -