Polynomial bounds for the grid-minor theorem

Chekuri, Chandra; Chuzhoy, Julia

doi:10.1145/2591796.2591813

Cited by 53 publications

(71 citation statements)

References 31 publications

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“…More precisely, we develop a polynomial-time algorithm that given an instance of Snake Game, outputs an equivalent instance of Snake Game where the treewidth of the graph is bounded by a polynomial in k. Our procedure is based on the irrelevant vertex technique [30]. First, we exploit the relatively recent breakthrough result by Chekuri and Chuzhoy [7] that states that, for any positive integer t ∈ N, any graph whose treewidth is at least d · t c (for some fixed constants c and d) 2 has a t × t-grid as a minor, and hence also a so called t-wall as a subgraph (see Section 5). We utilize this result to argue that if the treewidth of our input graph is too large, then it has a ck-wall as a subgraph (for some fixed constant c) such that no vertex of this ck-wall belongs to the initial or final positions of the snake.…”

Section: Kernelizationmentioning

confidence: 99%

“…Lemma 3. 7. Let SG = G, k, init, fin be an instance of Snake Game and t ∈ N. Let C be the (k − 1)-configuration graph and C be a (k − 1)-sparse configuration graph of SG.…”

Section: Proofmentioning

confidence: 99%

“…First, we present the procedure executed for each iteration, called TrRedAlgIter (see Algorithm 3). For the execution of TrRedAlgIter, we use as "black box" a known algorithm (Chekuri and Chuzhoy, 2016 [7]) that (in polynomial time) either finds an r-wall in a graph, or reports that its treewidth is k O(1) (see Proposition 5.1).…”

Section: The Algorithm For Reduction Of Treewidthmentioning

confidence: 99%

See 2 more Smart Citations

The Parameterized Complexity of Motion Planning for Snake-Like Robots

Gupta

Sa'ar

Zehavi

2019

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence

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We study the parameterized complexity of a variant of the classic video game Snake that models real-world problems of motion planning. Given a snake-like robot with an initial position and a final position in an environment (modeled by a graph), our objective is to determine whether the robot can reach the final position from the initial position without intersecting itself. Naturally, this problem models a wide-variety of scenarios, ranging from the transportation of linked wagons towed by a locomotor at an airport or a supermarket to the movement of a group of agents that travel in an "ant-like" fashion and the construction of trains in amusement parks. Unfortunately, already on grid graphs, this problem is PSPACEcomplete [Biasi and Ophelders, 2016]. Nevertheless, we prove that even on general graphs, the problem is solvable in time k O(k) |I| O(1) where k is the size of the snake, and |I| is the input size. In particular, this shows that the problem is fixed-parameter tractable (FPT). Towards this, we show how to employ color-coding to sparsify the configuration graph of the problem to have size k O(k) |I| O(1) rather than |I| O(k) . We believe that our approach will find other applications in motion planning. Additionally, we show that the problem is unlikely to admit a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion planning problems (where the intermediate configurations of the motion are of importance) has so far been largely overlooked. Thus, our work is pioneering in this regard. * Ben-Gurion University of the Negev, Israel. siddhart@post.bgu.ac.il † Ben-Gurion University of the Negev, Israel. saag@bgu.ac.il ‡ Ben-Gurion University of the Negev, Israel. meiravze@bgu.ac.il

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Section: Kernelizationmentioning

confidence: 99%

“…Lemma 3. 7. Let SG = G, k, init, fin be an instance of Snake Game and t ∈ N. Let C be the (k − 1)-configuration graph and C be a (k − 1)-sparse configuration graph of SG.…”

Section: Proofmentioning

confidence: 99%

Section: The Algorithm For Reduction Of Treewidthmentioning

confidence: 99%

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The Parameterized Complexity of Motion Planning for Snake-Like Robots

Gupta

Sa'ar

Zehavi

2019

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence

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“…Kawarabayashi and Kobayashi [24] proved that f (k) ≤ 2 O(k 2 log k) , and Leaf and Seymour [30] proved that f (k) ≤ 2 O(k log k) . The function f (k) was first shown to be polynomial by Chekuri and Chuzhoy [7], who showed f (k) ≤ O(k 98 polylog k). A recent result of Chuzhoy [8] improves this to f (k) ≤ O(k 36 polylog k).…”

Section: Grid Minorsmentioning

confidence: 99%

Parameters Tied to Treewidth

Harvey

Wood

2016

Journal of Graph Theory

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Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory. This article surveys several graph parameters tied to treewidth, including separation number, tangle number, well‐linked number, and Cartesian tree product number. We review many results in the literature showing these parameters are tied to treewidth. In a number of cases we also improve known bounds, provide simpler proofs, and show that the inequalities presented are tight.

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“…It was further improved to f (g) = 2 O(g 2 / log g) by Kawarabayashi and Kobayashi [KK12] and by Leaf and Seymour [LS15]. The first polynomial upper bound of f (g) = O(g 98 poly log g) was proved by Chekuri and Chuzhoy [CC16]. The proof is constructive and provides a randomized algorithm that, given an n-vertex graph G of treewidth k, finds a model of the (g × g)-grid minor in G, with g =Ω(k 1/98 ), in time polynomial in both n and k. Unfortunately, the proof itself is quite complex.…”

Section: Introductionmentioning

confidence: 99%

Towards Tight(er) Bounds for the Excluded Grid Theorem

Chuzhoy

Tan

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function f : Z + → Z + , such that for every integer g > 0, every graph of treewidth at least f (g) contains the (g ×g)-grid as a minor. For every integer g > 0, let f (g) be the smallest value for which the theorem holds. Establishing tight bounds on f (g) is an important graph-theoretic question. Robertson and Seymour showed that f (g) = Ω(g 2 log g) must hold. For a long time, the best known upper bounds on f (g) were super-exponential in g. The first polynomial upper bound of f (g) = O(g 98 poly log g) was proved by Chekuri and Chuzhoy. It was later improved to f (g) = O(g 36 poly log g), and then to f (g) = O(g 19 poly log g). In this paper we further improve this bound to f (g) = O(g 9 poly log g). We believe that our proof is significantly simpler than the proofs of the previous bounds. Moreover, while there are natural barriers that seem to prevent the previous methods from yielding tight bounds for the theorem, it seems conceivable that the techniques proposed in this paper can lead to even tighter bounds on f (g).

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Polynomial bounds for the grid-minor theorem

Cited by 53 publications

References 31 publications

The Parameterized Complexity of Motion Planning for Snake-Like Robots

The Parameterized Complexity of Motion Planning for Snake-Like Robots

Parameters Tied to Treewidth

Towards Tight(er) Bounds for the Excluded Grid Theorem

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