The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
Easy problems for tree-decomposable graphs
Journal of Algorithms
Discrete Mathematics - Topics on domination
Graph searching and a min-max theorem for tree-width
Journal of Combinatorial Theory Series B
Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
Quickly excluding a planar graph
Journal of Combinatorial Theory Series B
Approximation schemes for covering and packing problems in image processing and VLSI
Journal of the ACM (JACM)
Separators for sphere-packings and nearest neighbor graphs
Journal of the ACM (JACM)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
The expression of graph properties and graph transformations in monadic second-order logic
Handbook of graph grammars and computing by graph transformation
NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs
Journal of Algorithms
A partial k-arboretum of graphs with bounded treewidth
Theoretical Computer Science
Label placement by maximum independent set in rectangles
WADS '97 Selected papers presented at the international workshop on Algorithms and data structure
Three-dimensional orthogonal graph drawing algorithms
Discrete Applied Mathematics
Some APX-completeness results for cubic graphs
Theoretical Computer Science
Journal of the ACM (JACM)
Approximation algorithms for independent sets in map graphs
Journal of Algorithms
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
Polynomial-time approximation schemes for packing and piercing fat objects
Journal of Algorithms
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Robust algorithms for restricted domains
Journal of Algorithms - Special issue: Twelfth annual ACM-SIAM symposium on discrete algorithms
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Bidimensionality: new connections between FPT algorithms and PTASs
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Polynomial-Time Approximation Schemes for Geometric Intersection Graphs
SIAM Journal on Computing
Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs
ACM Transactions on Algorithms (TALG)
Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs
Journal of the ACM (JACM)
Fast and accurate algorithms for protein side-chain packing
Journal of the ACM (JACM)
Ad hoc networks beyond unit disk graphs
Wireless Networks
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Minimum Clique Partition in Unit Disk Graphs
Graphs and Combinatorics - The Japan Conference on Computational Geometry and Graphs (JCCGG2009)
Bidimensionality and geometric graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Efficient approximation schemes for geometric problems?
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Polynomial kernels for hard problems on disk graphs
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Parameterized Complexity and Approximation Algorithms
The Computer Journal
Bidimensionality and geometric graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Graph minors and parameterized algorithm design
The Multivariate Algorithmic Revolution and Beyond
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Bidimensionality theory was introduced by Demaine et al. [JACM 2005] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been successfully applied to yield subex-ponential time parameterized algorithms, EPTASs and linear kernels for many problems on families of graphs excluding a fixed graph H as a minor. In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, on map graphs and unit disk graphs, PTASs for Diamond Hitting Set on map graphs and unit disk graphs, and a PTAS and a subexponential time algorithm for Cycle Packing on unit disk graphs. To the best of our knowledge, these results were previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. Our results are based on the recent decomposition theorems proved by Fomin et al. in [SODA 2011] and novel grid-excluding theorems in unit disk and map graphs without large cliques. Our algorithms work directly on the input graph and do not require the geometric representations of the input graph. We also show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in Rd, d ≥ 3. Specifically, we prove that Feedback Vertex Set on unit-ball graphs in R3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on, fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in Rd for every fixed d.