Reut Levi
Abstract
Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient partition oracles. A partition oracle is a procedure that, given access to the incidence lists representation of a bounded-degree graph G= (V,E) and a parameter ϵ, when queried on a vertex v ∈ V, returns the part (subset of vertices) that v belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most ϵ |V|. In this work, we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/ϵ, improving on the result of Hassidim et al. (Proceedings of FOCS 2009), who gave a partition oracle with query complexity exponential in 1/ϵ. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.
- Noga Alon, Paul D. Seymour, and Robin Thomas. 1990. A separator theorem for graphs with an excluded minor and its applications. In Proceedings of the 22nd Annual ACM Symposium on the Theory of Computing. 293--299. Google Scholar
Digital Library
- Kazuoki Azuma. 1967. Weighted sums of certain dependent variables. Tôhoku Math 19, 3, 357--367.Google ScholarCross Ref
- Itai Benjamini, Oded Schramm, and Asaf Shapira. 2008. Every minor-closed property of sparse graphs is testable. In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing. 393--402. Google ScholarDigital Library
- John M. Boyer and Wendy J. Myrvold. 2004. On the cutting edge: simplified O(n) planarity by edge addition. Journal of Graph Algorithms and Applications 8, 3, 241--273.Google ScholarCross Ref
- Artur Czumaj, Asaf Shapira, and Christian Sohler. 2009. Testing hereditary properties of nonexpanding bounded-degree graphs. SIAM Journal on Computing 38, 6, 2499--2510. Google ScholarDigital Library
- Artur Czumaj, Oded Goldreich, Dana Ron, C. Seshadhri, Asaf Shapira, and Christian Sohler. 2014. Finding cycles and trees in sublinear time. Random Struct. Algorithms 45, 2, 139--184. DOI:http://dx.doi.org/10.1002/rsa.20462Google ScholarDigital Library
- Andrzej Czygrinow, Michal Hańćkowiak, and Wojciech Wawrzyniak. 2008. Fast distributed approximations in planar graphs. In Proceedings of the 22nd International Symposium on Distributed Computing (DISC). 78--92. Google ScholarDigital Library
- Hubert de Fraysseix, Patrice Ossona de Mendez, and Pierre Rosenstiehl. 2006. Trémaux trees and planarity. International Journal of Foundations of Computer Science 17, 5, 1017--1030.Google ScholarCross Ref
- Alan Edelman, Avinatan Hassidim, Huy N. Nguyen, and Krzysztof Onak. 2011. An efficientpartitioning oracle for bounded-treewidth graphs. In Proceedings of the 15th International Workshop on Randomization and Computation (RANDOM'11). 530--541. Google ScholarDigital Library
- Gábor Elek. 2006. The combinatorial cost. Technical Report math/0608474. ArXiv.Google Scholar
- Gábor Elek. 2008. L2-spectral invariants and convergent sequences of finite graphs. Journal of Functional Analysis 254, 10, 2667--2689.Google ScholarCross Ref
- Gábor Elek. 2010. Parameter testing in bounded degree graphs of subexponential growth. Random Structures and Algorithms 37, 2, 248--270. Google ScholarDigital Library
- Shimon Even and Robert Endre Tarjan. 1976. Computing an st-numbering. Theoretical Computer Science 2, 3, 339--344.Google ScholarCross Ref
- M. R. Garey, David S. Johnson, and Larry J. Stockmeyer. 1976. Some simplified NP-Complete graphs problems. Theoretical Computer Science 1, 3, 237--267.Google ScholarCross Ref
- Avinatan Hassidim, Jonathan A. Kelner, Huy N. Nguyen, and Krzysztof Onak. 2009. Local graph partitions for approximation and testing. In Proceedings of the 50th Annual Symposium on Foundations of Computer Science. 22--31. Google ScholarDigital Library
- John E. Hopcroft and Robert Endre Tarjan. 1974. Efficient planarity testing. Journal of the ACM 21, 4, 549--568. Google ScholarDigital Library
- John E. Hopcroft and J. K. Wong. 1974. Linear time algorithm for isomorphism of planar graphs (preliminary report). In Proceedings of the 6th Annual ACM Symposium on the Theory of Computing. 172--184. Google ScholarDigital Library
- W. Mader. 1967. Homomorphieeigenschaften und mittlere Kantendichte von Graphen. Mathematische Annalen 174, 4, 265--268. DOI:http://dx.doi.org/10.1007/BF01364272Google ScholarCross Ref
- Crispin St. J. A. Nash-Williams. 1964. Decomposition of finite graphs into forests. Journal of the London Mathematical Society s1-39, 1, 12. DOI:http://dx.doi.org/10.1112/jlms/s1-39.1.12Google ScholarCross Ref
- Ilan Newman and Christian Sohler. 2011. Every property of hyperfinite graphs is testable. In Proceedings of the 43rd Annual ACM Symposium on the Theory of Computing. 675--684. Google ScholarDigital Library
- Krzysztof Onak. 2010. New Sublinear Methods in the Struggle Against Classical Problems. Ph.D. Dissertation. MIT, Cambridge, MA. Google ScholarDigital Library
- Krzysztof Onak. 2012. On the Complexity of Learning and Testing Hyperfinite Graphs. Retrieved November 5, 2014 from http://onak.pl/download/publications/Onak-hyperfinite_complexity.pdf.Google Scholar
- Seth Pettie and Vijaya Ramachandran. 2008. Randomized minimum spanning tree algorithms using exponentially fewer random bits. ACM Transactions on Algorithms 4, 1. Google ScholarDigital Library
- Neil Robertson and Paul D. Seymour. 1995. Graph minors. XIII. The disjoint paths problem. Journal of Combinatorial Theory Series B 63, 1, 65--110. Google ScholarDigital Library
- Neil Robertson and Paul D. Seymour. 2004. Graph minors. XX. Wagner's conjecture. Journal of Combinatorial Theory Series B 92, 1, 325--357. Google ScholarDigital Library
- Wei Kuan Shih and Wen Lian Hsu. 1999. A new planarity test. Theoretical Computer Science 223, 1--2, 179--191. Google ScholarDigital Library
- Yuichi Yoshida and Hiro Ito. 2010. Testing outerplanarity of bounded degree graphs. In Proceedings of the 14th International Workshop on Randomization and Computation (RANDOM'10). 642--655. Google ScholarDigital Library
Index Terms
- A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor
Recommendations
A quasi-polynomial time partition oracle for graphs with an excluded minor
ICALP'13: Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part IMotivated by the problem of testing planarity and related properties, we study the problem of designing efficient partition oracles. A partition oracle is a procedure that, given access to the incidence lists representation of a bounded-degree graph G=(...
On the vertex-arboricity of K5-minor-free graphs of diameter 2
An induced forest k-partition of a graph G is a k-partition (V"1,V"2,...,V"k) of the vertex set of G such that, for each i with 1@?i@?k, the subgraph induced by V"i is a forest. The vertex-arboricity of a graph G is the minimum k such that G has an ...
Polynomial Kernels and Faster Algorithms for the Dominating Set Problem on Graphs with an Excluded Minor
Parameterized and Exact ComputationThe domination number of a graph G = ( V , E ) is the minimum size of a dominating set U V , which satisfies that every vertex in V U is adjacent to at least one vertex in U . The notion of a problem kernel refers to a polynomial time ...
Comments