Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
A separator theorem for graphs with an excluded minor and its applications
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Some results on tree decomposition of graphs
Journal of Graph Theory
Dynamic Programming on Graphs with Bounded Treewidth
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
Combinatorica
Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms
Theoretical Computer Science
On digraph coloring problems and treewidth duality
European Journal of Combinatorics
Every minor-closed property of sparse graphs is testable
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Constant-Time Approximation Algorithms via Local Improvements
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Approximating the distance to properties in bounded-degree and general sparse graphs
ACM Transactions on Algorithms (TALG)
European Journal of Combinatorics
An improved constant-time approximation algorithm for maximum~matchings
Proceedings of the forty-first annual ACM symposium on Theory of computing
Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs
SIAM Journal on Computing
Local Graph Partitions for Approximation and Testing
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Parameter testing in bounded degree graphs of subexponential growth
Random Structures & Algorithms
Testing outerplanarity of bounded degree graphs
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Every property of hyperfinite graphs is testable
Proceedings of the forty-third annual ACM symposium on Theory of computing
New sublinear methods in the struggle against classical problems
New sublinear methods in the struggle against classical problems
Combinatorial Optimization on Graphs of Bounded Treewidth
The Computer Journal
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 I
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Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constant-time algorithms. For any ε 0, a partitioning oracle provides query access to a fixed partition of the input bounded-degree minor-free graph, in which every component has size poly(1/ε), and the number of edges removed is at most εn, where n is the number of vertices in the graph. However, the oracle of Hassidim et al. makes an exponential number of queries to the input graph to answer every query about the partition. In this paper, we construct an efficient partitioning oracle for graphs with constant treewidth. The oracle makes only O(poly(1/ε)) queries to the input graph to answer each query about the partition. Examples of bounded-treewidth graph classes include k-outerplanar graphs for fixed k, series-parallel graphs, cactus graphs, and pseudoforests. Our oracle yields poly(1/ε)-time property testing algorithms for membership in these classes of graphs. Another application of the oracle is a poly(1/ε)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive en in graphs with bounded treewidth. Finally, the oracle can be used to test in poly(1/ε) time whether the input boundedtreewidth graph is k-colorable or perfect.