Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
A simpler proof of the excluded minor theorem for higher surfaces
Journal of Combinatorial Theory Series B
All structured programs have small tree width and good register allocation
Information and Computation
Zero knowledge and the chromatic number
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
A polynomial-time approximation scheme for weighted planar graph TSP
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
The extremal function for complete minors
Journal of Combinatorial Theory Series B
Light spanners and approximate TSP in weighted graphs with forbidden minors
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Efficient Approximation for Triangulation of Minimum Treewidth
UAI '01 Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence
An approximation scheme for planar graph TSP
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
Excluding any graph as a minor allows a low tree-width 2-coloring
Journal of Combinatorial Theory Series B
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximation algorithms for classes of graphs excluding single-crossing graphs as minors
Journal of Computer and System Sciences
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
A subset spanner for Planar graphs,: with application to subset TSP
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Approximation algorithms via contraction decomposition
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
A polynomial-time approximation scheme for Steiner tree in planar graphs
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Catalan structures and dynamic programming in H-minor-free graphs
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth
Proceedings of the forty-second ACM symposium on Theory of computing
Decomposition, approximation, and coloring of odd-minor-free graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Faster approximation schemes and parameterized algorithms on H-minor-free and odd-minor-free graphs
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
A simpler algorithm and shorter proof for the graph minor decomposition
Proceedings of the forty-third annual ACM symposium on Theory of computing
SOFSEM'05 Proceedings of the 31st international conference on Theory and Practice of Computer Science
The Bidimensionality Theory and Its Algorithmic Applications 1
The Computer Journal
Spanning closed walks and TSP in 3-connected planar graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Structure theorem and isomorphism test for graphs with excluded topological subgraphs
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Light spanners in bounded pathwidth graphs
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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We prove that any graph excluding a fixed minor can have its edges partitioned into a desired number k of color classes such that contracting the edges in any one color class results in a graph of treewidth linear in k. This result is a natural finale to research in contraction decomposition, generalizing previous such decompositions for planar and bounded-genus graphs, and solving the main open problem in this area (posed at SODA 2007). Our decomposition can be computed in polynomial time, resulting in a general framework for approximation algorithms, particularly PTASs (with k ∼ 1/ε), and fixed-parameter algorithms, for problems closed under contractions in graphs excluding a fixed minor. For example, our approximation framework gives the first PTAS for TSP in weighted H-minor-free graphs, solving a decade-old open problem of Grohe; and gives another fixed-parameter algorithm for k-cut in H-minor-free graphs, which was an open problem of Downey et al. even for planar graphs. To obtain our contraction decompositions, we develop new graph structure theory to realize virtual edges in the clique-sum decomposition by actual paths in the graph, enabling the use of the powerful Robertson--Seymour Graph Minor decomposition theorem in the context of edge contractions (without edge deletions). This requires careful construction of paths to avoid blowup in the number of required paths beyond 3. Along the way, we strengthen and simplify contraction decompositions for bounded-genus graphs, so that the partition is determined by a simple radial ball growth independent of handles, starting from a set of vertices instead of just one, as long as this set is tight in a certain sense. We show that this tightness property holds for a constant number of approximately shortest paths in the surface, introducing several new concepts such as dives and rainbows.