Approximation Algorithms via Structural Results for Apex-Minor-Free Graphs

Authors:
Erik D. Demaine;Mohammadtaghi Hajiaghayi;Ken-Ichi Kawarabayashi
Affiliations:
MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, USA 02139;AT&T Labs -- Research, Florham Park, USA 07932;National Institute for Informatics, Tokyo, Japan 101-8430
Venue:
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Year:
2009

Citing 27
Cited 4

Five-coloring maps on surfaces

Journal of Combinatorial Theory Series B
Approximation algorithms for NP-complete problems on planar graphs

Journal of the ACM (JACM)
Every planar graph is 5-choosable

Journal of Combinatorial Theory Series B
An Õ(n3/14)-coloring algorithm for 3-colorable graphs

Information Processing Letters
Color-critical graphs on a fixed surface

Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
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
The extremal function for complete minors

Journal of Combinatorial Theory Series B
Approximate TSP in Graphs with Forbidden Minors

ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
The chromatic number of a graph of girth 5 on a fixed surface

Journal of Combinatorial Theory Series B
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
Approximation schemes for minimum 2-edge-connected and biconnected subgraphs in planar graphs

SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Equivalence of local treewidth and linear local treewidth and its algorithmic applications

SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Local Tree-Width, Excluded Minors, and Approximation Algorithms

Combinatorica
Excluding any graph as a minor allows a low tree-width 2-coloring

Journal of Combinatorial Theory Series B
Bidimensional Parameters and Local Treewidth

SIAM Journal on Discrete Mathematics
Bidimensionality: new connections between FPT algorithms and PTASs

SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality

SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring

FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
A linear-time approximation scheme for planar weighted TSP

FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs

Journal of the ACM (JACM)
A subset spanner for Planar graphs,: with application to subset TSP

Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Object location using path separators

Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Approximation algorithms via contraction decomposition

SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation schemes for minimum 2-connected spanning subgraphs in weighted planar graphs

ESA'05 Proceedings of the 13th annual European conference on Algorithms
The Bidimensionality Theory and Its Algorithmic Applications 1

The Computer Journal

The Stackelberg Minimum Spanning Tree Game on Planar and Bounded-Treewidth Graphs

WINE '09 Proceedings of the 5th International Workshop on Internet and Network Economics
Sparse spanners vs. compact routing

Proceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures
Linear kernels for (connected) dominating set on H-minor-free graphs

Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
The Stackelberg minimum spanning tree game on planar and bounded-treewidth graphs

Journal of Combinatorial Optimization

Quantified Score

Hi-index	0.00

Visualization

Abstract

We develop new structural results for apex-minor-free graphs and show their power by developing two new approximation algorithms. The first is an additive approximation for coloring within 2 of the optimal chromatic number, which is essentially best possible, and generalizes the seminal result by Thomassen [32] for bounded-genus graphs. This result also improves our understanding from an algorithmic point of view of the venerable Hadwiger conjecture about coloring H -minor-free graphs. The second approximation result is a PTAS for unweighted TSP in apex-minor-free graphs, which generalizes PTASs for TSP in planar graphs and bounded-genus graphs [20,2,24,15]. We strengthen the structural results from the seminal Graph Minor Theory of Robertson and Seymour in the case of apex-minor-free graphs, showing that apices can be made adjacent only to vortices if we generalize the notion of vortices to "quasivortices" of bounded treewidth, proving a conjecture from [10]. We show that this structure theorem is a powerful tool for developing algorithms on apex-minor-free graphs, including for the classic problems of coloring and TSP. In particular, we use this theorem to partition the edges of a graph into k pieces, for any k , such that contracting any piece results in a bounded-treewidth graph, generalizing previous similar results for planar graphs [24] and bounded-genus graphs [15]. We also highlight the difficulties in extending our results to general H -minor-free graphs.