Topological graph theory
The graph genus problem is NP-complete
Journal of Algorithms
Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
A note on approximating graph genus
Information Processing Letters
Theoretical Computer Science - Special issue on computing and combinatorics
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Polynomial time approximation schemes for dense instances of NP -hard problems
Journal of Computer and System Sciences
What are the least tractable instances of max independent set?
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Vertex cover: further observations and further improvements
Journal of Algorithms
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Dominating sets in planar graphs: branch-width and exponential speed-up
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Improved Parameterized Algorithms for Planar Dominating Set
MFCS '02 Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science
Planarization of Graphs Embedded on Surfaces
WG '95 Proceedings of the 21st International Workshop on Graph-Theoretic Concepts in Computer Science
On the existence of subexponential parameterized algorithms
Journal of Computer and System Sciences - Special issue on Parameterized computation and complexity
Polynomial-time data reduction for dominating set
Journal of the ACM (JACM)
Parameterized complexity: exponential speed-up for planar graph problems
Journal of Algorithms
The dominating set problem is fixed parameter tractable for graphs of bounded genus
Journal of Algorithms
Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs
Journal of the ACM (JACM)
Tight lower bounds for certain parameterized NP-hard problems
Information and Computation
A refined search tree technique for Dominating Set on planar graphs
Journal of Computer and System Sciences
Strong computational lower bounds via parameterized complexity
Journal of Computer and System Sciences
Parameterized Complexity
When is weighted satisfiability FPT?
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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We study the fixed-parameter tractability, subexponential time computability, and approximability of the well-known NP-hard problems: independent set, vertex cover, and dominating set. We derive tight results and show that the computational complexity of these problems, with respect to the above complexity measures, is dependent on the genus of the underlying graph. For instance, we show that, under the widely-believed complexity assumption W[1] FPT, independent set on graphs of genus bounded by g"1(n) is fixed parameter tractable if and only ifg"1(n)=o(n^2), and dominating set on graphs of genus bounded by g"2(n) is fixed parameter tractable if and only ifg"2(n)=n^o^(^1^). Under the assumption that not all SNP problems are solvable in subexponential time, we show that the above three problems on graphs of genus bounded by g"3(n) are solvable in subexponential time if and only ifg"3(n)=o(n). We also show that the independent set, the kernelized vertex cover, and the kernelized dominating set problems on graphs of genus bounded by g"4(n) have PTAS if g"4(n)=o(n/logn), and that, under the assumption P NP, the independent set problem on graphs of genus bounded by g"5(n) has no PTAS if g"5(n)=@W(n), and the vertex cover and dominating set problems on graphs of genus bounded by g"6(n) have no PTAS if g"6(n)=n^@W^(^1^).