Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

@inproceedings{CohenAddad2019AlmostTL,
  title={Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs},
  author={Vincent Cohen-Addad and {\'E}ric Colin de Verdi{\`e}re and D{\'a}niel Marx and Arnaud de Mesmay},
  booktitle={SoCG},
  year={2019}
}
We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus g has a cut graph of length at most a given value. We… 
2 Citations

Figures from this paper

Parameterized Complexity of Small Weight Automorphisms and Isomorphisms
TLDR
An application of this yields an FPT algorithm for finding exact weight k nontrivial automorphisms in d-hypergraphs, d as second fixed parameter.
Modern Lower Bound Techniques in Database Theory and Constraint Satisfaction
TLDR
Different types of lower bounds are overviewed, and how they can be applied to problems in database theory and constraint satisfaction are seen.

References

SHOWING 1-10 OF 48 REFERENCES
Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs
TLDR
The main novel idea for the results is to understand what structures instead of grids are needed if the authors want to exploit optimally a certain value  G of the genus, Reductions to planar problems usually involve a gridlike structure.
A Separator Theorem for Graphs of Bounded Genus
Tight conditional lower bounds for counting perfect matchings on graphs of bounded treewidth, cliquewidth, and genus
TLDR
It is proved that, assuming the counting version of the Strong Exponential-Time Hypothesis (#SETH), the problem of counting perfect matchings • has no (2 --- e)knO(1) time algorithm for any e > 0 on graphs of treewidth k (but it can be solved in time O(nk+1) if a k-expression is given).
A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs
TLDR
An algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds and allows us to derive matching 2Ω(n1−1/d) lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs.
Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs
We introduce a new framework for designing fixed-parameter algorithms with subexponential running time---2O(&kradic;) nO(1). Our results apply to a broad family of graph problems, called
Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals
TLDR
A polynomial-time algorithm for the minimum multicut problem, which asks for a minimum-weight set of edges such that, after deleting these edges, the two terminals of each pair belong to different connected components of the graph.
Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering
TLDR
All the results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.
On Subexponential Parameterized Algorithms for Steiner Tree and Directed Subset TSP on Planar Graphs
TLDR
Steiner Tree is shown to be the first "genuinely planar" problem for which the square root phenomenon does not appear, and the combination of the results for Steiner Tree and Subset Traveling Salesman is the first to admit a parameter-preserving polynomial kernel on planar graphs unless ETH fails.
Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)
TLDR
The main hardness result is a matching lower bound for the algorithm, and it is shown that planar SCSS does not have an f(k) · no(√k) algorithm for any computable function f, unless the Exponential Time Hypothesis (ETH) fails.
...
1
2
3
4
5
...