Dynamic programming for graphs on surfaces

Question

1. What are the contributions in "Dynamic programming for graphs on surfaces" ?

2. What is the challenge in the design of such algorithms?

3. What is the common step for performing dynamic programming?

4. How can one obtain a branch decomposition of G and G?

Accepted Answer

Rué et al. this paper proposed a dynamic programming for graphs on surfaces ( DP-SOM ) framework, which is based on the Courcelle theorem that graph problems expressible in Monadic Second Order Logic can be solved in f ( bw ) · n steps.

Accepted Answer

A challenge in the design of such algorithms is to reduce the contribution of branchwidth to the size of their tables and therefore to simplify f as much as possible.

Accepted Answer

For performing dynamic programming, their approach resides in a common preprocessing step that is to construct a surface cut decomposition.

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Given a carving decomposition (T, µ) of MG (or equivalently, a radial decomposition (T ∗, µ∗) of RG), one can obtain in a natural way branch decompositions of G and G∗ by redefining the bijection µ from the leaves of T to the edges of G (or G∗) that correspond to the faces of RG.

Accepted Answer

For instance, surface cut decompositions have recently been used in [Adler et al. 2010] for minor containment problems, where tables encode partitions of packings of the middle sets.

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Every connected packing-encodable problem whose input graph G is embedded in a surface of Euler genus γ, and has branchwidth at most k, can be solved in γO(k) · kO(γ) · γO(γ) · nO(1) steps.

Accepted Answer

Every connected packing-encodable problem whose input graph G is embedded in a surface of Euler genus γ, and has branchwidth at most k, can be solved by a dynamic programming algorithm on a surface cut decomposition of G with tables of size γO(k) · kO(γ) · γO(γ) · nO(1).

Accepted Answer

Given branch decompositions {(TH , µH) | H ∈ G)}, the authors can compute in linear time a branch decomposition (T, µ) of G such that w(T, µ) ≤ max{2, {w(TH , µH) | H ∈ G}}.

Accepted Answer

the number of faces that G determines over Σs is smaller than 2 + |N |+ γ. G defines a map on Σs (i.e., all faces are contractible), and consequently the authors can apply Euler’s formula.

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For i = 1, 2, the authors take an arbitrary edge {ai, bi} of T vi , the authors subdivide it by adding a new vertex wi, and then the authors build a tree T from T1 and T2 by adding the edge {w1, w2}.

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In Section 7, the authors define the concept of a surface cut decomposition, which is a key tool for the main result of this paper, summarized as follows.

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Then for every e ∈ E(T ), |ΨGe(mid(e))| = γO(γ) · kO(γ) · γO(k).ACM Transactions on Algorithms, Vol. -, No. -, Article -, Publication date: -.Before the authors give the proof of the above lemma, the authors first need to define formally the notion of non-crossing partitions on surfaces with boundary and then to prove some lemmata that combine elements from topology and combinatorics.

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A wider family of problems are those where the tables of dynamic programming encode connected packings of S into sets, i.e., collections of subsets of S that are pairwise disjoint and where each subset is a connected part of a partial solution (see Section 3 for the formal definitions).

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the authors merge the branch decompositions of the polyhedral components, using Lemma 5.5, and finally the authors add the edges of G with at least one endvertex in A, using Observation 5.1, to obtain a branch decomposition (T, µ) of G.Combining the discussion above with Lemmata 5.2 and 5.5 and Observation 5.1, and using that |A| = O(γ), the authors get thatw(T, µ) ≤ max{2, {w(TH , µH) | H ∈ G}}+ |A| ≤ 27k +O(γ) + |A| = 27k +O(γ).

Accepted Answer

As finding an optimal branch decomposition of a surface-embedded graph in polynomial time is open, it may be even possible that computing an optimal surface cut decomposition can be done in polynomial time.

Dynamic programming for graphs on surfaces

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Most frequently asked questions

1. What are the contributions in "Dynamic programming for graphs on surfaces" ?

2. What is the challenge in the design of such algorithms?

3. What is the common step for performing dynamic programming?

4. How can one obtain a branch decomposition of G and G?

5. What is the way to solve a problem where tables encode partitions of the middle sets?

6. How many packing-encodable problems can be solved in a linear manner?

7. How can the authors solve a packing-encodable problem?

8. What is the simplest way to compute branch decompositions of graphs?

9. What is the number of faces that G determines over s?

10. how do the authors build a tree from a graph with a vertex?

11. What is the key tool for the main result of this paper?

12. What is the first step in the proof of the above lemma?

13. What are the common problems in dynamic programming?

14. How do the authors get the branch decomposition of the polyhedral components?

15. How can the authors find an optimal graph in polynomial time?

Figures

Citations

Parameterized Algorithms

Parameterized Algorithms

Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions

Computational topology of graphs on surfaces

Hitting minors on bounded treewidth graphs

References

Analytic Combinatorics

Parameterized complexity theory

Analytic Combinatorics: RANDOM STRUCTURES

Invitation to fixed-parameter algorithms

Which Problems Have Strongly Exponential Complexity

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