Dynamic programming for graphs on surfaces
Juanjo Rué,Ignasi Sau,Dimitrios M. Thilikos +2 more
- 06 Jul 2010
- pp 372-383
TL;DR: This work uses singularity analysis over expressions obtained by the symbolic method to prove that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations, which proves that, when applied on surface cut decompositions, dynamic programming runs in 2O(k) n steps.
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Abstract: We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(kċlog k)ċ n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called surface cut decomposition, capturing how partial solutions can be arranged on a surface. Then we use singularity analysis over expressions obtained by the symbolic method to prove that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2O(k) ċ n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve all previous results in this direction.
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Figures
Fig. 2. Tree-cotree partition (T,C,X) of the complete graph K5 on vertices {1, 2, 3, 4, 5} embedded in the torus. White circles correspond to the vertices of its dual K∗5 . For simplicity, not all edges of K ∗ 5 are drawn, only a spanning tree C∗. The corresponding spanning cotree C of K5 is drawn with dashed edges. A spanning tree T of K5 is drawn with bold edges. Finally, the set X is given by the two edges {2, 4} and {3, 5}. 
Fig. 3. Non-crossing partitions on a disk, which enumerate the number of partial solutions on planar graphs when using sphere cut decompositions. 
Fig. 1. Merging branch decompositions (T1, µ1) and (T2, µ2) of two components H1 and H2 in a polyhedral decomposition (G, A) ofG = (V,E). There are three cases: (a)H1 andH2 share two vertices v1, v2 and the edge e = {v1, v2} is in E; (b) H1 and H2 share two vertices v1, v2 and e = {v1, v2} is not in E; (c) H1 and H2 share one vertex v. 
Fig. 5. Example of the construction of Σ′ and G′ in the proof of Lemma 8.9. On the left, we have a graph G (depicted with thick lines) embedded in a pseudo-surface Σ whose boundary is given by the set of nooses N = {N1, N2, N3, N4, N5} (in grey) pairwise intersecting at vertices of G, with θ(N ) = 4. On the right, the corresponding graph G′ embedded in a pseudo-surface Σ′ with boundary given byN ′ = {N ′1, N2, N3, N4, N5}, and such that θ(N ′) = 3. In this example, we have that |S| = 6 and |S′| = 7. 
Fig. 4. An example of an extended partition family. The surface Σ is the closure of one of the two connected components obtained by a torus after cutting along two disjoint nooses N1 and N2. Given that S = {s1, . . . , s7}, it holds that BΣ = {B1, B2} where B1 = {s1, s7, s6} and B2 = {s2, s5, s4, s3}. The set CA contains the four bold edges. Notice that, in this case, BΣA contains only one set consisting of the union of all the elements of BΣ and CA.
Citations
Catalan structures and dynamic programming in H-minor-free graphs
TL;DR: The approach builds on a combination of Demaine-Hajiaghayi's bounds on the size of an excluded grid in such graphs with a novel combinatorial result on certain branch decompositions of H-minor-free graphs to bound the number of ways vertex disjoint paths can be routed through the separators of such decomposition.
61
Faster parameterized algorithms for minor containment
TL;DR: This work improves the dependence on k of Hicks' result by showing that checking if H is a minor of G can be done in time O, and sets up an approach based on a combinatorial object called rooted packing, which captures the properties of the subgraphs of H that the authors seek in their dynamic programming algorithm.
59
Dynamic programming for graphs on surfaces
TL;DR: It is proved that, when applied on surface cut decompositions, dynamic programming runs in 2O(k) ṡ n steps, and the class of problems that can be solved in running times with a single-exponential dependence on branchwidth is considerably extended.
Faster parameterized algorithms for minor containment
Isolde Adler,Frederic Dorn,Fedor V. Fomin,Ignasi Sau,Dimitrios M. Thilikos +4 more
- 21 Jun 2010
TL;DR: This work improves the dependence on k of Hicks' result by showing that checking if H is a minor of G can be done in time, and obtains the first single-exponential algorithm for minor containment testing.
35
Fast Minor Testing in Planar Graphs
TL;DR: This work gives an algorithm that, given a planar n-vertex graph G and an h- Vertex graph H, either finds in time a model of H in G, or correctly concludes that G does not contain H as a minor.
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TL;DR: A new framework for designing fixed-parameter algorithms with subexponential running time---2O(&kradic;) nO(1) is introduced, which applies to a broad family of graph problems, called bidimensional problems, which includes many domination and problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominate set, disk dimension, and many others restricted to bounded-genus graphs.
Algorithms for Vertex Partitioning Problems on Partial k -Trees
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Efficient approximation for triangulation of minimum treewidth
Eyal Amir
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TL;DR: In this paper, the authors presented four approximation algorithms for finding triangulation of minimum treewidth for large graphs associated with real-world problems, and reported on experimental results confirming the effectiveness of their algorithms.
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