Dynamic Programming for H-minor-free Graphs

Abstract

Abstract. We provide a framework for the design and analysis of dynamic programming algorithms for H-minor-free graphs with branchwidth at most k. Our technique applies to a wide family of problems where standard (deterministic) dynamic programming runs in 2 O(k·log k) · n O(1) steps, with n being the number of vertices of the input graph. Extending the approach developed by the same authors for graphs embedded in surfaces, we introduce a new type of branch decomposition for H-minor-free graphs, called an H-minor-free cut decomposition, and we show that they can be constructed in O h (n 3 ) steps, where the hidden constant depends exclusively on H. We show that the separators of such decompositions have connected packings whose behavior can be described in terms of a combinatorial object called -triangulation. Our main result is that when applied on H-minor-free cut decompositions, dynamic programming runs in 2steps. This broadens substantially the class of problems that can be solved deterministically in single-exponential time for H-minor-free graphs.

Hitting Minors on Bounded Treewidth Graphs. I. General Upper Bounds

“…Indeed, otherwise, for any solution S ⊆ V (G) of size at most k, G\S would contain as a minor a large enough grid, as a function of F, which would contain a planar graph in F as a minor, contradicting the fact that S is a solution. Therefore, when F contains a planar graph, Theorem 3.4 yields an algorithm to solve F-M-Deletion in time 2 O F ,H ( √ k•log k) • n when the input graph is H-minor-free, and, by Theorem 10.2, in time 2 O F ,g ( √ k) • n when the input graph has genus at most g. Plausibly, using the results in [45], the running time 2 O F ,g ( √ k) • n could be achieved also for H-minor-free graphs. To be best of our knowledge, subexponential algorithms for F-M-Deletion on these classes of graphs were not known before.…”

“…In both cases, the key tool is a special type of branch decomposition with nice topological properties. It seems plausible that this result could be extended to input graphs excluding a fixed graph H as a minor, by using the so-called H-minor-free cut decompositions introduced by Rué et al [45].…”

Dynamic programming for graphs on surfaces

“…Therefore, the existence of deterministic single-exponential algorithms parameterized by treewidth for connected packing-encodable problems in general graphs remains wide open. Our results for graphs on surfaces, as well as their generalization to any proper minor-free graph family [34], can be seen as an intermediate step towards an eventual positive answer to this question.…”

“…Extending these results for connected packing-encodable problems (where tables encode subsets of the middle sets) using the planarization approach of [15] appears to be a quite complicated task. We believe that our surface-oriented approach could be more successful in this direction and we find it an interesting, but non-trivial task [34].…”

Μερικές Διατάξεις Και Αλγόριθμοι Σε Γραφήματα

“…Τα αποτελέσματα της θεωρίας αυτής ξεκαθάρισαν αρκετά το τοπίο όσον αφορά τις κλάσεις των γραφημάτων που αποκλείουν κάποιο συγκεκριμενοποιημένο γράφημα σαν ελάσσον, και αποτέλεσαν/αποτελούν πηγή έμπνευσης αλλά και ισχυρό εργαλείο απόδειξης πολλών ακόμα αποτελεσμάτων στη Δομική [83,[123][124][125]196], αλλά και την Αλγοριθμική [5,51,86,90,212], Θεωρία Γραφημάτων. ΄Ενα από τα σημαντικότερα αποτελέσματα, και στους δύο αυτούς κλάδους, ήταν το πόρισμα ότι κάθε κλάση γραφημάτων που είναι κλειστή ως προς ελάσσονα, δηλαδή, τέτοια ώστε εάν ένα γράφημα ανήκει στην κλάση τότε και όλα τα ελάσσονά του επίσης ανήκουν στην κλάση, μπορεί να χαρακτηριστεί από ένα πεπερασμένο σύνολο απαγορευμένων γραφημάτων.…”