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DAIA: a decompose and improve algorithm for treedepth decomposition

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HAL Id: hal-03133843

https://hal-amu.archives-ouvertes.fr/hal-03133843

Submitted on 7 Feb 2021

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DAIA: a decompose and improve algorithm for

treedepth decomposition

Stéphane Grandcolas

To cite this version:

Stéphane Grandcolas. DAIA: a decompose and improve algorithm for treedepth decomposition.

[Re-search Report] Aix-Marseille Université (AMU); LIS. 2020. �hal-03133843�

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DAIA: a decompose and improve algorithm for

treedepth decomposition

Stéphane Grandcolas

Laboratoire d’Informatique et des Systèmes, Aix-Marseille université stephane.grandcolas@lis-lab.fr

Abstract

DAIA is a two-phases heuristic algorithm that searches for good treedepth decompositions of graphs. First it builds a treedepth decomposition partitionning recursively the vertices. Then it modifies the resulting tree in order to reduce its height.

Keywords and phrases Treedepth

Funding Stéphane Grandcolas: https://www.lis-lab.fr/

DOI 10.5281/zenodo.3884054

https://gitlab.lis-lab.fr/stephane.grandcolas/pace-2020

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Approach

Given a connected graph G the problem is to build a rooted tree whose nodes correspond to the vertices of G and such that, for each edge (u, v) of G, the nodes corresponding to u and v are in the same branch (that is either u is an ancestor of v or v is an ancestor of u). The objective is to minimize the height of the tree. We propose a method that consists in first building recursively a treedepth decomposition, then in improving the tree relocating subtrees. No preprocessing is applied to simplify the graph or to detect particular structures. The process can be reiterated as long as a given time limit is not reached.

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Initial decomposition

The process is based on the partitioning of the vertices of subgraphs of G. If G0= (V0, E0) is a subgraph of G, separation consists in partitioning V0 in three sets A, B and S, such that there is no edge of E0between A and B. S is called the separator. A treedepth decomposition of G0is obtained decomposing recursively the subgraphs induced by A and B, and connecting the resulting trees as subtrees of a path-branch built with the vertices of S. H. Althoby et al. [1] describe a simple greedy heuristic to solve the vertex separation problem (VSP). The idea is to transfer the vertices from S or B to A, starting with all the vertices in B but one, which is in S. At each step a vertex of S with as few neighbors in B as possible is chosen and moved to A, and its neighbors which are in B are moved to S. For the VSP the objective is to minimize the size of the separator. For treedepth decomposition the goal is also that the decomposition of the subgraphs induced by A and B give low trees, preferably of the same heights. We have developed the greedy heuristic as follows:

several flushes are made, moving the vertices from B to A, then from A to B, and so on, during the process the separation is evaluated and the best one is memorized.

We use two evaluations. The first one estimates roughly the height of the tree:

evalCG(S, A, B) = size(S) +pmax{#Edges(A), #Edges(B)}

The other aims at generating sets of similar sizes and a small separator:

evalSV (S, A, B) = card(S) + 1 min{size(A), size(B)} + 1

(3)

2

Before each partitioning an evaluation function is chosen at random. Then a number of separations are performed, each with a number of flushes and the same evaluation function.

3

Improvement

Two operations are implemented to make the tree lower. Both concern critical nodes, that is nodes whose subtrees contain all the nodes of the tree which are at the maximal depth (thus removing a critical node gives a lower tree). The first operation consists in pulling up subtrees that are independent of their nearest ancestor, see figure 1 (a subtree is independent of a given node if it contains no neighbor of the node).

... x A s A ... ... s ... x

Figure 1 Pullup: there is no neighbor of x in A, A is moved up.

The second operation consists in ejecting a critical node x to a lower position, carrying with it the subtrees with which it is dependent (figure 2, the blue subtrees are dependent of

x). It requires to search x critical correspondant cc, the lowest critical node that has subtrees

dependent of x. The height of the tree decreases if the maximal height of the subtrees dependent of x plus two is not greater than the height of the subtree rooted in cc before the operation. A2 1 ... A1 1 ... Ak 1 ... A2 1 ... A1 1 ... ... Ak 1 a1 cc ... cc . . . . a1 s x s x Ak nk A2 n2 A1 n1 Ak nk ... A1 n1 A2 n2 ...

Figure 2 Ejection: node x is moved under its critical correspondent with its dependent subtrees

To improve a given treedepth decomposition, we propose first to pullup the subtrees whose nearest ancestor is a critical node while there are some, then to eject critical nodes as long as it reduces the height of the tree.

References

1 Haeder Y. Althoby, Mohamed [Didi Biha], and André Sesboüé. Exact and heuristic methods for the vertex separator problem. Computers & Industrial Engineering, 139:106135, 2020. URL: http://www.sciencedirect.com/science/article/pii/S0360835219306047, doi:https:// doi.org/10.1016/j.cie.2019.106135.

Figure

Figure 2 Ejection: node x is moved under its critical correspondent with its dependent subtrees

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