On the Size of Minimal Separators for Treedepth Decomposition

08/22/2020
by   Zijian Xu, et al.
0

Treedepth decomposition has several practical applications and can be used to speed up many parameterized algorithms. There are several works aiming to design a scalable algorithm to compute exact treedepth decompositions. Those include works based on a set of all minimal separators. In those algorithms, although a number of minimal separators are enumerated, the minimal separators that are used for an optimal solution are empirically very small. Therefore, analyzing the upper bound on the size of minimal separators is an important problem because it has the potential to significantly reduce the computation time. A minimal separator S is called an optimal top separator if td(G) = |S| + td(G \ S), where td(G) denotes the treedepth of G. Then, we have two theoretical results on the size of optimal top separators. (1) For any G, there is an optimal top separator S such that |S| ≤ 2tw(G), where tw(G) is the treewidth of G. (2) For any c < 2, there exists a graph G such that any optimal top separator S of G have |S| > c · tw(G), i.e., the first result gives a tight bound on the size of an optimal top separator.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

04/05/2014

Nearly Optimal Minimax Tree Search?

Knuth and Moore presented a theoretical lower bound on the number of lea...
06/25/2018

Minimal Unimodal Decompositions on Trees

The decomposition of a density function on a domain into a minimal sum o...
06/12/2020

SMS in PACE 2020

We describe SMS, our submission to the exact treedepth track of PACE 202...
11/04/2021

Finding All Leftmost Separators of Size ≤ k

We define a notion called leftmost separator of size at most k. A leftmo...
06/01/2018

Joint Size and Depth Optimization of Sorting Networks

Sorting networks are oblivious sorting algorithms with many interesting ...
03/17/2021

Vertex Deletion Parameterized by Elimination Distance and Even Less

We study the parameterized complexity of various classic vertex deletion...
12/15/2017

Optimal top dag compression

It is shown that for a given ordered node-labelled tree of size n and wi...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.