Subexponential time algorithms and lower bounds for finding path and tree decompositions with few bags Joint work (in progress) with Hans Bodlaender. Jesper Nederlof Technical University Eindhoven Treewidth , , : All containing induce a
"Subexponential time algorithms and lower bounds" is the property of its rightful owner. Permission is granted to
download and print the materials on this website for personal, non-commercial use only, and to display it
on your personal computer provided you do not modify the materials and that you retain all copyright
notices contained in the materials. By downloading content from our website, you accept the terms of this
agreement.
Presentation Transcript
01
Subexponential time algorithms and lower bounds for finding path and tree decompositions with few bags Joint work (in progress) with Hans Bodlaender. Jesper Nederlof
Technical University Eindhoven<br>
02
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition<br>
03
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition<br>
04
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D F G H E<br>
05
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D F G H E<br>
06
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D F G H E<br>
07
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D F G H E G B E<br>
08
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D F G H E G B E<br>
09
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D F G H E G B E Separates A, F and CEH.<br>
10
Treewidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Definition The width of a treedecomposition is . The treewidth of a graph is the minimum width among all possible tree decompositions of G.<br>
11
Pathwidth ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Definition The width of a treedecomposition is . The treewidth of a graph is the minimum width among all possible tree decompositions of G. path path path path path<br>
12
Pathwidth Example A B C D F G H E G B E ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition path path<br>
13
Pathwidth Example A B D G H E G B E ,
,
: All containing induce a connected subtree. A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition path path Length: 4<br>
14
Bounded length Path decompositions n denotes |V|<br>
15
The algorithm<br>
16
Naïve branching algorithm Most naïve algorithm you can think of<br>
17
Naïve branching algorithm Most naïve algorithm you can think of<br>
18
Naïve branching algorithm Most naïve algorithm you can think of A B D G H E<br>
19
Naïve branching algorithm Most naïve algorithm you can think of A B D G H E<br>
20
Naïve branching algorithm Most naïve algorithm you can think of
Plus a bit of memorization A B D G<br>
21
A bit of memorization its incident edges are contained in the right bags<br>
22
A bit of memorization<br>
23
A bit of memorization<br>
24
The lower bound<br>
25
Reducing disj.strings to min length pathwidth K=199, length 5<br>
26
1 0 0 1 0 0 0 1 0 1<br>
27
1 0 0 1 0 0 0 1 0 1 8
9
10<br>
28
1 0 0 1 0 0 0 1 0 1 0
1<br>
29
Concluding Remarks Upper bound is quite similar to algorithm of Bodlaender and van Rooij (TAPAS’11).
Similar results for minimum length tree decompositions and intervalizing colored graphs.