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Solving connectivity problems parameterized by treewidth in single exponential time
Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk
Jesper Nederlof, Dagstuhl seminar ‘Exploiting graph structure to cope with hard problems’, 2011
Johan van Rooij and Jakub Onufry Wojtaszszyk
Joint work with:
Outline
1. Definition of treewidth and -graphs.2. Dynamic programming on -graphs for local
problems.3. Introduction of our main result.4. The Isolation Lemma.5. on -graphs using “Cut&Count’’
and the Isolation Lemma in time.
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
DefinitionRefer to as a bag.
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
Definition
ExampleA B C
D
F G H
E
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
Definition
ExampleA B C
D
F G H
E
DBA
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
Definition
ExampleA B C
D
F G H
E
DBA
DGB
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
Definition
ExampleA B C
D
F G H
ED
GB
DBA
F GD
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
Definition
ExampleA B C
D
F G H
ED
GB
DBA
F GD
GB E
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
Definition
ExampleA B C
D
F G H
ED
GB
DBA C
EB
F GD
GB E
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
Definition
ExampleA B C
D
F G H
ED
GB
DBA C
EB
G HE
F GD
GB E
Treewidth
1. 2. 3. : All containing induce a connected subtree
A treedecomposition of graph is a pair where with and a tree with vertex set such that:
Definition
DefinitionThe width of a treedecomposition is . The treewidth of a graph is the minimum width among all possible tree decompositions of G.
Example (with and ):
graphsA simplification of graphs of treewidth ~ made up for this occassion.
DefinitionA graph is a graph with arranged in columns and .
So the treewidth of a graph is at most .
on graphsTheorem[Folklore]
on graphs can be solved in time.
on graphsTheorem[Folklore]
on graphs can be solved in time.
Proof Idea
on graphsTheorem[Folklore]
on graphs can be solved in time.
Proof
.
For and define as the maximum size of anindependent set of such that . Then,
Use DP to compute all and return
• Works for most ‘’local’’ problems.• Also extends to counting solutions.
on graphsDefinition( )
Given: graph , set terminals and integer Asked: Does there exist of size at most with
and connected?
Example ( ).
on graphs
Example ( ).
Definition( )Given: graph , set terminals and integer Asked: Does there exist of size at most with
and connected?
on graphs
Example ( ).
Definition( )Given: graph , set terminals and integer Asked: Does there exist of size at most with
and connected?
The straightforward dynamic programming approach needs atleast timesteps!
Cut&Count for
Theorem on graphs can be solved in time.
TheoremActually, the orginal and more precise version is:
There exists a Monte-Carlo algorithm that given a graph and tree decomposition of width solves in time. The algorithm cannot give false positives and givesfalse negatives with probability at most .
The Isolation Lemma
Given a set family over a universe and weight function , isolates if there is a unique such that .
Definition denotes
The Isolation Lemma
Given a set family over a universe and weight function , isolates if there is a unique such that .
Definition denotes
Lemma[Mulmuley et al., STOC 87]For every element , choose independently and uniformly at random, then:
.
The Isolation Lemma
Given a set family over a universe and weight function , isolates if there is a unique such that .
Definition denotes
Lemma[Mulmuley et al., STOC 87]For every element , choose independently and uniformly at random, then:
.
The Isolation Lemma
Proof• Define .
• So then .• This happens with probability at most by union bound.
• Notice does not depend on .• Thus, for a fixed .
• Now assume are both minimizers, and :
Lemma[Mulmuley et al., STOC 87]For every element , choose independently and uniformly at random, then:
.
Definition (The “Cut” part)
Cut&Count for Theorem
on graphs can be solved in time.
The set of relaxed solutions :
the set of solutions :
,
.
Let be an arbitrarily terminal. The set of cut solutions is
.
Example of A cut solution
Example ( ).
Example of A cut solution
Example ( ).
Example of A cut solution
Example ( ).
Red: in Green: in
Example of A cut solution
Example ( ).
Red: in Green: in
Cut&Count for
Observation .
Recall the set of cut solutions :
.
Aha! So suppose we just want to know where there is an even number of solutions, does it help to count ?
Proof
The count partLemma
can be computed in time.
Proof sketchFor and such that , define as the number of cut solutions of .
Write a recurrence relation expressing in terms of .
Compute all table entries using dynamic programming and read of the answer from the entries.
Corollary can be computed in time.
Using Isolation lemmaDefinition
.
Given a set family and weight , let be
• Recall , and .
1. Let be chosen independently and uniformly at random
2. For every compute and return iff a one is encountered.
Algorithm
• If is odd for some , we know is non-empty and can safely return .
• If was non-empty, the isolation lemma tells that (namely the smallest for which ) for some with probability at least .
Using Isolation lemma
• Recall , and .
1. Let be chosen independently and uniformly at random
2. For every compute and return iff a one is encountered.
• If is odd for some , we know is non-empty and can safely return .
• If was non-empty, the isolation lemma tells that (namely the smallest for which ) for some with probability at least .
Algorithm
Lemma(recalled)For every element , choose independently and uniformly at random, then:
.
Adding weights
Observation .
Recall the set of cut solutions :
.
Proof
Adding weightsLemma
For any , can be computed in time.
Proof sketchFor , and such that , define as the number of cut solutions of weight of .Write a recurrence relation expressing in terms of . Compute all table entries using dynamic programming and read off the answer from the entries.
ConclusionsTheorem
on graphs can be solved in time.
• In the paper we obtained lots of other results.• We ask many open problems, but maybe the best one is:
Is it possible to solve connectivity problems parameterized by treewidth in single-exponential time in a deterministic, more intuitive way?
For example, is there some structure present that allows us to always ignore many partial solutions?
Thanks for listening!
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