Cut & Count
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Cut & Count
Steiner Tree
Steiner Tree
Input A graph G, a set of terminals T, and a number k.
Steiner Tree
Input
QuestionIs there a set X of size at most k such that T is contained in X and G[X] is connected?
A graph G, a set of terminals T, and a number k.
Steiner Tree
Input
QuestionIs there a set X of size at most k such that T is contained in X and G[X] is connected?
A graph G, a set of terminals T, and a number k.
parameterized by treewidth
Promise
The input either has no solution or has exactly one solution.
Promise
The input either has no solution or has exactly one solution.
Want: All connected sets of size at most kthat contain the terminals.
Want: All connected sets of size at most kthat contain the terminals.
U: All sets of size at most k that contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Our universe consists of pairs (X,f) where !!!
X is a subset of size at most k containing the terminals, !
and f: X —> {Red,Green} is a 2-coloring of the components of X,
where f(v1) = Red.
|U| is odd if, and only if, G is a YES-instance.
For a bag x, store Tx[green,red; i]
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
For a bag B of a tree decomposition, guess how solution intersects the bag,
and further, guess the labeling.
For a bag B of a tree decomposition, guess how solution intersects the bag,
and further, guess the labeling.
For each i in {1,…,k}, we maintain:
For a bag B of a tree decomposition, guess how solution intersects the bag,
and further, guess the labeling.
For each i in {1,…,k}, we maintain:
T[green,red; i] = #of solutions of size i thatintercept the bag according to green/red.
Terminal in gray zone?
v1 in the green zone?
Introduce Bag
Introduce Bag
Introduce Bag
if v is not a terminal: Tx[green,red; i] = Ty[green,red; i]
Introduce Bag
if v is not a terminal: Tx[green,red; i] = Ty[green,red; i]
otherwise: zero
Introduce Bag
Introduce Bag
Introduce Bag
Tx[green,red; i] = Ty[green*,red*; i-1], provided v is not v1!
Introduce Bag
Tx[green,red; i] = Ty[green*,red*; i-1], provided v is not v1!
otherwise: zero
Introduce Bag
Introduce Bag
Introduce Bag
Tx[green,red; i] = Ty[green*,red*; i-1]
Forget Bag
Forget Bag
Forget Bag
Tx[green,red; i] = Ty[green,red; i]
Forget Bag
Tx[green,red; i] = Ty[green,red; i]
Sum over all relevant configurations in child bag.
Forget Bag
Tx[green,red; i] = Ty[green,red; i]
Sum over all relevant configurations in child bag.
Introduce Edge Bag
Tx[green,red; i] = Ty[green,red; i]
Introduce Edge Bag
Tx[green,red; i] = Ty[green,red; i]
Filter out entries with this edge crossing green to red.
Join Bag
Join Bag
Tp[green,red; r] x Tq[green,red; s]
Join Bag
Tp[green,red; r] x Tq[green,red; s]
Sum over all r,s such that:r+s - (#green + #red) = i
Want: All connected sets of size at most k that contain the terminals.
U: All sets of size at most k that contain the terminals.
Highlight some solution so it’s easier to find.
Universe
Family
Universe
Family
Universe
Family
Universe
Family
Isolation Lemma
Let F be a family over an universe U. For each element in the universe, assign a weight from
{1,2,…,N} uniformly and independently at random. !
Then, the probability that F is isolated by this weight function is at least
!
1 �|U|
N
Want: All connected sets of size at most kthat contain the terminals.
Want: All connected sets of size at most k that contain the terminals.
Focus: All connected sets of size at most kthat contain the terminals,
and have weight W.
U: All sets of size at most k and weight Wthat contain the terminals.
Focus: All connected sets of size at most kthat contain the terminals,
and have weight W.
Introduce Bag
Introduce Bag
Introduce Bag
Tx[green,red;i,w] = Ty[green*,red*; i-1,w-w(v)]
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