Information about Cut and Count

Steiner Tree

Input A graph G, a set of terminals T, and a number k. Steiner Tree

Input A graph G, a set of terminals T, and a number k. Is there a set X of size at most k such that T is contained in X and G[X] is connected? Steiner Tree Question

Input A graph G, a set of terminals T, and a number k. Is there a set X of size at most k such that T is contained in X and G[X] is connected? Steiner Tree parameterized by treewidth Question

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 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.

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 that intercept 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 k that contain the terminals.

Want: All connected sets of size at most k that contain the terminals.

Focus: All connected sets of size at most k that contain the terminals, and have weight W.

Focus: All connected sets of size at most k that contain the terminals, and have weight W. U: All sets of size at most k and weight W that contain the terminals.

Introduce Bag

Introduce Bag

Introduce Bag Tx[green,red;i,w] = Ty[green*,red*; i-1,w-w(v)]

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