A polynomial time algorithm for bounded directed pathwidth Hisao - - PowerPoint PPT Presentation

Hisao Tamaki Meiji University

A polynomial time algorithm for bounded directed pathwidth

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Undirected *𝑣, 𝑤+

• some 𝑗

𝑣, 𝑤 ∈ 𝑌𝑗

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

• 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽𝑤 = 𝑗 𝑤 ∈ 𝑌𝑗+ is a single non-

empty interval

• 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such

that 𝑣 ∈ 𝑌𝑗 and 𝑤 ∈ 𝑌

𝑘

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

Directed pathwidth/decomposition

The width of a directed path-decomposition is 𝑛𝑏𝑦𝑗 |𝑌𝑗| - 1. The directed pathwidth of 𝐻 is the minimum 𝑥 such that there is a directed path-decomposition of 𝐻 of width 𝑥.

b e a d c

𝐻

a c b f g e g d

A directed path-decomposition of 𝐻

𝑌1 𝑌2 𝑌3 𝑌4 𝑌5

f g g g

width = 1 𝐞𝐪𝐱(𝐻) = 1

𝐻: undirected graph 𝐻’: digraph with a pair of anti-parallel edges for each edge of 𝐻 The condition for a path-decomposition of 𝐻 = the condition for a directed path-decomposition of 𝐻′

Observation 1 𝐻 𝐻’

𝑌𝑗 𝑌

𝑘

𝑣 𝑤

𝑘 ≤ 𝑗 𝑗 ≤ 𝑘 The problem of deciding the directed-pathwidth is a generalization of that of deciding the pathwidth.

Observation 2

𝑌𝑗

A directed path decomposition represents a linear system

• f dicuts.

𝑌𝑗+1 𝑌

Observation 2

𝑌≤𝑗

A directed path-decomposition represents a linear system

• f dicuts of size at most the width.

𝑌𝑗+1 𝑌 𝑌>𝑗 𝑌𝑗 𝑣 𝑤

Some facts on directed pathwidth

Introduced by Reed, Seymour, and Thomas in mid 90’s. Relates to directed treewidth [Johnson, Robertson, Seymour and Thomas 01], D-width [Safari 05], Dag-width [ Berwanger, Dawar, Hunter & Kreutzer 05, Obdrzalek 06], and Kelly-width[Hunter & Kreutzer 07] as pathwidth relates to treewidth. For digraphs of directed pathwidth 𝑥, some problems including directed Hamiltonian cycle can be solved in 𝑜𝑃(𝑥) time [JRST01] . Used in a heuristic algorithm for enumerating attractors of boolean networks [Tamaki 10].

Complexity

Input: positive integer k and graph (digraph) G Question: Is the (directed) pathwidth of G at most k ?

NP-complete for the undirected case [Kashiwabara & Fujisawa 79] and hence for the directed case.

Undirected pathwidth is fixed parameter tractable:

𝑔 𝑙 𝑜𝑃(1) time: graph minor theorem 2𝑃 𝑙3 𝑜 time: [Bodlaender 96, Bodlaender & Kloks 96]

Directed pathwidth is open for FPT

Even for k = 2, no polynomial time was previously known.

Result

An 𝑃 𝑛𝑜𝑙+1 time algorithm for deciding if the directed pathwidth is ≤ 𝑙 (and constructing the associated decomposition) for a digraph of 𝑜 vertices and 𝑛 edges. Note This algorithm is extremely simple, easy to implement, and useful even for undirected pathwidth/-decomposition (the linear time algorithm of Bodlaender depends exponentially on k 3)

Notation

𝐻 : digraph, fixed 𝑜 = 𝑊 𝐻 𝑛 = 𝐹 𝐻 𝑂

– 𝑌 = 𝑣 ∈ 𝑌 𝑣, 𝑤 ∈ 𝐹 𝐻 , 𝑤 ∈ 𝑌 +

: set of in-neighbors of 𝑌 𝑊(𝐻) 𝑒

– (𝑌) = |𝑂 – (𝑌)| : in-degree of 𝑌 𝑊(𝐻)

Σ(𝐻 ): the set of all non-duplicating sequences of vertices of 𝐻 𝑊 (σ) : the set of vertices appearing in σ S(G )

Directed vertex separation number

A vertex sequence 𝜏 ∈ Σ 𝐻 is 𝑙-feasible if 𝑒− 𝑊 τ ≤ 𝑙 for every prefix τ of σ. The directed vertex separation number of G : dvsn(G ) = min 𝑙 𝜏 ∈ Σ 𝐻 : 𝑊 𝜏 = 𝑊(𝐻)and σ is 𝑙-feasible+ Fact: dvsn(G ) = dpw(G )

The conversion from a vertex separation sequence to a directed path-decomposition is straightforward.

Search tree for k-feasible sequences Vertex sets in the search tree are feasible. Those leading to a solution are strongly feasible.

b e a d c

𝐻 𝑙 = 2 {a} 2 {} 0 {b} 1 {c} 2 {e} 2 {d} 2 {a,b} 2 {a,c} 4 {a,d} 3 {a,e} 3 {a,b,c} 3 {a,b,c} 3 {a,b,d} 3 {a,b,e} 3 {d,e} 2 {c,d,e} 1 {b,c,d,e} 0 {a,b,c,d,e} 0

Commitment: a special case Is it safe to commit to this child? In other words, is it true that if 𝑉 is strongly feasible then 𝑉 ∪ 𝑤 is? 𝑉 𝑒 𝑉 ∪ 𝑤 ≤𝑒

Adding 𝑤 does not increase the indegree.

Commitment: a special case Is it safe to commit to this child? In other words, is it true that if 𝑉 is strongly feasible then 𝑉 ∪ 𝑤 is? YES, in a more genral form 𝑉 𝑒 𝑉 ∪ 𝑤 ≤𝑒

Adding 𝑤 does not increase the in-degree.

Commitment in general form Commitment lemma Let 𝑉 ⊂ 𝑊 both feasible and suppose:

• 1. 𝑒− 𝑊 ≤ 𝑒− 𝑉 ,
• 2. 𝑒− 𝑊′ > 𝑒− 𝑉 for every feasible proper

superset 𝑊′ of 𝑉 strictly smaller than 𝑊, and

• 3. 𝑉 is strongly feasible.

Then 𝑊 is strongly feasible. 𝑉 𝑒 𝑊 ≤𝑒 𝑊′ >𝑒

First decsendant with the same or smaller in-degree.

Search tree pruning based on commitment

When a node has a descendant to which it can commit, all other descendants are removed from the tree. Effectively, branching occurs only when the in-degree increases. The pruned tree behaves like a depth 𝑙 tree in a fuzzy sense. Can show, with some technicality, that the size of the pruned tree is at most 𝑜𝑙+1 .

Proof of the commitment lemma

Fact:

The in-degree function 𝑒

– is submodular:

for every pair of subsets 𝑌, 𝑍 ⊆ 𝑊 (𝐻 ),

𝑒

– 𝑌 + 𝑒 – 𝑍 ≥ 𝑒 – (𝑌 ∩ 𝑍 ) + 𝑒 – (𝑌 ∪ 𝑍 )

Proof of the commitment lemma

Step 2:

Using the condition established in Step 1, derive the strong feasibility of 𝑊 from that of 𝑉.

Step 1:

Let 𝑉 and 𝑊 as in the lemma. Then, 𝑒

– 𝑊 ≤ 𝑒 – (𝑌)

holds for every 𝑌 such that 𝑉  𝑌  𝑊. (Even if 𝑌 is not feasible.)

Step 2 of the proof

Assumptions:

𝑉 ⊂ 𝑊, 𝑉 is strongly feasible, 𝑊 is feasible, and

𝑒

– 𝑊 ≤ 𝑒 – (𝑌) for every 𝑌 such that 𝑉  𝑌  𝑊.

Goal: 𝑊 is also strongly feasible

𝑉 𝑊

Step 2 of the proof

Assumptions:

𝑉 is strongly feasible, 𝑊 is feasible, and

𝑒

– 𝑊 ≤ 𝑒 – (𝑌) for every 𝑌 such that 𝑉  𝑌  𝑊.

Goal: 𝑊 is also strongly feasible

𝑉 𝑊

𝜏 : 𝑙-feasible

Step 2 of the proof

Assumptions:

𝑊 is feasible, and

𝑒

– 𝑊 ≤ 𝑒 – (𝑌) for every 𝑌 such that 𝑉  𝑌  𝑊.

Goal: 𝛽 is 𝑙-feasible

𝑉 𝑊

𝜏 : 𝑙-feasible 𝛽

Step 2 of the proof

Assumptions:

𝑊 is feasible, and

𝑒

– 𝑊 ≤ 𝑒 – (𝑌) for every 𝑌 such that 𝑉  𝑌  𝑊.

Goal: 𝛽 is 𝑙-feasible

𝑉 𝑊

𝜏 : 𝑙-feasible 𝛽 𝑙-feasible

Step 2 of the proof

𝑒

– 𝑊 ≤ 𝑒 – (𝑌) for every 𝑌 such that 𝑉  𝑌  𝑊.

Goal:𝑒

– 𝐵 ≤ 𝑙 for each superset 𝐵 of 𝑊 that is the vertex

set of some prefix of 𝛽

𝑉 𝑊

𝜏 : 𝑙-feasible 𝛽

𝐵

Step 2 of the proof

𝑒

– 𝑊 ≤ 𝑒 – (𝑌) for every 𝑌 such that 𝑉  𝑌  𝑊.

Goal: 𝑒

– 𝐵 ≤ 𝑙

𝜏 : 𝑙-feasible 𝛽

𝐵

𝐶: the vertex set of the minimal prefix of𝜏 such that

𝐶 ∖ 𝑊 = 𝐵 ∖ 𝑊

𝐶

Step 2 of the proof

𝑒

– 𝑊 ≤ 𝑒 – (𝑌) for every 𝑌 such that 𝑉  𝑌  𝑊.

Goal: 𝑒

– 𝐵 ≤ 𝑙

𝜏 : 𝑙-feasible 𝛽

𝐵

By submodularity: 𝑒

– 𝑊 + 𝑒 – 𝐶 ≥ 𝑒 – (𝑊 ∩ 𝐶) + 𝑒 – (𝑊 ∪ 𝐶)

𝐶 𝐵 𝑌

Step 2 of the proof

𝑒

– 𝑊 ≤ 𝑒 – (𝑌) for every 𝑌 such that 𝑉  𝑌  𝑊.

Goal: 𝑒

– 𝐵 ≤ 𝑙

𝜏 : 𝑙-feasible 𝛽

𝐵

By submodularity: 𝑒

– 𝑊 + 𝑒 – 𝐶 ≥ 𝑒 – (𝑊 ∩ 𝐶) + 𝑒 – (𝑊 ∪ 𝐶)

𝐶 𝐵 𝑌

≤ 𝑙 ≥ 𝑒

– 𝑊

Step 2 of the proof

Goal: 𝑒

– 𝐵 ≤ 𝑙 : established

⇒ 𝛽 is 𝑙-feasible ⇒ 𝑊 is strongly feasible

𝜏 : 𝑙-feasible 𝛽

𝐵

By submodularity: 𝑒

– 𝑊 + 𝑒 – 𝐶 ≥ 𝑒 – (𝑊 ∩ 𝐶) + 𝑒 – (𝑊 ∪ 𝐶)

𝐶 𝐵 𝑌

≤ 𝑙 ≥ 𝑒

– 𝑊

Open problems

Is directed-pathwidth FPT, i.e., has an 𝑔 𝑙 𝑜𝑃(1) time algorithm? Are directed-treewidth, D-width, Dag-width, and Kelly-width in XP, i.e., has an 𝑜𝑃 𝑙 time algorithm? More applications?