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

A polynomial time algorithm for bounded directed pathwidth Hisao Tamaki Meiji University

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Undirected Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval some 𝑗 *𝑣, 𝑤+ 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such ---------- 𝑣, 𝑤 ∈ 𝑌 𝑗 that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 1. for each 𝑤 ∈ 𝑊(𝐻) , 𝐽 𝑤 = 𝑗 𝑤 ∈ 𝑌 𝑗 + is a single non- empty interval 2. for each directed edge (𝑣, 𝑤) there is a pair 𝑗 ≤ 𝑘 such that 𝑣 ∈ 𝑌 𝑗 and 𝑤 ∈ 𝑌 𝑘

Directed pathwidth/decomposition A directed path-decomposition of 𝐻 𝐻 𝑌 1 𝑌 2 𝑌 3 𝑌 4 𝑌 5 f b f c d a e e c g g g g g b a d 𝐞𝐪𝐱(𝐻) = 1 width = 1 The width of a directed path-decomposition is 𝑛𝑏𝑦 𝑗 |𝑌 𝑗 | - 1. The directed pathwidth of 𝐻 is the minimum 𝑥 such that there is a directed path-decomposition of 𝐻 of width 𝑥 .

Observation 1 The problem of deciding the directed-pathwidth is a generalization of that of deciding the pathwidth. 𝐻 : undirected graph 𝐻 ’: digraph with a pair of anti -parallel edges for each edge of 𝐻 𝑌 𝑗 𝐻 𝐻 ’ 𝑣 𝑗 ≤ 𝑘 𝑘 ≤ 𝑗 𝑤 𝑌 𝑘 The condition for a path-decomposition of 𝐻 = the condition for a directed path-decomposition of 𝐻′

Observation 2 A directed path decomposition represents a linear system of dicuts. 𝑌 𝑗 𝑌 𝑗 +1 𝑌

Observation 2 A directed path-decomposition represents a linear system of dicuts of size at most the width. 𝑌 ≤𝑗 𝑌 >𝑗 𝑌 𝑗 𝑌 𝑗 +1 𝑣 𝑤 𝑌

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 𝑜 𝑃(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 Q uestion : 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 𝑜 𝑃(1) time: graph minor theorem 2 𝑃 𝑙 3 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 𝑛𝑜 𝑙+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 – 𝑌 = 𝑣 ∈ 𝑌 𝑣, 𝑤 ∈ 𝐹 𝐻 , 𝑤 ∈ 𝑌 + 𝑂 : set of in-neighbors of 𝑌  𝑊(𝐻) – (𝑌) = |𝑂 – (𝑌)| : in-degree of 𝑌  𝑊(𝐻) 𝑒 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 𝐻 {} 0 𝑙 = 2 b d {a} 2 {b} 1 {c} 2 {d} 2 {e} 2 c a {d,e} 2 {a,b} 2 {a,c} 4 {a,d} 3 {a,e} 3 e {a,b,c} 3 {a,b,c} 3 {a,b,d} 3 {a,b,e} 3 {c,d,e} 1 {b,c,d,e} 0 Vertex sets in the search tree are feasible. {a,b,c,d,e} 0 Those leading to a solution are strongly feasible.

Commitment: a special case 𝑉 𝑒 Adding 𝑤 does not increase the indegree. 𝑉 ∪ 𝑤 ≤𝑒 Is it safe to commit to this child? In other words, is it true that if 𝑉 is strongly feasible then 𝑉 ∪ 𝑤 is?

Commitment: a special case 𝑉 𝑒 Adding 𝑤 does not increase the in-degree. 𝑉 ∪ 𝑤 ≤𝑒 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

Commitment in general form 𝑉 𝑒 𝑊′ >𝑒 First decsendant with the same or smaller 𝑊 ≤𝑒 in-degree. 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.

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 𝑜 𝑙+1 .

Proof of the commitment lemma Fact: – is submodular : The in-degree function 𝑒 for every pair of subsets 𝑌, 𝑍 ⊆ 𝑊 (𝐻 ), – 𝑌 + 𝑒 – 𝑍 ≥ 𝑒 – (𝑌 ∩ 𝑍 ) + 𝑒 – (𝑌 ∪ 𝑍 ) 𝑒

Proof of the commitment lemma Step 1 : – 𝑊 ≤ 𝑒 – (𝑌) Let 𝑉 and 𝑊 as in the lemma. Then , 𝑒 holds for every 𝑌 such that 𝑉  𝑌  𝑊 . (Even if 𝑌 is not feasible.) Step 2 : Using the condition established in Step 1, derive the strong feasibility of 𝑊 from that of 𝑉 .