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Parameterized Algorithms for Book-Embedding Problems Sujoy Bhore , - - PowerPoint PPT Presentation

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Parameterized Algorithms for Book-Embedding Problems Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N ollenburg Graph Drawing September 19, 2019 1/33 The Problem v 9 v 8 v 4 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 5 v 3 v 7 v 6 v

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Parameterized Algorithms for Book-Embedding Problems

Sujoy Bhore, Robert Ganian, Fabrizio Montecchiani, Martin N¨

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Graph Drawing · September 19, 2019

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The Problem

v1 v2 v3 v4 v5 v6 v7 v8 v9

G = (V, E)

v1 v2 v3 v4 v5 v6 v7 v8 v9

v1 v2 v3 v4 v5 v6 v7 v8 v9

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The Problem v1 v2 v3 v4 v5 v6 v7 v8 v9 v1 v2 v3 v4 v5 v6 v7 v8 v9 v1 v2 v3 v4 v5 v6 v7 v8 v9

G has 3-Page Book-Embedding Page 3 Page 2 Page 1

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Book Thickness

Book Thickness (bt(G)): the minimum k such that G admits a k-page book-embedding. Alternatively, known as Stack Number.

Applications:

Bioinformatics VLSI Parallel Computing

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What we know ...

Every planar graph has book thickness at most four. [ Yannakakis – J. Comput. Syst. Sci., 89] Given a graph G and a positive integer k, determining whether bt(G) ≤ k is NP-complete (even for k ≥ 2). [Bernhart et al. – J. Comb. Theory, Ser. B, 79]

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What happens if the linear order ≺ of the vertices is fixed?

v1 v3 v4 v2 v5 v1 v3 v2 v4 v5

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v4 v5 v1 v2 v3 Fixed-order book-thickness (fo-bt(G) = 3): v4 v5 v1 v3 v2 Book-thickness (bt(G) = 2): v1 v3 v4 v2 v5

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Deciding whether fo-bt(G, ≺) ≤ 2 is Polynomial, since equiv- alent to testing the bipartiteness of a suitable conflict graph.

What we know ...

Deciding if fo-bt(G, ≺) ≤ 4 is NP-Complete, since equivalent to finding a 4-coloring of a circle graph which is NP-complete [W. Unger – STACS 1992].

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A problem is fixed-parameter tractable (FPT) with respect to parameter k if there exists a solution run- ning in f(k) · nO(1) time, where f is a computable function of k which is independent of n.

Problem + Parameter

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Results:

Fixed-Order Book Thickness parameterized by the vertex cover number of the graph FPT-algorithms : Fixed-Order Book Thickness parameterized by the pathwidth of the graph w.r.t the vertex order Book Thickness parameterized by the vertex cover num- ber of the graph

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If the answer is ‘YES’ we shall return a corresponding k-page book embedding as a witness. Input: Graph G = (V, E), a linear order ≺ of V , and a positive integer k. Task: Decide if there is a page assignment σ: E → [k] such that ≺, σ is a k-page book embedding of G, that is whether fo-bt(G, ≺) ≤ k.

Algorithms for Fixed-Order Book-Thickness ...

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Parameterization by the Vertex Cover number (τ)... Vertex Cover C of a graph G can be computed in time O(2τ + τ · n) [TCS, 10 - Chen et al.]

Vertex Cover

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Observation 1 Every graph G with a vertex cover C of size τ admits a τ-page book embedding with any vertex order ≺. 4-page book embedding ...

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Set-up for the Algorithm v∞ If τ ≤ k - Yes! Else ... v∞ Compute set of all valid page assignments S of G[C] |S| < τ τ 2

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Towards the dynamic program ... v∞ c1 c2 c3 cτ Consider an assignment s ∈ S v∞ c1 c2 c3 c4 c5 cτ 2-page assignment

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Notion of Visibility ... v∞ c1 c2 c3 c4 c5 cτ u1 valid page assignment... v∞ c1 c2 c3 c4 c5 cτ u1 s ∈ S

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Building visibility matrix ... v∞ c1 c2 c3 c4 c5 cτ u1

for an index a ∈ [n − τ], a k × τ visibility matrix Mi(a, α, s)...

c2 c3 c4 c5 c6 c7 u1 c1 u2 u3 . . .

s ∈ S

1 1 1 1 1 1 1 1 M2(2, α, s) =

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High-level Idea ...

Dynamically process the vertices in U(non vertex cover ver- tices) from left to right ... For each vertex, a bounded size snapshot of its visibility vertices ... Store one (arbitarily) chosen valid partial edge assignment ... All valid partial page assignments lead to the same visibility matrices are interchangeable ...

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Record set ... Some Observations ...

For a vertex ui ∈ U, Ri(s) = {

  • Mi(i, α, s), Mi(x1, α, s), . . . , Mi(xz, α, s)
  • | ∃ valid partial page assignment α: Ei → [k]}

|Ri(s)| ≤ 2τ 3+τ 2 If Rn−τ(s) = ∅ for some s

(un−τ is a dummy vertex) then there is a valid partial page assignment α: En−τ → [k] s.t. s ∪ α is a non-crossing page assignment of all edges in G.

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It suffices to compute Rn−τ(s) for each s ∈ S.

u∞

Observation 2 If for all s ∈ S it holds that Rn−τ(s) = ∅, then (G, ≺, k) is a NO-instance of Fixed-Order Book Thickness.

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Dynamic Step ...

If it is NOT a valid partial page assignment - discard! Branch over each page assignment β of the edges(≤ τ) incident to ui−1, and each tuple ρ ∈ Ri−1(s) ... Assume we have computed Ri−1(s) ... Compute R1(s) ... Else, compute the visibility matrices add the corresponding tuple into Ri(s).

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Runtime is upper-bounded by - Theorem 1 There is an algorithm which takes as input an n-vertex graph G with a vertex order ≺, runs in time 2O(τ 3)· n where τ is the vertex cover number of G, and computes a page assignment σ such that (≺, σ) is a (fo-bt(G, ≺))-page book embedding of G.

Lemma 1 The procedure correctly computes Ri(s) from Ri−1(s).

(τ τ 2) · n · (2τ 3+τ 2τ τ)

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Fixed-Order Book Thickness parameterized by the vertex cover number of the graph FPT-algorithms : Fixed-Order Book Thickness parameterized by the pathwidth of the graph w.r.t a vertex order Book Thickness parameterized by the vertex cover num- ber of the graph.

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Parameterization by the pathwidth ...

v1 v3 v2 v4 v5

v1 v3 v4 v2 v5

i/p : G = (V, E), ≺

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Lemma 2 Every graph G = (V, E) with a linear order ≺ of V such that (G, ≺) has pathwidth k admits a k-page book embedding ≺, σ, which can be computed in O(n + k · n) time. red page is free ...

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Dynamic guard sets ...

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Set-up for the algorithm ... v0 v0 concept of guards, BUT, in reverse order ... gi

1

gi

k

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Concept of (α, i, p) important edge ...

Observation 3 If va has no (α, i, p)-important edge, then every vertex vx with x < a is α-visible to va. If the (α, i, p)-important guard of va is vc, then vx (x < a) is α-visible to va if and only if x ≥ c.

va vc vi vd

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Theorem 2 There is an algorithm which takes as input an n-vertex graph G = (V, E) with a vertex ordering ≺ and computes a page assignment σ of E such that (≺, σ) is a (fo-bt(G, ≺))-page book em- bedding of G. The algorithm runs in n · κO(κ2) time where κ is the pathwidth of (G, ≺).

Runtime is upper bounded by O(n · (κ + 2)κ2 · κκ)

Lemma 3 The procedure correctly computes Qi−1 from Qi.

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Fixed-Order Book Thickness parameterized by the vertex cover number of the graph FPT-algorithms : Fixed-Order Book Thickness parameterized by the pathwidth of the graph w.r.t a vertex order Book Thickness parameterized by the vertex cover number of the graph

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Algorithm for Book-Thickness

v1 v3 v4 v2 v5 Book-thickness (bt(G)):

Theorem 3 Given a graph graph G = (V, E) with vertex cover number τ and a positive integer k, there is an algorithm that runs in time O(τ τ O(τ) + 2τ · n) (τ = τ(G) is the vertex cover number of G), and decides whether bt(G) ≤ k.

v4 v5 v1 v2 v3

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Conclusion ...

for fixed-order book-thickness

FPT algorithms

(parameter: vertex cover, pathwidth) for book-thickness (parameter: vertex cover) what if we allow constant number of crossings ??? ...

Issues with Treewidth ...

t1 t2 t1 tx t2 t1 t2

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Thank You