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 1 v 2 G = ( V, E ) v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 2/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
The Problem Page 1 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 Page 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 Page 3 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 G has 3 -Page Book-Embedding 3/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
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 4/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
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] 5/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
What happens if the linear order ≺ of the vertices is fixed? v 1 v 2 v 5 v 1 v 3 v 2 v 4 v 5 v 3 v 4 6/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Fixed-order book-thickness (fo-bt ( G ) = 3 ): v 2 v 3 v 4 v 5 v 1 v 1 v 2 v 5 Book-thickness (bt ( G ) = 2 ): v 3 v 4 v 3 v 2 v 4 v 5 v 1 7/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
What we know ... Deciding whether fo-bt ( G, ≺ ) ≤ 2 is Polynomial, since equiv- alent to testing the bipartiteness of a suitable conflict graph. 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]. 8/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Problem + Parameter A problem is fixed-parameter tractable (FPT) with respect to parameter k if there exists a solution run- ning in f ( k ) · n O (1) time, where f is a computable function of k which is independent of n . 9/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Results: FPT-algorithms : Fixed-Order Book Thickness parameterized by the vertex cover number of the graph 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 10/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Algorithms for Fixed-Order Book-Thickness ... 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 . If the answer is ‘YES’ we shall return a corresponding k -page book embedding as a witness. 11/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Parameterization by the Vertex Cover number ( τ )... Vertex Cover Vertex Cover C of a graph G can be computed in time O (2 τ + τ · n ) [TCS, 10 - Chen et al.] 12/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
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 ... 13/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
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 14/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Towards the dynamic program ... c 3 c τ c 2 c 1 v ∞ Consider an assignment s ∈ S c 5 c τ c 2 c 3 c 4 c 1 v ∞ 2 -page assignment 15/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Notion of Visibility ... s ∈ S c 2 c 3 c 4 c 5 c τ u 1 c 1 v ∞ c 2 c 3 c 4 c 5 c τ u 1 c 1 v ∞ valid page assignment... 16/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Building visibility matrix ... s ∈ S c 2 c 3 c 4 c 5 c τ u 1 c 1 v ∞ for an index a ∈ [ n − τ ] , a k × τ visibility matrix M i ( a, α, s ) ... 1 1 0 0 0 1 1 M 2 (2 , α, s ) = u 1 c 2 c 3 c 4 c 5 c 6 u 3 c 7 . . . 1 1 1 1 0 0 0 c 1 u 2 17/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
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 ... 18/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Record set ... For a vertex u i ∈ U , � � R i ( s ) = { M i ( i, α, s ) , M i ( x 1 , α, s ) , . . . , M i ( x z , α, s ) | ∃ valid partial page assignment α : E i → [ k ] } Some Observations ... |R i ( s ) | ≤ 2 τ 3 + τ 2 If R n − τ ( s ) � = ∅ for some s ( u n − τ is a dummy vertex) then there is a valid partial page assignment α : E n − τ → [ k ] s.t. s ∪ α is a non-crossing page assignment of all edges in G . 19/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Observation 2 If for all s ∈ S it holds that R n − τ ( s ) = ∅ , then ( G, ≺ , k ) is a NO-instance of Fixed-Order Book Thickness . u ∞ It suffices to compute R n − τ ( s ) for each s ∈ S . 20/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Dynamic Step ... Compute R 1 ( s ) ... Assume we have computed R i − 1 ( s ) ... Branch over each page assignment β of the edges( ≤ τ ) incident to u i − 1 , and each tuple ρ ∈ R i − 1 ( s ) ... If it is NOT a valid partial page assignment - discard! Else, compute the visibility matrices add the corresponding tuple into R i ( s ) . 21/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Lemma 1 The procedure correctly computes R i ( s ) from R i − 1 ( s ) . Runtime is upper-bounded by - ( τ τ 2 ) · n · (2 τ 3 + τ 2 τ τ ) Theorem 1 There is an algorithm which takes as input an n -vertex graph G with a vertex order ≺ , runs in time 2 O ( 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 . 22/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
FPT-algorithms : Fixed-Order Book Thickness parameterized by the vertex cover number of the graph 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. 23/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Parameterization by the pathwidth ... v 1 v 2 v 5 v 3 v 4 i/p : G = ( V, E ) , ≺ v 1 v 3 v 2 v 4 v 5 24/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
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 ... 25/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Dynamic guard sets ... 26/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
Set-up for the algorithm ... v 0 concept of guards, BUT , in reverse order ... v 0 g i g i 1 k 27/33 Sujoy Bhore , Robert Ganian, Fabrizio Montecchiani, Martin N¨ ollenburg · Parameterized Algorithms for Book-Embedding Problems
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