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Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface by Petr Hlineny, Markus Chimani
Shiyu Hu
April 26, 2011
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Problem Description
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Problem Description
Problem Given a graph, how can we embed it in a planar surface so as to minimize the number of edge-crossing? NP-Complete. Even for graphs with bounded degree. (by M.R Garey and D.S. Johnson, 1983) Fixed Parameter Tractable. Test whether a graph has a crossing number at most k in linear time. (by K. Kawarabayashi and B. Reed in STOC 2007) Approximate |V (G )| + cr (G ) with ratio log 3 |V (G )|. (by G. Even, S. Guha, and B. Schieber, 2000) cr (G ): minimum crossing number
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Problem Description
Two constrains Bounded maximum degree “Densely enough” embeddable in any fixed orientable surface
Problem Given a maximum degree bounded and “densely enough” embeddable graph, how can we embed it in a planar surface so as to minimize the number of edge-crossing?
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Main Idea and Results
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Main Idea and Results
Main idea
Main idea: 1
Embed graph G on surface Σ as a starting point
2
Iteratively “cut and open” handles of Σ to a sphere, greedily removing the fewest affected edges
3
“Cheaply” insert edges back to G , counting the crossing number
4
Bound the crossing number
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Main Idea and Results
Results: Theorem 1.1 Let G be a graph embeddable in an orientable surface of genus g ≥ 1 with nonseparating dual edge-width at least 2g +2 ∆, where ∆ is the maximum degree of G . The algorithm in this paper computes a drawing of G in the plane with at most 3 · 23g +2 · ∆2 · cr (G ) crossings. Constant Ratio: 3 · 23g +2 · ∆2 Running time: O(nlogn), n = |V (G )| + |E (G )|
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Algorithm: 1. Construct an embedding G1 of G in Sg . 2. For i = 1, 2, . . . g ; 2.1 we compute a shortest non-separating dual cycle γi in the dual graph Gi∗ . O(nlogn) 2.2 Construct an embedding Gi+1 = Gi /γi by cutting Gi along γi . 3. Gg +1 is a planar graph. For any edge e ∈ E (G1 )/E (Gg +1 ) with ends v1 , v2 , compute the shortest dual path π(f (v1 ), f (v2 )) between faces f (v1 ) and f (v2 ) in Gg∗+1 . 4. Draw every edge e “along” the dual path π(f (v1 ), f (v2 )) and recover the rotations at v1 and v2
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Ratio Analysis Ratio Analysis: Let Fi = E (Gi )/E (Gi+1 ) be the set of edges cut by γi at step i. ci : the length of the shortest dual cycle γi in Gi∗ . ∗ li : the length of the shortest dual path in Gi+1 between the two cut faces. Add each e ∈ Fi , the crossing number contains two parts: 1. with edges in Gi+1 : ≤ li + li+1 + · · · + lg . 2. with re-inserted edges: ≤ ci + ci+1 + · · · + cg . We can easily prove that 2li ≥ ci , otherwise we can achieve smaller ci .
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Ratio Analysis Ratio Analysis: g g g g g j X X X X X X (ck · (cj + lj )) ≤ (ck · (3lj )) = 3 (lj · (ci )) k=1
j=k
k=1
j=k
j=1
i=1
We can also easily prove that ci ≤ 2ci+1 .
3
g X
(lj ·
j=1
=3 =
g X
j X i=1
(ci )) ≤ 3
g X (lj cj (2j−1 + · · · + 21 + 20 ) j=1
lj cj (2j − 1) ≤ 3M · (21 + 22 + · · · + 2g − g )
j=1 3(2g +1
− 2 − g ) · M; where M = max{ci li : i = 1, 2, . . . , g }
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Ratio Analysis Ratio Analysis: It is safe to assume c1 l1 = max{ci li : i = 1, 2, . . . , g }. Proof. If cj lj = max{ci li : i = 1, 2, . . . , g }, we assume g 0 = g + 1 − j. 0 0 0 cj ≥ 21−j c1 = 2g −g c1 ≥ 2g −g 2g +2 ∆ = 2g +2 ∆. Claim: cr (G ) ≥ 2−2g −1 · ∆(G )−2 · c1 l1 . If it is true, we have crossing number ≤ 3 · 2g +1 · c1 l1 ≤ 3 · 23g +2 · ∆(G )−2 · cr (G ) Done!
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Lower Bound Proofs Concepts: α and β are in one-leaping position, if α ∩ β have exactly one connection component, and α crossing β. Note that α and β are both non-separating cycles stretch(G ): the smallest value len(α) · len(β) in G , where len(·) is the length of the cycle, where α and β are in one-leaping position. Existing Result (by P. Hlineny and Salazar in 2007) Let G be a graph embeddable in the torus such that c ≥ 8∆(G ). Then cr (G ) ≥ 81 ∆(G )−2 · stretch(G ).
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Lower Bound Proofs
It is easy to prove stretch(G ) ≤ 2c1 l1 . Proof. Let δ be a non-separating dual cycle in G , and ζ as a δ-switching ear in G ∗ , then stretch(G ) ≤ len(ζ) + 21 len(δ), since ζ separate δ into two sub-paths and one of them is ≤ 12 len(δ). stretch(G ) ≤ c1 (l + 12 c1 ) ≤ c1 (l1 + l1 ) = 2c1 l1 .
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Main Lemma Let a graph G be embeddable in an orientable surface, and assume (1) there is a bipolar dual subgraph δ in G ∗ (2) there exists a closed walk in G ∗ that is odd-leaping (3) the shortest δ-polarity switching ear in G ∗ has length h. Let α, β be a one-leaping pair of dual cycles in G ∗ such that len(α) ≤ len(β) and streth(G ) = len(α) · len(β). We denote by G0 = G /α the embedded subgraph of G obtained by cutting H along α. Unless len(β) ≥ h, the following hold (1’) there is a bipolar dual subgraph δ0 in G0∗ (2’) there exists a closed walk in G0∗ that is odd-leaping (3’) the shortest δ0 -polarity switching ear in G0∗ has length h0 ≥ h − 21 len(α).
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Main Lemma If len(β) ≥ h stretch(G ) ≥ h · c, where c is the shortest non-separating dual cycle in G ∗ . Assume T is a subgraph of G , which is embeddable in the torus, then cT ≥ 21−g c, where cT is the shortest non-separating dual cycle in T . It is easy to prove that stretch(G0 ) ≥ 14 stretch(G ). Then, we have stretch(T0 ) ≥ 22−2g stretch(G ) ≥ 22−2g · h · c ≥ 22−2g · cl Combine existing result, we have cr (G ) ≥ 2−2g −1 · ∆(G )−1 · cl.
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Main Lemma
If len(β) ≤ h the new embedded graph G0 = G /α has genus g − 1, then h0 ≥ h − 21 len(α) ≥ h − 21 len(β) ≥ 12 h In torus, we know stretch(T ) ≥ cT lT , then stretch(T ) ≥ 21−g · l1 · 21−g · c1 = 22−2g · c1 l1
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Questions
Questions
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface by Petr Hlineny, Markus Chimani
Shiyu Hu
April 26, 2011
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Problem Description
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Problem Description
Problem Given a graph, how can we embed it in a planar surface so as to minimize the number of edge-crossing? NP-Complete. Even for graphs with bounded degree. (by M.R Garey and D.S. Johnson, 1983) Fixed Parameter Tractable. Test whether a graph has a crossing number at most k in linear time. (by K. Kawarabayashi and B. Reed in STOC 2007) Approximate |V (G )| + cr (G ) with ratio log 3 |V (G )|. (by G. Even, S. Guha, and B. Schieber, 2000) cr (G ): minimum crossing number
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Problem Description
Two constrains Bounded maximum degree “Densely enough” embeddable in any fixed orientable surface
Problem Given a maximum degree bounded and “densely enough” embeddable graph, how can we embed it in a planar surface so as to minimize the number of edge-crossing?
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Main Idea and Results
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Main Idea and Results
Main idea
Main idea: 1
Embed graph G on surface Σ as a starting point
2
Iteratively “cut and open” handles of Σ to a sphere, greedily removing the fewest affected edges
3
“Cheaply” insert edges back to G , counting the crossing number
4
Bound the crossing number
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Main Idea and Results
Results: Theorem 1.1 Let G be a graph embeddable in an orientable surface of genus g ≥ 1 with nonseparating dual edge-width at least 2g +2 ∆, where ∆ is the maximum degree of G . The algorithm in this paper computes a drawing of G in the plane with at most 3 · 23g +2 · ∆2 · cr (G ) crossings. Constant Ratio: 3 · 23g +2 · ∆2 Running time: O(nlogn), n = |V (G )| + |E (G )|
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Algorithm: 1. Construct an embedding G1 of G in Sg . 2. For i = 1, 2, . . . g ; 2.1 we compute a shortest non-separating dual cycle γi in the dual graph Gi∗ . O(nlogn) 2.2 Construct an embedding Gi+1 = Gi /γi by cutting Gi along γi . 3. Gg +1 is a planar graph. For any edge e ∈ E (G1 )/E (Gg +1 ) with ends v1 , v2 , compute the shortest dual path π(f (v1 ), f (v2 )) between faces f (v1 ) and f (v2 ) in Gg∗+1 . 4. Draw every edge e “along” the dual path π(f (v1 ), f (v2 )) and recover the rotations at v1 and v2
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Ratio Analysis Ratio Analysis: Let Fi = E (Gi )/E (Gi+1 ) be the set of edges cut by γi at step i. ci : the length of the shortest dual cycle γi in Gi∗ . ∗ li : the length of the shortest dual path in Gi+1 between the two cut faces. Add each e ∈ Fi , the crossing number contains two parts: 1. with edges in Gi+1 : ≤ li + li+1 + · · · + lg . 2. with re-inserted edges: ≤ ci + ci+1 + · · · + cg . We can easily prove that 2li ≥ ci , otherwise we can achieve smaller ci .
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Ratio Analysis Ratio Analysis: g g g g g j X X X X X X (ck · (cj + lj )) ≤ (ck · (3lj )) = 3 (lj · (ci )) k=1
j=k
k=1
j=k
j=1
i=1
We can also easily prove that ci ≤ 2ci+1 .
3
g X
(lj ·
j=1
=3 =
g X
j X i=1
(ci )) ≤ 3
g X (lj cj (2j−1 + · · · + 21 + 20 ) j=1
lj cj (2j − 1) ≤ 3M · (21 + 22 + · · · + 2g − g )
j=1 3(2g +1
− 2 − g ) · M; where M = max{ci li : i = 1, 2, . . . , g }
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Approximation Algorithm and Ratio Analysis
Ratio Analysis Ratio Analysis: It is safe to assume c1 l1 = max{ci li : i = 1, 2, . . . , g }. Proof. If cj lj = max{ci li : i = 1, 2, . . . , g }, we assume g 0 = g + 1 − j. 0 0 0 cj ≥ 21−j c1 = 2g −g c1 ≥ 2g −g 2g +2 ∆ = 2g +2 ∆. Claim: cr (G ) ≥ 2−2g −1 · ∆(G )−2 · c1 l1 . If it is true, we have crossing number ≤ 3 · 2g +1 · c1 l1 ≤ 3 · 23g +2 · ∆(G )−2 · cr (G ) Done!
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Overview
1 Problem Description
2 Main Idea and Results
3 Approximation Algorithm and Ratio Analysis
4 Lower Bound Proofs
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Lower Bound Proofs Concepts: α and β are in one-leaping position, if α ∩ β have exactly one connection component, and α crossing β. Note that α and β are both non-separating cycles stretch(G ): the smallest value len(α) · len(β) in G , where len(·) is the length of the cycle, where α and β are in one-leaping position. Existing Result (by P. Hlineny and Salazar in 2007) Let G be a graph embeddable in the torus such that c ≥ 8∆(G ). Then cr (G ) ≥ 81 ∆(G )−2 · stretch(G ).
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Lower Bound Proofs
It is easy to prove stretch(G ) ≤ 2c1 l1 . Proof. Let δ be a non-separating dual cycle in G , and ζ as a δ-switching ear in G ∗ , then stretch(G ) ≤ len(ζ) + 21 len(δ), since ζ separate δ into two sub-paths and one of them is ≤ 12 len(δ). stretch(G ) ≤ c1 (l + 12 c1 ) ≤ c1 (l1 + l1 ) = 2c1 l1 .
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Main Lemma Let a graph G be embeddable in an orientable surface, and assume (1) there is a bipolar dual subgraph δ in G ∗ (2) there exists a closed walk in G ∗ that is odd-leaping (3) the shortest δ-polarity switching ear in G ∗ has length h. Let α, β be a one-leaping pair of dual cycles in G ∗ such that len(α) ≤ len(β) and streth(G ) = len(α) · len(β). We denote by G0 = G /α the embedded subgraph of G obtained by cutting H along α. Unless len(β) ≥ h, the following hold (1’) there is a bipolar dual subgraph δ0 in G0∗ (2’) there exists a closed walk in G0∗ that is odd-leaping (3’) the shortest δ0 -polarity switching ear in G0∗ has length h0 ≥ h − 21 len(α).
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Main Lemma If len(β) ≥ h stretch(G ) ≥ h · c, where c is the shortest non-separating dual cycle in G ∗ . Assume T is a subgraph of G , which is embeddable in the torus, then cT ≥ 21−g c, where cT is the shortest non-separating dual cycle in T . It is easy to prove that stretch(G0 ) ≥ 14 stretch(G ). Then, we have stretch(T0 ) ≥ 22−2g stretch(G ) ≥ 22−2g · h · c ≥ 22−2g · cl Combine existing result, we have cr (G ) ≥ 2−2g −1 · ∆(G )−1 · cl.
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Main Lemma
If len(β) ≤ h the new embedded graph G0 = G /α has genus g − 1, then h0 ≥ h − 21 len(α) ≥ h − 21 len(β) ≥ 12 h In torus, we know stretch(T ) ≥ cT lT , then stretch(T ) ≥ 21−g · l1 · 21−g · c1 = 22−2g · c1 l1
Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface Lower Bound Proofs
Questions
Questions
