Approximation Algorithms for Contact Representations of Rectangles - - PowerPoint PPT Presentation
Approximation Algorithms for Contact Representations of Rectangles Michalis Bekos Thomas van Dijk Martin Fink Philipp Kindermann Stephen Kobourov Sergey Pupyrev Joachim Spoerhase Alexander Wolff Universit at T ubingen University of
Michalis Bekos Thomas van Dijk Martin Fink Philipp Kindermann Stephen Kobourov Sergey Pupyrev Joachim Spoerhase Alexander Wolff
Universit¨ at T¨ ubingen University of Arizona Universit¨ at W¨ urzburg
Heather Williams, June 2008
Coalition treaty 2013
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Coalition treaty 2013 Words that were more important in the 2013 than in the 2009 treaty
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extra contacts: not counted, not forbidden
extra contacts: not counted, not forbidden
extra contacts: not counted, not forbidden corner contacts don’t count
extra contacts: not counted, not forbidden
corner contacts don’t count
extra contacts: not counted, not forbidden
corner contacts don’t count
[Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
– Every planar graph w/o sep. triangles has a touching rectangle representation
[Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
– Every planar graph w/o sep. triangles has a touching rectangle representation
[Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
– Every planar graph has a touching cube representation.
[Felsner & Francis, SoCG’11] [Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
– Every planar graph has a touching cube representation.
[Felsner & Francis, SoCG’11]
[Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
– Every planar graph has a touching cube representation.
[Felsner & Francis, SoCG’11]
– introduced by Raisz [1934] – area-universal rectangular layouts
[Eppstein et al., SICOMP’12] [Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
– Every planar graph has a touching cube representation.
[Felsner & Francis, SoCG’11]
– introduced by Raisz [1934] – area-universal rectangular layouts
[Eppstein et al., SICOMP’12] [Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
– Every planar graph has a touching cube representation.
[Felsner & Francis, SoCG’11]
– introduced by Raisz [1934] – area-universal rectangular layouts
[Eppstein et al., SICOMP’12] [Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
– Every planar graph has a touching cube representation.
[Felsner & Francis, SoCG’11]
– introduced by Raisz [1934] – area-universal rectangular layouts
[Eppstein et al., SICOMP’12] [Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
– Every planar graph has a touching cube representation.
[Felsner & Francis, SoCG’11]
– introduced by Raisz [1934] – area-universal rectangular layouts
[Eppstein et al., SICOMP’12]
[Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
– Every planar graph has a touching cube representation.
[Felsner & Francis, SoCG’11]
– introduced by Raisz [1934] – area-universal rectangular layouts
[Eppstein et al., SICOMP’12]
– edge weights prescribe length of contact
[N¨
[Ko´ zmin´ nski & Kin- nen, Networks’85; He, SICOMP’93; He & Kant, TCS’97]
(which can be computed in linear time). – Every planar graph w/o sep. triangles has a touching rectangle representation
Weighted Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Weighted Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Weighted Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Weighted Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
1 2 j j j j
1 2 j j j j
[Fleischer et al., MOR’11]
– eight bins (for the 4 sides and the 4 corners of B1)
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the capacity of side bins is their “free” length
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi item i has size 1/2 in corner bins, wi in top/bottom side bins, hi in left/right side bins – – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi item i has size 1/2 in corner bins, wi in top/bottom side bins, hi in left/right side bins – – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi item i has size 1/2 in corner bins, wi in top/bottom side bins, hi in left/right side bins – – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf Assume that the 4 corner rectangles have contacts
2 in a fixed optimal solution.
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi item i has size 1/2 in corner bins, wi in top/bottom side bins, hi in left/right side bins – – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf Assume that the 4 corner rectangles have contacts
2 in a fixed optimal solution.
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi item i has size 1/2 in corner bins, wi in top/bottom side bins, hi in left/right side bins – – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf Assume that the 4 corner rectangles have contacts
2 in a fixed optimal solution.
Each contact may be horizontal or vertical.
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi item i has size 1/2 in corner bins, wi in top/bottom side bins, hi in left/right side bins – – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf Assume that the 4 corner rectangles have contacts
2 in a fixed optimal solution.
Each contact may be horizontal or vertical. Try all 24 possibilities
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi item i has size 1/2 in corner bins, wi in top/bottom side bins, hi in left/right side bins – – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf Assume that the 4 corner rectangles have contacts
2 in a fixed optimal solution.
Each contact may be horizontal or vertical. Try all 24 possibilities
by calling α-approx. for Gap.
– eight bins (for the 4 sides and the 4 corners of B1) – corner bins have capacity 1/2 – the value of item i is p(v1vi), the profit of edge v1vi item i has size 1/2 in corner bins, wi in top/bottom side bins, hi in left/right side bins – – the capacity of side bins is their “free” length – items 2, . . . , n; one for each leaf Assume that the 4 corner rectangles have contacts
2 in a fixed optimal solution.
Each contact may be horizontal or vertical. Try all 24 possibilities
by calling α-approx. for Gap. ⇒ α-approx. algorithm for Max-Crown on stars
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Weighted Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
3α
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
Apply α1-approx. to G1 and α2-approx. to G2.
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
For G, G1, G2,
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2,
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg.
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi.
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2.
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2. Assume OPT1 /α1 ≥ OPT2 /α2.
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2. Assume OPT1 /α1 ≥ OPT2 /α2. Then
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2. Assume OPT1 /α1 ≥ OPT2 /α2. Then ALG ≥
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2. Assume OPT1 /α1 ≥ OPT2 /α2. Then ALG ≥ ALG1 ≥
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2. Assume OPT1 /α1 ≥ OPT2 /α2. Then ALG ≥ ALG1 ≥ OPT1 α1 ≥
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2. Assume OPT1 /α1 ≥ OPT2 /α2. Then ALG ≥ ALG1 ≥ OPT1 α1 ≥ OPT1 + OPT2 α1 + α2 ≥
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2. Assume OPT1 /α1 ≥ OPT2 /α2. Then ALG ≥ ALG1 ≥ OPT1 α1 ≥ OPT1 + OPT2 α1 + α2 ≥ OPT α1 + α2 .
Apply α1-approx. to G1 and α2-approx. to G2. Return result with larger profit for G.
– let OPT, OPT1, OPT2 be the optimum profits, For G, G1, G2, – let ALG, ALG1, ALG2 be the profits of the approx. alg. By def., ALGi > OPTi /αi. Clearly, OPT ≤ OPT1 + OPT2. Assume OPT1 /α1 ≥ OPT2 /α2. Then ALG ≥ ALG1 ≥ OPT1 α1 ≥ OPT1 + OPT2 α1 + α2 ≥ OPT α1 + α2 .
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Weighted Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
3α
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
[Briest, Krysta V¨
[Briest, Krysta V¨
[Chekuri & Khanna: SIAM J. Comput.’05]
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2.
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2.
Idea: Realize stars!
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1:
Idea: Realize stars!
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers,
Idea: Realize stars!
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves.
Idea: Realize stars!
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
Idea: Realize stars!
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
Idea: Realize stars!
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-(
Idea: Realize stars!
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-(
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
⇒ ALG1 ≥ 3/4 · ALG′
1
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
⇒ ALG1 ≥ 3/4 · ALG′
1 ≥ 3/4 · OPT′ 1/α
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
⇒ ALG1 ≥ 3/4 · ALG′
1 ≥ 3/4 · OPT′ 1/α ≥ 3/4 · OPT1/α.
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
⇒ ALG1 ≥ 3/4 · ALG′
1 ≥ 3/4 · OPT′ 1/α ≥ 3/4 · OPT1/α.
Analogously, find solution of profit ALG2 ≥ 3/4 · OPT2 /α.
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
⇒ ALG1 ≥ 3/4 · ALG′
1 ≥ 3/4 · OPT′ 1/α ≥ 3/4 · OPT1/α.
Analogously, find solution of profit ALG2 ≥ 3/4 · OPT2 /α. Take better one!
Idea: Realize stars! No slack?
Let G = (V1˙ ∪V2, E) with E ⊆ V1 ∪ V2. First, find a good solution with all star centers in V1: – for each u ∈ V1, make 8 bins as for star centers, – for each v ∈ V2, make 1 item as for star leaves. Gap yields a solution of profit ALG′
1 ≥ OPT′ 1 /α,
where OPT′
1 is profit of an opt. sol. with centers in V1.
This solution may have corner contacts :-( ⇒ Remove two cheapest items from (3 top and 3 bottom bins) or (3 left and 3 right) bins.
⇒ ALG1 ≥ 3/4 · ALG′
1 ≥ 3/4 · OPT′ 1/α ≥ 3/4 · OPT1/α.
Analogously, find solution of profit ALG2 ≥ 3/4 · OPT2 /α. Take better one!
Idea: Realize stars! No slack? ⇒ profit ALG = max{ALG1, ALG2}.
We know: ALG = max{ALG1, ALG2}
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2}
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution!
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph.
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆).
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4.
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64]
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2 such that the star centers of Si1 are in V1 and those of Si2 are in V2
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2 such that the star centers of Si1 are in V1 and those of Si2 are in V2 W.l.o.g., S11 has profit ≥ OPT /4.
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2 such that the star centers of Si1 are in V1 and those of Si2 are in V2 W.l.o.g., S11 has profit ≥ OPT /4. On the other hand, p(S11) ≤ OPT1.
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2 such that the star centers of Si1 are in V1 and those of Si2 are in V2 W.l.o.g., S11 has profit ≥ OPT /4. On the other hand, p(S11) ≤ OPT1. ⇒ ALG ≥
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2 such that the star centers of Si1 are in V1 and those of Si2 are in V2 W.l.o.g., S11 has profit ≥ OPT /4. On the other hand, p(S11) ≤ OPT1. ⇒ ALG ≥ ALG1 ≥
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2 such that the star centers of Si1 are in V1 and those of Si2 are in V2 W.l.o.g., S11 has profit ≥ OPT /4. On the other hand, p(S11) ≤ OPT1. ⇒ ALG ≥ ALG1 ≥ 3/4 · OPT1/α ≥
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2 such that the star centers of Si1 are in V1 and those of Si2 are in V2 W.l.o.g., S11 has profit ≥ OPT /4. On the other hand, p(S11) ≤ OPT1. ⇒ ALG ≥ ALG1 ≥ 3/4 · OPT1/α ≥ 3/4 · p(S11)/α ≥
and ALGi ≥ 3/4 · OPTi/α. We know: ALG = max{ALG1, ALG2} Now, compare with a fixed optimum solution! Let G ⋆ = (V , E ⋆) be its profit graph. , i.e., OPT = p(E ⋆). G ⋆ is bipartite & planar ⇒ |E ⋆| ≤ 2n − 4. ⇒ E ⋆ can be decomposed into two forests F1 and F2. [Nash-Williams, JLMS’64] ⇒ Fi can be decomposed into two star forests Si1 and Si2 such that the star centers of Si1 are in V1 and those of Si2 are in V2 W.l.o.g., S11 has profit ≥ OPT /4. On the other hand, p(S11) ≤ OPT1. ⇒ ALG ≥ ALG1 ≥ 3/4 · OPT1/α ≥ 3/4 · p(S11)/α ≥ 3/16 · OPT /α.
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
Let G = (V , E) be any graph.
Let G = (V , E) be any graph. Idea: Reduce to bipartite case!
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Let G = (V , E) be any graph. Idea: Reduce to bipartite case!
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Let G = (V , E) be any graph. Idea: Reduce to bipartite case!
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2.
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case!
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′.
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case!
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α).
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case!
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α). Let G ⋆ = (V , E ⋆) be a fixed optimum solution.
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case!
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α). Let G ⋆ = (V , E ⋆) be a fixed optimum solution.
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case! Any edge of G ⋆ is contained in G ′ with probability 1/2.
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α). Let G ⋆ = (V , E ⋆) be a fixed optimum solution. Let OPT = p(E ⋆ ∩ E ′).
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case! Any edge of G ⋆ is contained in G ′ with probability 1/2.
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α). Let G ⋆ = (V , E ⋆) be a fixed optimum solution. Let OPT = p(E ⋆ ∩ E ′). Then E[OPT] = OPT /2.
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case! Any edge of G ⋆ is contained in G ′ with probability 1/2.
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α). Let G ⋆ = (V , E ⋆) be a fixed optimum solution. Let OPT = p(E ⋆ ∩ E ′). Then E[OPT] = OPT /2. ⇒ E[ALG] ≥
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case! Any edge of G ⋆ is contained in G ′ with probability 1/2.
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α). Let G ⋆ = (V , E ⋆) be a fixed optimum solution. Let OPT = p(E ⋆ ∩ E ′). Then E[OPT] = OPT /2. ⇒ E[ALG] ≥ 3E[OPT′]/(16α)
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case! Any edge of G ⋆ is contained in G ′ with probability 1/2.
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α). Let G ⋆ = (V , E ⋆) be a fixed optimum solution. Let OPT = p(E ⋆ ∩ E ′). Then E[OPT] = OPT /2. ⇒ E[ALG] ≥ 3E[OPT′]/(16α) ≥ 3E[ OPT ]/(16α) =
{v1v2 ∈ E | v1 ∈ V1, v2 ∈ V2} =
Let G = (V , E) be any graph. Idea: Reduce to bipartite case! Any edge of G ⋆ is contained in G ′ with probability 1/2.
Partition V randomly into V1 and V2 with Pr[v ∈ V1] = 1/2. Consider the bipartite graph G ′ = (V , E ′) induced by V1 and V2. Apply previous theorem to G ′. ⇒ solution for G of profit ALG ≥ 3 OPT′/(16α). Let G ⋆ = (V , E ⋆) be a fixed optimum solution. Let OPT = p(E ⋆ ∩ E ′). Then E[OPT] = OPT /2. ⇒ E[ALG] ≥ 3E[OPT′]/(16α) ≥ 3E[ OPT ]/(16α) = 3 OPT/(32α).
=
Let G = (V , E) be any graph. Idea: Reduce to bipartite case! Any edge of G ⋆ is contained in G ′ with probability 1/2.
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
Let G = (V , E) be any graph.
Let G = (V , E) be any graph. Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance. Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. Use α-approx. alg. for Gap. Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ Use α-approx. alg. for Gap. Use Gap – as in bipartite case!
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ Use α-approx. alg. for Gap. Use Gap – as in bipartite case! OPTGap /α ≥
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap. Use Gap – as in bipartite case! OPTGap /α ≥
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Use Gap – as in bipartite case! OPTGap /α ≥
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Use Gap – as in bipartite case! OPTGap /α ≥
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Use Gap – as in bipartite case! OPTGap /α ≥
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. Use Gap – as in bipartite case! OPTGap /α ≥
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. All contacts in Si can be realized – with corner contacts! Use Gap – as in bipartite case! OPTGap /α ≥
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. All contacts in Si can be realized – with corner contacts! Use Gap – as in bipartite case! OPTGap /α ≥ Choose heavier of S1 and S2.
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. All contacts in Si can be realized – with corner contacts! Use Gap – as in bipartite case! OPTGap /α ≥ Choose heavier of S1 and S2. Remove corner contacts.
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. All contacts in Si can be realized – with corner contacts! ⇒ ALG ≥ Use Gap – as in bipartite case! OPTGap /α ≥ Choose heavier of S1 and S2. Remove corner contacts.
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. All contacts in Si can be realized – with corner contacts! ⇒ ALG ≥ Use Gap – as in bipartite case! OPTGap /α ≥ Choose heavier of S1 and S2. Remove corner contacts. ALGGap
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. All contacts in Si can be realized – with corner contacts! ⇒ ALG ≥ 1/2 · 3/4 · ALGGap ≥ Use Gap – as in bipartite case! OPTGap /α ≥ Choose heavier of S1 and S2. Remove corner contacts.
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. All contacts in Si can be realized – with corner contacts! ⇒ ALG ≥ 1/2 · 3/4 · ALGGap ≥ Use Gap – as in bipartite case! OPTGap /α ≥ Choose heavier of S1 and S2. Remove corner contacts.
Let G = (V , E) be any graph. New: for every vertex, we construct both 8 bins and 1 item. Let OPTGap be the value of an opt. sol. of our Gap instance.
Any star forest is a feasible solution to our Gap instance. ⇒ OPTGap ≥ OPT /5. ⇒ ALGGap ≥ OPT /(5α). Use α-approx. alg. for Gap.
Partition each into star forest S1 and star forest + cycle S2. All contacts in Si can be realized – with corner contacts! ⇒ ALG ≥ 1/2 · 3/4 · ALGGap ≥ 3 OPT /(40α). Use Gap – as in bipartite case! OPTGap /α ≥ Choose heavier of S1 and S2. Remove corner contacts.
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
⋆) [Barth, Fabrikant, Kobourov, Lubiw, N¨
≈ 13.4
Unweighted Graph class
new◦ new◦ cycle, path 1 star α 1 + ε tree 2α, NP-hard 2 + ε 2 max-degree ∆ ⌊(∆ + 1)/2⌋ planar max-deg. ∆ 1 + ε
3 + ε planar 5α 5 + ε bipartite 16α/3 ≈ 8.4 APX-hard general rand.: 32α/3 ≈ 16.9 5 + 16α/3 det.: 40α/3 ≈ 21.1 α = e/(e − 1) ≈ 1.58
[He, SICOMP’93; He & Kant, TCS’97]
[He, SICOMP’93; He & Kant, TCS’97]