tetris hardness of approximation - mitcourses.csail.mit.edu/6.890/fall14/lectures/l12_images.pdf ·...
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Tetris Hardness of Approximation [Breukelaar, Demaine, Hohenberger, Hoogeboom, Kosters, Liben-Nowell 2003]
Max E3-X(N)OR-SAT → Max E3SAT[Håstad 2001]
food
verb
transportation
animal
verb
furniture
Projection Puzzles [Moshkovitz]
b
a
t
b a t
food
verb
transportation
animal
verb
furniture
Score: 3
Projection Puzzles [Moshkovitz]
b
cab
t
b a t
food
verb
transportation
animal
verb
furniture
c a b
Score: 6
Projection Puzzles [Moshkovitz]
b
cab
t
b a t
food
verb
transportation
animal
verb
furniture
c a b
Score: 6
Projection Puzzles [Moshkovitz]
b
cabet
b a t
food
verb
transportation
animal
verb
furniture
c a b
b e d
Score: 8
Projection Puzzles [Moshkovitz]
b
cabety
b a t
food
verb
transportation
animal
verb
furniture
c a b
b e d
Score: 10
c r y
Projection Puzzles [Moshkovitz]
b
cabety
b a t
food
verb
transportation
animal
verb
furniture
c a b
b e d
Score: 10
c r y
? a y
Projection Puzzles [Moshkovitz]
b
cabety
b a t
food
verb
transportation
animal
verb
furniture
c a b
b e d
Score: 10
c r y
? e y
? a y
Projection Puzzles [Moshkovitz]
bpcabety
b a t
food
verb
transportation
animal
verb
furniture
c a b
b e d
Score: 15
c r y
p e a
p a y
Projection Puzzles [Moshkovitz]
furniture
cscaroty
c a t
s a y
c a r
s o y
c o t
c r y
food
verb
transportation
animal
verb
Score: 18
Projection Puzzles [Moshkovitz]
Typical Approximation FactorsApproximation Factor Minimization Problems Maximization Problems
1 + 𝜀𝜀 Planar/𝐻𝐻-minor-free/2D… e.g. dominating set
Planar/𝐻𝐻-minor-free/2D… e.g. independent set
Θ(1) Steiner tree, Steiner forest, Traveling Salesman, …
Maximum coverage,Max cut
Θ(log∗𝑛𝑛) Asymmetric 𝑘𝑘-center
Θ log𝑛𝑛Set cover, Dominating set,Node-weighted Steiner tree…
Unique coverage,Domatic number
Θ log2 𝑛𝑛 Group Steiner treeΩ log2 𝑛𝑛 ∩ 𝑂𝑂(𝑛𝑛𝜀𝜀) Directed Steiner tree
Ω 2log1−𝜀𝜀 𝑛𝑛 ∩ 𝑂𝑂 𝑛𝑛𝑐𝑐 Label cover (MinRep),Directed Steiner forest
Label cover (MaxRep)
Ω 𝑛𝑛1−𝜀𝜀 ∩ �𝑂𝑂 𝑛𝑛 Chromatic number Independent set = clique𝑐𝑐 = 4
5+ 𝜀𝜀
𝑐𝑐 = 13
Reductions to Steiner Problems