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