0/1 Polytopes with Quadratic Chv atal Rank Thomas Rothvo and - PowerPoint PPT Presentation

0/1 Polytopes with Quadratic Chv´ atal Rank Thomas Rothvoß and Laura Sanit` a 3rd Cargese Workshop on Combinatorial Optimization

b b b b b b b b b b b b b b b b Gomory Chv´ atal Cuts ◮ Given: Polytope P ⊆ R n P

b b b b b b b b b b b b b b b b Gomory Chv´ atal Cuts ◮ Given: Polytope P ⊆ R n ◮ Want: Integral hull P P I := conv { P ∩ Z n } P I

b b b b b b b b b b b b b b b b Gomory Chv´ atal Cuts ◮ Given: Polytope P ⊆ R n ◮ Want: Integral hull P P I := conv { P ∩ Z n } ◮ Idea: Let cx ≤ β valid inequality for P ( c ∈ Z n ) P I cx ≤ β

b b b b b b b b b b b b b b b b Gomory Chv´ atal Cuts ◮ Given: Polytope P ⊆ R n ◮ Want: Integral hull P P I := conv { P ∩ Z n } ◮ Idea: Let cx ≤ β valid inequality for P ( c ∈ Z n ) ◮ The Gomory Chv´ atal cut cx ≤ ⌊ β ⌋ valid for P I P I cx ≤ β cx ≤ ⌊ β ⌋

b b b b b b b b b b b b b b b b Gomory Chv´ atal Cuts ◮ Given: Polytope P ⊆ R n ◮ Want: Integral hull P P I := conv { P ∩ Z n } ◮ Idea: Let cx ≤ β valid inequality for P ( c ∈ Z n ) P ′ ◮ The Gomory Chv´ atal cut cx ≤ ⌊ β ⌋ valid for P I P I cx ≤ β cx ≤ ⌊ β ⌋ ◮ Gomory Chv´ atal closure � � P ′ = { all GC cuts for P } = { x | cx ≤ ⌊ max { cy | y ∈ P }⌋} c ∈ Z n

b b b b b b b b b b b b b b b b Gomory Chv´ atal Cuts ◮ Given: Polytope P ⊆ R n ◮ Want: Integral hull P P I := conv { P ∩ Z n } ◮ Idea: Let cx ≤ β valid inequality for P ( c ∈ Z n ) P ′ ◮ The Gomory Chv´ atal cut cx ≤ ⌊ β ⌋ valid for P I P I cx ≤ β cx ≤ ⌊ β ⌋ ◮ Gomory Chv´ atal closure � � P ′ = { all GC cuts for P } = { x | cx ≤ ⌊ max { cy | y ∈ P }⌋} c ∈ Z n ◮ k th closure P ( k ) := P ′′′ . . . ′ � �� � k times

b b b b b b b b b b b b b b b b Gomory Chv´ atal Cuts ◮ Given: Polytope P ⊆ R n ◮ Want: Integral hull P P I := conv { P ∩ Z n } ◮ Idea: Let cx ≤ β valid inequality for P ( c ∈ Z n ) P ′ ◮ The Gomory Chv´ atal cut cx ≤ ⌊ β ⌋ valid for P I P I cx ≤ β cx ≤ ⌊ β ⌋ ◮ Gomory Chv´ atal closure � � P ′ = { all GC cuts for P } = { x | cx ≤ ⌊ max { cy | y ∈ P }⌋} c ∈ Z n ◮ k th closure P ( k ) := P ′′′ . . . ′ � �� � k times atal rank: P (rk( P )) = P I ◮ Chv´

What’s known

What’s known ◮ For each rational polyhedron P , rk( P ) < ∞

b b b b b b b b b b b b b b b b b b b b What’s known ◮ For each rational polyhedron P , rk( P ) < ∞ ◮ But for every k , there is P ⊆ R 2 with rk( P ) ≥ k (0 , 1) ( k, 1 2 ) P (0 , 0)

b b b b b b b b b b b b b b b b b b b b What’s known ◮ For each rational polyhedron P , rk( P ) < ∞ ◮ But for every k , there is P ⊆ R 2 with rk( P ) ≥ k (0 , 1) ( k, 1 2 ) P (0 , 0) ◮ For the rest of the talk assume P ⊆ [0 , 1] n

What’s known — if P ⊆ [0 , 1] n ◮ rk( P ) ≤ O ( n 3 log n ) [Bockmayer, Eisenbrand, Hartmann, Schulz ’98]

What’s known — if P ⊆ [0 , 1] n ◮ rk( P ) ≤ O ( n 3 log n ) [Bockmayer, Eisenbrand, Hartmann, Schulz ’98] O ( n 2 log n ) ◮ rk( P ) ≤ [Eisenbrand, Schulz ’99]

What’s known — if P ⊆ [0 , 1] n ◮ rk( P ) ≤ O ( n 3 log n ) [Bockmayer, Eisenbrand, Hartmann, Schulz ’98] O ( n 2 log n ) ◮ rk( P ) ≤ [Eisenbrand, Schulz ’99] ◮ For some P , rk( P ) ≥ (1 + ε ) n [Eisenbrand, Schulz ’99]

What’s known — if P ⊆ [0 , 1] n ◮ rk( P ) ≤ O ( n 3 log n ) [Bockmayer, Eisenbrand, Hartmann, Schulz ’98] O ( n 2 log n ) ◮ rk( P ) ≤ [Eisenbrand, Schulz ’99] ◮ For some P , rk( P ) ≥ (1 + ε ) n [Eisenbrand, Schulz ’99] ◮ For some P , rk( P ) ≥ 1 . 36 n [Pokutta, Stauffer ’11]

What’s known — if P ⊆ [0 , 1] n ◮ rk( P ) ≤ O ( n 3 log n ) [Bockmayer, Eisenbrand, Hartmann, Schulz ’98] ◮ rk( P ) ≤ O ( n 2 log n ) [Eisenbrand, Schulz ’99] ◮ For some P , rk( P ) ≥ (1 + ε ) n [Eisenbrand, Schulz ’99] ◮ For some P , rk( P ) ≥ 1 . 36 n [Pokutta, Stauffer ’11]

What’s known — if P ⊆ [0 , 1] n ◮ rk( P ) ≤ O ( n 3 log n ) [Bockmayer, Eisenbrand, Hartmann, Schulz ’98] ◮ rk( P ) ≤ O ( n 2 log n ) [Eisenbrand, Schulz ’99] ◮ For some P , rk( P ) ≥ (1 + ε ) n [Eisenbrand, Schulz ’99] ◮ For some P , rk( P ) ≥ 1 . 36 n [Pokutta, Stauffer ’11] Theorem (Sanit` a, R. ’12) There exists a family of polytopes P ⊆ [0 , 1] n with Chv´ atal rank rk( P ) ≥ Ω( n 2 ).

The polytope cx = 1 2 � c � 1

The polytope ◮ Let c ∈ Z n ≥ 0 be a vector � x ∈ { 0 , 1 } n : cx ≤ � c � 1 � P ( c ) := conv 2 � �� � Knapsack solutions cx = 1 2 � c � 1

The polytope ◮ Let c ∈ Z n ≥ 0 be a vector � x ∈ { 0 , 1 } n : cx ≤ � c � 1 � P ( c ) := conv 2 � �� � Knapsack solutions ( 1 2 , . . . , 1 2 ) cx = 1 2 � c � 1

The polytope ◮ Let c ∈ Z n ≥ 0 be a vector � � x ∈ { 0 , 1 } n : cx ≤ � c � 1 � � { x ∗ ( ε ) } P ( c, ε ) := conv ∪ 2 � �� � � �� � special vertex Knapsack solutions x ∗ ( ε ) = ( 1 2 + ε, . . . , 1 2 + ε ) P ( 1 2 , . . . , 1 2 ) P I cx = 1 2 � c � 1

Critical vectors ◮ Call ˜ c maximized at x ∗ c critical ⇔ ˜ x ∗ ( ε ) P

Critical vectors ◮ Call ˜ c maximized at x ∗ c critical ⇔ ˜ ˜ c x ∗ ( ε ) P

Critical vectors ◮ Call ˜ c maximized at x ∗ c critical ⇔ ˜ cx ≤ ⌊ β ⌋ cuts of x ∗ = ◮ ˜ ⇒ ˜ c critical ˜ c x ∗ ( ε ) P

b b b b b b b b b b b b b b b b b b Critical vectors ◮ Call ˜ c maximized at x ∗ c critical ⇔ ˜ cx ≤ ⌊ β ⌋ cuts of x ∗ = ◮ ˜ ⇒ ˜ c critical 0 P

b b b b b b b b b b b b b b b b b b Critical vectors ◮ Call ˜ c maximized at x ∗ c critical ⇔ ˜ cx ≤ ⌊ β ⌋ cuts of x ∗ = ◮ ˜ ⇒ ˜ c critical ˜ c 0 P

b b b b b b b b b b b b b b b b b b Critical vectors ◮ Call ˜ c maximized at x ∗ c critical ⇔ ˜ cx ≤ ⌊ β ⌋ cuts of x ∗ = ◮ ˜ ⇒ ˜ c critical ˜ c ∼ ε 0 P

Overview

Overview ⇒ Ω( n 2 ) rank critical vectors are long =

Overview critical vectors are good Simultaneous Diophantine Approximations to c ⇒ Ω( n 2 ) rank critical vectors are long =

Overview random vector has no short, good SDA critical vectors are good Simultaneous Diophantine Approximations to c ⇒ Ω( n 2 ) rank critical vectors are long =

A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) .

A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 P (0)

A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 b ε 1 P (1)

A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 b ε 1 b ε 2 P (2)

b A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 b ε 1 b ε 2

b b A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 b ε 1 b ε 2

b b A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 b ε 1 b ε 2 b ε k

b b b b A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 x ∗ ( ε i ) x ∗ ( ε i +1 ) b ε k

b b b b b A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 x ∗ ( ε i ) x ∗ ( ε i +1 ) b ε k cx ≤ β ˜ ◮ Let ˜ cx ≤ β be the GC cut “cutting furthest” in it. i

b b b b b b A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 x ∗ ( ε i ) x ∗ ( ε i +1 ) b ε k cx ≤ β ˜ cx ≤ ⌊ β ⌋ ˜ ◮ Let ˜ cx ≤ β be the GC cut “cutting furthest” in it. i

b b b b b b A lower bound strategy Theorem Assume: ∀ ε ∈ [( 1 2 ) Θ( n ) , Θ(1)] : c � 1 ≥ Ω( n c critical = ˜ ⇒ � ˜ ε ) . Then rk ( P ) ≥ Ω( n 2 ) . b ε 0 x ∗ ( ε i ) x ∗ ( ε i +1 ) b ε k cx ≤ β ˜ cx ≤ ⌊ β ⌋ ˜ ◮ Let ˜ cx ≤ β be the GC cut “cutting furthest” in it. i cx ∗ ( ε i ) − ˜ cx ∗ ( ε i +1 ) ˜