Advanced Algorithms (IV) Chihao Zhang Shanghai Jiao Tong University - PowerPoint PPT Presentation

Advanced Algorithms (IV) Chihao Zhang Shanghai Jiao Tong University Mar. 18, 2019 Advanced Algorithms (IV) 1/11

Review n u v E w T u w v s.t. w u u max V w T u w u u V Advanced Algorithms (IV) e Vector Program MaxCut Integer Program Input: An undirected graph G V E . Problem: A set S V that maximizes E S S . max V e u v E x u x v s.t. x u u 2/11

Review w u max e u v E w T u w v s.t. n V u V w T u w u u V Advanced Algorithms (IV) Vector Program u MaxCut max Input: Problem: x u Integer Program 2/11 e u v E x u x v s.t. An undirected graph G = ( V , E ) . � � � E ( S , ¯ � . A set S ⊆ V that maximizes S )

Review w u e u v E w T u w v s.t. n Vector Program u V w T u w u u V Advanced Algorithms (IV) max 2/11 MaxCut s.t. Input: Problem: max Integer Program An undirected graph G = ( V , E ) . � � � E ( S , ¯ � . A set S ⊆ V that maximizes S ) ∑ 1 (1 − x u x v ) 2 e = { u , v }∈ E x u ∈ {− 1 , 1 } , ∀ u ∈ V

Review max Advanced Algorithms (IV) w T s.t. u w v max Vector Program MaxCut s.t. 2/11 Integer Program Input: Problem: An undirected graph G = ( V , E ) . � � � E ( S , ¯ � . A set S ⊆ V that maximizes S ) ( ) ∑ 1 ∑ 1 − w T 1 2 (1 − x u x v ) e = { u , v }∈ E 2 e = { u , v }∈ E w u ∈ R n , ∀ u ∈ V x u ∈ {− 1 , 1 } , ∀ u ∈ V u w u = 1 , ∀ u ∈ V

Let w v v V be an optimal solution. Task : Round w v v V to a cut max s.t. Advanced Algorithms (IV) polynomial-time) w T 3/11 u w v ( ) ∑ 1 1 − w T 2 e = { u , v }∈ E w u ∈ R n , ∀ u ∈ V u w u = 1 , ∀ u ∈ V is equivalent to a positive semi-definite programming (solvable in

Task : Round w v v V to a cut max s.t. Advanced Algorithms (IV) polynomial-time) w T 3/11 u w v ( ) ∑ 1 1 − w T 2 e = { u , v }∈ E w u ∈ R n , ∀ u ∈ V u w u = 1 , ∀ u ∈ V is equivalent to a positive semi-definite programming (solvable in Let { � w v } v ∈ V be an optimal solution.

max u w v Advanced Algorithms (IV) polynomial-time) w T s.t. 3/11 ( ) ∑ 1 1 − w T 2 e = { u , v }∈ E w u ∈ R n , ∀ u ∈ V u w u = 1 , ∀ u ∈ V is equivalent to a positive semi-definite programming (solvable in Let { � w v } v ∈ V be an optimal solution. Task : Round { � w v } v ∈ V to a cut

1. Pick a random hyperplane crossing the origin; r n where each r i Goemans–Williamson Rounding 2. The plane separates V into two sets. Implementation 1. Choose a vector r r i.i.d. 2. Let S u V r T w u . Advanced Algorithms (IV) 4/11

r n where each r i Goemans–Williamson Rounding 2. The plane separates V into two sets. Implementation 1. Choose a vector r r i.i.d. 2. Let S u V r T w u . Advanced Algorithms (IV) 4/11 1. Pick a random hyperplane crossing the origin;

Goemans–Williamson Rounding 2. The plane separates V into two sets. Implementation . Advanced Algorithms (IV) 4/11 1. Pick a random hyperplane crossing the origin; 1. Choose a vector r = ( r 1 , . . . , r n ) where each r i ∼ N (0 , 1) i.i.d. { } 2. Let S ≜ u ∈ V : r T � w u ≥ 0

r is a point on S n T w v . Analysis Proposition r uniformly at random. Proposition An edge u v E is separated with probability arccos w u Proposition Random hyperplane rounding is a -approximation of MaxCut. Advanced Algorithms (IV) 5/11

T w v . Analysis Proposition r Proposition An edge u v E is separated with probability arccos w u Proposition Random hyperplane rounding is a -approximation of MaxCut. Advanced Algorithms (IV) 5/11 ∥ r ∥ is a point on S n − 1 uniformly at random.

Analysis Proposition r Proposition w u Proposition Random hyperplane rounding is a -approximation of MaxCut. Advanced Algorithms (IV) 5/11 ∥ r ∥ is a point on S n − 1 uniformly at random. An edge { u , v } ∈ E is separated with probability 1 π arccos ( � T � w v ) .

Analysis Proposition r Proposition w u Proposition Advanced Algorithms (IV) 5/11 ∥ r ∥ is a point on S n − 1 uniformly at random. An edge { u , v } ∈ E is separated with probability 1 π arccos ( � T � w v ) . Random hyperplane rounding is a 0 . 878 -approximation of MaxCut.

i j n is positive semi-definite. Qvadratic Program We try to apply Goemans–Williamson rounding to general quadratic programs. Qvadratic Program max i j n a i j x i x j s.t. x i i n We assume A a i j Advanced Algorithms (IV) 6/11

i j n is positive semi-definite. Qvadratic Program We try to apply Goemans–Williamson rounding to general quadratic programs. Qvadratic Program max i j n a i j x i x j s.t. x i i n We assume A a i j Advanced Algorithms (IV) 6/11

i j n is positive semi-definite. Qvadratic Program We try to apply Goemans–Williamson rounding to general quadratic programs. Qvadratic Program max s.t. We assume A a i j Advanced Algorithms (IV) 6/11 ∑ a i , j x i x j 1 ≤ i , j ≤ n x i ∈ {− 1 , +1 } , i = 1 , . . . , n .

Qvadratic Program We try to apply Goemans–Williamson rounding to general quadratic programs. Qvadratic Program max s.t. Advanced Algorithms (IV) 6/11 ∑ a i , j x i x j 1 ≤ i , j ≤ n x i ∈ {− 1 , +1 } , i = 1 , . . . , n . We assume A = ( a i , j ) 1 ≤ i , j ≤ n is positive semi-definite.

i v j Rounding i n . Advanced Algorithms (IV) otherwise. ; x i T r if v i 3. x i . 2. Pick a vector r u.a.r on S n n 1. Compute v i We simply follow G-W… i n v i s.t. a i j v T i j n max Vector Program 7/11

Rounding i n . Advanced Algorithms (IV) otherwise. ; x i T r if v i 3. x i . 2. Pick a vector r u.a.r on S n 1. Compute v i We simply follow G-W… s.t. max Vector Program 7/11 ∑ i v j a i , j v T 1 ≤ i , j ≤ n v i ∈ R n , i = 1 , . . . , n .

Rounding s.t. Advanced Algorithms (IV) v i We simply follow G-W… 7/11 max Vector Program ∑ i v j a i , j v T 1 ≤ i , j ≤ n v i ∈ R n , i = 1 , . . . , n . 1. Compute { � v i } 1 ≤ i ≤ n . 2. Pick a vector r u.a.r on S n − 1 . 3. ˆ x i = 1 if � T r ≥ 0 ; ˆ x i = − 1 otherwise.

Analysis Proposition E x i x j arcsin v T i v j Proposition Random hypergraph rounding is a -approximation of QP. Proof. Use Schur producet theorem. Advanced Algorithms (IV) 8/11

Analysis Proposition Advanced Algorithms (IV) Use Schur producet theorem. Proof. -approximation of QP. Random hypergraph rounding is a Proposition v T x j E 8/11 [ ] = 2 ˆ x i ˆ π arcsin (ˆ i · ˆ v j ) .

Analysis Proposition Advanced Algorithms (IV) Use Schur producet theorem. Proof. Proposition v T x j E 8/11 [ ] = 2 ˆ x i ˆ π arcsin (ˆ i · ˆ v j ) . Random hypergraph rounding is a 2 π -approximation of QP.

Analysis Proposition Advanced Algorithms (IV) Use Schur producet theorem. Proof. Proposition v T E x j 8/11 [ ] = 2 ˆ x i ˆ π arcsin (ˆ i · ˆ v j ) . Random hypergraph rounding is a 2 π -approximation of QP. □

two weights w e w e S k of V . Correlation Clustering edges between clusters. Advanced Algorithms (IV) w e e E w e e E The goal is to maximize E edge in a cluster; E Given a undirected graph G S Find a partition . E has V E in which each e 9/11

S k of V . Correlation Clustering edges between clusters. Advanced Algorithms (IV) w e e E w e e E The goal is to maximize edge in a cluster; E E S Find a partition 9/11 ▶ Given a undirected graph G = ( V , E ) in which each e ∈ E has two weights w + e , w − e ≥ 0 .

E Correlation Clustering edge in a cluster; E edges between clusters. The goal is to maximize e E w e e E w e Advanced Algorithms (IV) 9/11 ▶ Given a undirected graph G = ( V , E ) in which each e ∈ E has two weights w + e , w − e ≥ 0 . ▶ Find a partition S = ( S 1 , . . . , S k ) of V .

Correlation Clustering The goal is to maximize e E w e e E w e Advanced Algorithms (IV) 9/11 ▶ Given a undirected graph G = ( V , E ) in which each e ∈ E has two weights w + e , w − e ≥ 0 . ▶ Find a partition S = ( S 1 , . . . , S k ) of V . ▶ E + ( S ) ≜ edge in a cluster; E − ( S ) ≜ edges between clusters.

Correlation Clustering Advanced Algorithms (IV) 9/11 ▶ Given a undirected graph G = ( V , E ) in which each e ∈ E has two weights w + e , w − e ≥ 0 . ▶ Find a partition S = ( S 1 , . . . , S k ) of V . ▶ E + ( S ) ≜ edge in a cluster; E − ( S ) ≜ edges between clusters. ▶ The goal is to maximize ∑ ∑ w + w − e + e . e ∈ E − ( S ) e ∈ E + ( S )

n , let e k be the k -th unit vector. w u v x T w u v x T Vector Program V w u v x T u x v s.t. x T v x v v u x v x T u v V x u n u V Advanced Algorithms (IV) u x v E For u x v k max u v E u x v w u v x T s.t. u v x u e e n u V Relaxation max 10/11

w u v x T w u v x T x T x T u x v s.t. x T v x v v V u x v u x v u v V x u n u V Advanced Algorithms (IV) w u v Vector Program 10/11 s.t. max u v E u x v w u v x T u x v x u E e e n u V Relaxation max u v For 1 ≤ k ≤ n , let e k be the k -th unit vector.

w u v x T x T x T u x v s.t. x T v x v v V u x v u x v u v V x u n u V Advanced Algorithms (IV) w u v Vector Program 10/11 s.t. u v max Relaxation max E For 1 ≤ k ≤ n , let e k be the k -th unit vector. ( ) ∑ w + u x v ) + w − u , v ( x T u , v (1 − x T u x v ) { u , v }∈ E x u ∈ { e 1 , . . . , e n } , ∀ u ∈ V .