The 21st International Computing and Combinatorics Conference - PowerPoint PPT Presentation

Introduction Formulation Algorithm Analysis Future work The 21st International Computing and Combinatorics Conference (COCOON’15) will be held in Beijing, China, during August 4-6, 2015. Special issue: Algorithmica Theoretical Computer Science Journal of Combinatorial Optimization Web site: cocoon2015.bjut.edu.cn . . . . . .

Introduction Formulation Algorithm Analysis Future work Important date: Submission Deadline: February 15, 2015 Notification of acceptance: April 5, 2015 Camera Ready: April 25, 2015 Conference Dates: August 4-6, 2015 . . . . . .

Introduction Formulation Algorithm Analysis Future work . . A complex semidefinite programming rounding approximation algorithm for the balanced Max- 3 -Uncut problem . . . . . Dachuan Xu (Joint work with Chenchen Wu, Donglei Du, and Wen-qing Xu) Email: xudc@bjut.edu.cn Beijing University of Technology September 2014, Peking University . . . . . . Balanced Max- 3 -Uncut

. .. . .. . .. . .. Introduction Formulation Algorithm Analysis Future work . Outline . . .. Introduction 1 . . . . . . Balanced Max- 3 -Uncut

. .. . .. . .. Introduction Formulation Algorithm Analysis Future work . Outline . . .. Introduction 1 . .. Formulation 2 . . . . . . Balanced Max- 3 -Uncut

. .. . .. Introduction Formulation Algorithm Analysis Future work . Outline . . .. Introduction 1 . .. Formulation 2 . .. Algorithm 3 . . . . . . Balanced Max- 3 -Uncut

. .. Introduction Formulation Algorithm Analysis Future work . Outline . . .. Introduction 1 . .. Formulation 2 . .. Algorithm 3 . .. Analysis 4 . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Outline . . .. Introduction 1 . .. Formulation 2 . .. Algorithm 3 . .. Analysis 4 . .. Future work 5 . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Introduction . Graph Partition Problem The most famous problem is Max Cut. Max Cut . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Introduction . Graph Partition Problem The most famous problem is Max Cut. There are also some other variant problems of Max Cut problem: Max Bisection(balanced version of Max-Cut): adding equal cardinality constraint Max- n 2 -Uncut: balanced version and calculating the weight not in the cut. · · · . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Introduction . We study the balanced Max- 3 -Uncut. Problem description Given a weighted graph G = ( V, E ) (Assume | V | is a multiple of 3 ) weight function w : E → R + Goal: partition V into three subsets S 1 , S 2 , and S 3 with equal cardinality such that the total weight of the edges from the same subsets is maximized Balanced Max- 3 -UnCut . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Introduction . Methods Approximation algorithm(attractive algorithm with bounded solution) linear program → Semidefinite program The real space R → The complex space C Results Based on the complex semdefinite programming rounding technique, we proposed a 0 . 3456 -approximation algorithm for the balanced Max- 3 -Uncut. . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Introduction . Literature review Based on semidefinite programming in R Goemans and Williamson(J. ACM, 1995) for the Max-Cut: 0 . 87856 , semidefinite programming rounding using randomly hyperplane; Frieze and Jerrum(Algorithmica, 2006) for the Max-Bisection: 0 . 6514 , semidefinite programming rounding + greedy swapping; Austin et al. (SODA, 2013) for the Max-Bisection: 0 . 8776 , semidefinite programming hierarchies rounding.(Best until now) Halperin and Zwick(Random Structures and Algorithms, 2002) for the Balanced Max- 2 -Uncut: 0 . 6436 , semidefinite programming rounding using randomly hyperplane. Wu et al.(J Combin Opt, 2013) for the Balanced Max- 2 -Uncut: 0 . 8776 , semidefinite programming rounding using randomly hyperplane. . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Introduction . Literature review Based on semidefinite programming in C Goemans and Williamson(J. Comput. Syst. Sci., 2004) for the ( 7 4 π 2 arccos 2 ( − 1 / 4) − ϵ 3 ) Max- 3 -Cut: ≈ (0 . 8360 − ϵ ) , for 12 any given ϵ > 0 , complex semidefinite programming rounding. Ling (COCOA, 2009) for Max- 3 -Section: 0 . 6733 , complex semidefinite programming rounding + greedy swapping. . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Formulation . The Balanced Max- 3 -Uncut can be described by ∑ ∑ ∑ max w ij + w ij + w ij , ( S 1 , S 2 , S 3 ) ∈ P ( V ) i,j ∈ S 1 i,j ∈ S 2 i,j ∈ S 3 | S 1 | = | S 2 | = | S 3 | where P ( V ) := { ( S 1 , S 2 , S 3 ) : S 1 ∪ S 2 ∪ S 3 = V, and S k ∩ S l = ∅ for all k ̸ = l } . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Formulation . x 2 = 1( − 1 , 1) Max − Cut x 3 = 1(1 , ω = e − 2 3 πi , ω 2 ) Balaced Max − 3 − Uncut 1 ∑ w ij (1 + y i · y j + y j · y i ) max 3 i<j ∑ s . t . y i = 0 , i ∈ V y i ∈ { 1 , ω, ω 2 } , ∀ i ∈ V. 1 ω 2 ω Introduction the variable y i ∈ { 1 , ω, ω 2 } for each i ∈ V . . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Formulation . Relaxation: y i ∈ C → v i ∈ C n , ∥ v i ∥ = 1 Tighter relaxation: Since y i ∈ { 1 , ω, ω 2 } for all i ∈ V , we must have y i · y j + y j · y i ≥ − 1 , ∀ i, j ∈ V, ω · ( y i · y j ) + ω 2 · ( y j · y i ) ≥ − 1 , ∀ i, j ∈ V, ω 2 · ( y i · y j ) + ω · ( y j · y i ) ≥ − 1 , ∀ i, j ∈ V, which can be rewritten as Re( y i · y j ) ≥ − 1 ∀ i, j ∈ V, 2 , Re( ω · ( y i · y j )) ≥ − 1 ∀ i, j ∈ V, 2 , Re( ω 2 · ( y i · y j )) ≥ − 1 ∀ i, j ∈ V. 2 , . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Formulation . By adding the above extra inequalities into the above program, we get the complex semidefinite programming relaxations as follows. 1 ∑ max w ij (1 + v i · v j + v j · v i ) 3 i<j Re( v i · v j ) ≥ − 1 ∀ i, j ∈ V, s . t . 2 , Re( ω · ( v i · v j )) ≥ − 1 ∀ i, j ∈ V, 2 , (2.1) Re( ω 2 · ( v i · v j )) ≥ − 1 ∀ i, j ∈ V, 2 , ∑ v i · v j = 0 , i,j ∥ v i ∥ = 1 , ∀ i ∈ V, v i ∈ C n , ∀ i ∈ V. . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Algorithm . Step 1 Solve complex semidefinite programming Solve the (2.1) to obtain an optimal solution { v i } , leading to a complex semidefinite matrix V := ( v i · v j ) . Step 2 Generate random complex variable For a given parameter θ ∈ [0 , 1] , choose a random vector ξ ∼ N (0 , θV + (1 − θ ) I ) , where I is the n × n identity matrix. Step 3 Obtain solution for the Max- 3 -Uncut Arg( ξ i ) ∈ [0 , 2  1 , 3 π );  Arg( ξ i ) ∈ [ 2 3 π, 4 y i = ˆ ω, 3 π ); Arg( ξ i ) ∈ [ 4 ω 2 , 3 π, 2 π ) .  Let S 1 := { i : ˆ y i = 1 } , S 2 := { i : ˆ y i = ω } , and y i = ω 2 } . S 3 := { i : ˆ . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Algorithm . Step 4 Swap greedy to obtain solution for the balanced Max- 3 -Uncut Assume, without loss of generality, | S 1 | ≥ | S 2 | ≥ | S 3 | . Initialize ˆ S ℓ = S ℓ ( ℓ = 1 , 2 , 3 ). Denote the final partition with equal cardinality as ˜ S 1 , ˜ S 2 , and ˜ S 3 . Case 4.1. If | S 1 | ≥ | S 2 | ≥ n 3 ≥ | S 3 | , then iteratively, perform the following operations (i)-(ii) until | ˆ S ℓ | = n 3 for each ℓ = 1 , 2 : (i) Sort the vertices in ˆ S ℓ such that S ℓ | ) where δ ( i ) = ∑ δ ( i 1 ) ≥ . . . ≥ δ ( i | ˆ S ℓ w i ′ i i ′ ∈ ˆ ( i ∈ ˆ S ℓ ). S ℓ | from ˆ S ℓ to ˆ (ii) Move the point i | ˆ S 3 ; namely, { } S ℓ = ˆ ˆ S 1 | } , and ˆ S 3 = ˆ S ℓ \{ i | ˆ S 3 ∪ . i | ˆ S ℓ | Case 4.2. Operate similarly for the case of | S 1 | ≥ n 3 ≥ | S 2 | ≥ | S 3 | . . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Algorithm . . . . . . . Balanced Max- 3 -Uncut

Introduction Formulation Algorithm Analysis Future work . Analysis . First, we define z ( γ ) := W ( S 1 , S 2 , S 3 ) + γ C C ∗ , W ∗ where W ( S 1 , S 2 , S 3 ) is the weight of the partition is S 1 , S 2 , S 3 . W ∗ is the optimal solution of the complex semidefinite relaxation of the Max- 3 -Uncut. C = | S 1 || S 2 | + | S 1 || S 3 | + | S 2 || S 3 | . C ∗ = n 2 3 . ( S 1 C n , S 2 n , S 3 ) µ ( x ) = C ∗ , where x = . n . . . . . . Balanced Max- 3 -Uncut