shudianzhao / ADMM-GP

ADMM-GP

This package is written based on the following papers:

Angelika Wiegele， Shudian Zhao: SDP-based bounds for graph partition via extended ADMM in arXiv at https://arxiv.org/abs/2105.09075.

Folders summary

• rand_data: benchmark instances used in numerical experiments;
• extended ADMM: codes for extended ADMM and post processing methods;
• form_model: codes to form SDP problems;
• heuristics: codes to build upper bounds for graph paritioning problems.

Generate graphs via Rudy in terminal:

source ./rudy_code_randGraphs.sh

Matlab functions:

• Form SDP problems

  kequi_form.m : form k-equipartition SDP relaxation

  make_GPKC.m : form GPKC SDP relaxation

• Extended ADMM aadmm_3b.m : extended ADMM for 3 blocks with adaptive method.

mprw_ineq_general.m : extended ADMM for SDP with inequalities and nonnegative constraints.

• Post-processing methods

post_proc_2.m: the post-processing method for aadmm_3b.m outputs.

post_proc_3.m: the post-processing nethod for mprw_ineq_general.m outputs.

• Heuristics

kequi_local.m: 2-opt method for k-equipartition.

kequi_rounding.m: Vector-clustering for k-equipartition.

kequi_random.m: Hyperplane rounding method for k-equipartition.

GPKC_local.m: 2-opt method for GPKC.

GPKC_rounding_flex_rand.m: Vertices-clustering for GPKC.

Example script:

• k-equipartition

  1. Form the input data (A,b,C) for extended ADMM with the adjacency matrix Ac and the partition number k

  % matlab code
  % Form the SDP relaxation 
  % <C，X> s.t. A(X) =b, X >= 0, X is postive semidefinite.
  
  [A,b,C] = kequi_form(Ac,k);
  

  2. Call the extended ADMM fucntion to solve the SDP relaxation with nonnegative constraints

  % maximum number of iteration is 100000
  % X1 is the primal solution, (y1,Z1,S1) is the dual solution
  [ X1, y1, Z1,S1] =  aadmm_3b(A, b, C,10000);
  
  

  3. Get the safe lower bound

  % tuning dual variable y and S to get a feasible solution for the dual SDP problem
  % ynew is the new y variable, LB is the safe lower bound
   [ynew1,LB1] = post_proc_2(Z1, A', C,b, S1);
  

  4. Add violated triangle inequalities
  • Find violated triangle indices and form the corresponding inequalties

  % initialize hash,f
  hash =[];
  f = []
  % max_ineq is the maximum number of violated triangle inequalties
  max_ineq = 100;
  [f0, T, hash,gamma,brk]=separation_kequi(X,f,T,hash,max_ineq);
  % form data for SDP with inqualtieis 
   [B,f]=formtri_kc(T,f0,n1);
  L = zeros(n1,n1);
  U = Inf*ones(n1,n1);
  
  

  • Solve the new SDP relaxation with inequalities with extended ADMM and get the safe lower bound

  %intialize sigma
  sigma= 0.1;
  % tolarence for infeasibility
  tol_err = 1e-5;
  % initalize X, Z with preivious ADMM results X1, Z1
  
  [ X2, y2,y2_bar, Z2,S2,s2,v2] = mprw_ineq_general(A, B, b, C ,f,L,U,10000,sigma,tol_err,X1,Z1);
  
  [ynew2,LB2] = post_proc_3(Z2,A',B',C,b,f,S2); 
  

  5. Build upper bounds from the SDP solution by heuristics
  • Vector clustering method for the solution of DNN relaxtion

  % rndseed: random seed
   [part1, newX1,ub1,partcell1] = kequi_rounding(rndseed,X1,k,C);
  

  • Hyperplane rounding method

  % rndseed: random seed
  [part1, newX1,ub1,partcell1] = kequi_random(rndseed,X1,k,C);
  
  

  • 2-opt method on output from vector clustering method/ hyperplane rounding method

    [partcell1_1,part1_1,newX1_1,ub1_1]= kequi_local(L,k,partcell1,part1,newX1,ub1);
  

  • GPKC
  1. Form the GPKC problem with adjacency matrix Ac, vertex weights a and capacity weight W

     [A,B,b,f,C]=make_GPKC(Ac,a,W);
     

  2. Solve the SDP relaxation with nonnegative constraints

  ```
  n1 = size(Ac,1);
  L = zeros(n1,n1);
  U = Inf*ones(n1,n1);
  [ X1 y1, y1_bar, Z1,S1,s1,v1] = mprw_ineq_general(A, B, b, C ,f,L,U,10000)；
  ```
  

  3. Get the safe lower bound

  ```
    [ynew1,LB1] = post_proc_3(Z1,A',B', C,b,f, S1); 
  ```
  

  4. Build upper bounds from the SDP solution
  • vector clustering method

  [part1, newX1,ub1,partcell1] = GPKC_rounding_flex_rand(rnd_seed,X1,C,W,a);
  

  • 2-opt method

  [partcell1_1,part1_1,newX1_1,ub1_1]= GPKC_local(Ac,W0,a,partcell1,part1,newX1,ub1);