Scale Reduction Techniques for Computing Maximum Induced Bicliques
Abstract
:1. Introduction
2. Integer Programming Formulations
3. Exact Algorithm Based on the Proposed Scale Reduction
 Find a lower bound. Use a heuristic algorithm to obtain a lower bound on the optimal solution.
 Apply scale reduction. Given a heuristic solution, recursively identify and remove vertices that cannot be included in a globally optimal solution, until no further reduction is possible.
 Solve. Apply a standard exact algorithm to find a globally optimal solution in the residual graph.
3.1. Finding a Lower Bound
Algorithm 1 Greedy induced star construction heuristic. 

3.2. Scale Reduction Techniques
3.2.1. Scale Reduction for the MB Problem
 1.
 ${P}^{{\mathcal{L}}^{\ast}}(E,k+1)\subseteq {P}^{{\mathcal{L}}^{\ast}}(E,k)\subseteq {P}^{\mathcal{L}}(E,k).$
 2.
 ${E}^{\ast}\subseteq {P}^{\mathcal{L}}(E,k)$ for all k.
Algorithm 2 Scale reduction algorithm. 

3.2.2. Scale Reduction for the MEB Problem
4. Computational Experiments
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Abbreviations
IP  integer programming 
MIP  mixed integer programming 
MB  maximum biclique 
MEB  maximum edge biclique 
RDS  Russian Doll Search 
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Graph  $\left\mathit{V}\right$  $\left\mathit{E}\right$  LB  ${\mathit{V}}^{\prime}$  ${\mathit{E}}^{\prime}$  SRCPU(s)  Opt. Obj.  CPU (s) 

jazz  198  2742  18  85  704  0.44  20  0.67 
1133  5451  34  35  35  0.61  34  0.64  
netscience  1589  2742  16  26  42  0.02  16  0.03 
add20  2395  7462  30  68  186  464.66  30  464.73 
data  2851  15,093  8  41  99  1.09  8  1.36 
as19971108  3015  5347  540  583  740  1531.77  540  1602.59 
add32  4960  9462  16  66  108  0.17  16  0.42 
CAGrQC  5242  14,490  29  35  41  0.39  29  0.42 
as19991204  6296  12,830  1294  1407  2234  3461.69  1294  3582.89 
p2pGnutella08  6301  20,777  88  88  87  1.36  88  1.48 
as20000102  6474  13,233  1338  1454  2430  1440.82  1340  19,645.00 
p2pGnutella09  8114  26,013  98  98  97  1.09  98  1.47 
hepth  8361  15,751  22  81  151  0.40  23  0.66 
p2pGnutella06  8717  31,525  104  113  121  0.21  104  0.44 
p2pGnutella05  8846  31,839  87  88  88  0.76  87  1.11 
CAHepTH  9877  25,973  32  43  59  4.18  32  4.42 
PGPgiantcompo  10,680  24,316  105  114  122  3223.57  105  3223.85 
p2pGnutella04  10,876  39,994  97  100  102  0.12  97  0.42 
oregon1010519  11,051  22,724  2203  2384  4289  1466.45  2207  2085.01 
oregon1010526  11,173  23,409  2199  2385  4308  1619.41  2203  3852.24 
oregon2010526  11,460  16,365  2230  2428  4676  1652.65  2234  4762.25 
condmat  16,726  47,594  41  55  72  5257.99  41  5258.05 
p2pGnutella25  22,687  54,705  64  66  67  0.22  64  0.33 
as22july06  22,963  48,436  2243  2387  4125  1678.23  2245  10,722.34 
p2pGnutella24  26,518  65,369  304  356  426  1040.93  304  1043.72 
p2pGnutella30  36,682  88,328  54  54  53  0.59  54  0.70 
p2pGnutella31  62,586  147,892  90  95  100  0.41  90  0.70 
delaunayn14  16,384  49,122  9  26  39  4.15  9  4.23 
delaunayn16  65,536  196,575  9  18  34  61.05  9  61.13 
delaunayn17  131,072  393,176  9  67  123  243.29  10  244.00 
delaunayn18  262,144  786,396  11  65  123  541.73  12  542.29 
delaunayn19  524,288  1,572,823  11  54  92  2205.50  11  2206.09 
delaunayn20  1,048,576  3,145,686  12  24  45  5086.75  13  5087.06 
Graph  $\left\mathit{V}\right$  $\left\mathit{E}\right$  LB  ${\mathit{V}}^{\prime}$  ${\mathit{E}}^{\prime}$  SRCPU(s)  Opt. Obj.  CPU (s) 

jazz  198  2742  17  85  704  0.67  19  2.67 
1133  5451  33  35  35  1.42  33  1.45  
netscience  1589  2742  15  26  42  0.01  15  0.03 
add20  2395  7462  29  68  186  306.95  29  307.45 
data  2851  15,093  7  1944  11,290  0.78    >11,000 
as19971108  3015  5347  539  583  740  1430.57  539  1462.28 
add32  4960  9462  15  66  108  0.17  15  0.89 
CAGrQC  5242  14,490  28  35  41  0.47  28  0.51 
as19991204  6296  12,830  1293  1407  2234  2081.83  1293  9542.64 
p2pGnutella08  6301  20,777  87  448  3194  9.19  87  8177.89 
as20000102  6474  13,233  1337  1454  2430  1201.03  1339  10,044.40 
p2pGnutella09  8114  26,013  97  98  97  12.37  97  12.51 
hepth  8361  15,751  21  81  151  0.50  22  1.26 
p2pGnutella06  8717  31,525  103  305  1283  2.53  138  654.47 
p2pGnutella05  8846  31,839  86  136  203  5.74  92  8.38 
CAHepTH  9877  25,973  31  43  59  6.19  31  6.51 
PGPgiantcompo  10,680  24,316  104  114  122  3899.91  104  3900.27 
p2pGnutella04  10,876  39,994  96  100  102  0.56  96  0.78 
oregon1010519  11,051  22,724  2202  2384  4289  1262.38  2206  10,494.20 
oregon1010526  11,174  23,409  2198  2385  4308  1415.75  2202  10,750.50 
oregon2010526  11,460  16,365  2229  2428  4676  1579.16    >11,000 
condmat  16,726  47,594  40  55  72  7703.95  40  7704.02 
p2pGnutella25  22,687  54,705  63  66  67  0.31  63  0.44 
as22july06  22,963  48,436  2242  2387  4125  3735.30    >11,000 
p2pGnutella24  26,518  65,369  303  356  426  220.50  303  225.52 
p2pGnutella30  36,682  88,328  53  54  53  2.06  53  2.12 
p2pGnutella31  62,586  147,892  89  95  100  0.98  89  1.30 
delaunayn14  16,384  49,122  8  26  39  5.57  8  5.70 
delaunayn16  65,536  196,575  8  18  34  80.41  8  80.54 
delaunayn17  131,072  393,176  8  67  123  315.68  9  316.55 
delaunayn18  262,144  786,396  10  65  123  373.68  11  374.53 
delaunayn19  524,288  1,572,823  10  54  92  1478.44  10  1479.16 
delaunayn20  1,048,576  3,145,686  11  24  45  3044.67  12  3045.02 
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Shahinpour, S.; Shirvani, S.; Ertem, Z.; Butenko, S. Scale Reduction Techniques for Computing Maximum Induced Bicliques. Algorithms 2017, 10, 113. https://doi.org/10.3390/a10040113
Shahinpour S, Shirvani S, Ertem Z, Butenko S. Scale Reduction Techniques for Computing Maximum Induced Bicliques. Algorithms. 2017; 10(4):113. https://doi.org/10.3390/a10040113
Chicago/Turabian StyleShahinpour, Shahram, Shirin Shirvani, Zeynep Ertem, and Sergiy Butenko. 2017. "Scale Reduction Techniques for Computing Maximum Induced Bicliques" Algorithms 10, no. 4: 113. https://doi.org/10.3390/a10040113