1. Applying bitwise operations for solving combinatorial optimization problems Mikhail Batsyn 1, Larisa Komosko 1, Alexey Nikolaev 1, Pablo San Segundo 2 1 LATNA Laboratory, Higher School of Economics, Russia 2 Centro de Automática y Robótica (CAR), UPM-CSIC, Spain

2. Bitwise operations Binary data can be represented as bit strings CPU has such bitwise operations as: V – or & – and – xor popcnt – population count (number of 1-bits) lsb/msb – least/most significant bit index 128 bits in 1 operation (fraction of nanosecond) 2

3. Bitwise operations Find the first neighbour: lsb Leave only neighbours: & Add neighbours: V Hamming distance:, popcnt Crossover in GA: V 3

4. Vertex colouring problem Independent set partition problem Clique partition problem SF (Smallest-First) algorithm GIS (Greedy Independent Sets) algorithm BnB (Branch and Bound) algorithm 4

5. Bitwise SF algorithm c i – the colour of vertex i F – the matrix of forbidden colours. If F ki =1 then colour k is forbidden for vertex i. F = 0 for i = 1..n c i = min{k | F ki = 0} Fc i = Fc i V A i end 5

6. Bitwise GIS algorithm ____ B = A – the matrix of non-neighbours c i = 0, k = 1 while ∃ i c i = 0 while B i ≠ 0 j * = min{j | B ij = 1} c j * = k B i = B i & B j * end k = k + 1 end 6

7. Bitwise branch and bound Same operations as in bitwise SF algorithm Different colours are checked for every vertex 7

8. Computational results for SF 8 |V|50010005000 DensitySpeedup 0.013.23.4 0.103.03.33.7 0.203.53.94.5 0.304.04.55.1 0.404.24.95.5 0.504.34.95.2 0.603.84.5 0.703.33.93.8 0.802.73.33.4 0.902.22.53.0 0.991.81.92.9

9. Computational results for SF 9 G(V,E)|V||E|Speedup brock200_1200148342.8 brock200_220098763.2 brock200_3200120485.3 brock200_4200130892.7 brock400_1400597233.0 brock400_2400597862.8 brock400_3400596813.0 brock400_4400597652.7 brock800_18002075053.9 brock800_28002081664.0 brock800_38002073333.8 brock800_48002076434.4 c-fat200-120015341.6 c-fat200-220032352.7 c-fat200-520084732.0

10. Computational results for SF On Erdos-Renyi graphs the average speedup is greater for bigger graphs: 3.3 times for 500 vertices, 3.7 – for 1000, and 4.1 – for 5000. Average speedup on DIMACS is 2.6 times. 10

11. Maximum clique problem 11 1 9 2 83 4 5 7 6 10 12 16 13 14 15 Objective: find a complete subgraph of maximum size

12. Maximum clique problem 12 1 9 2 11 83 4 5 7 6 10 12 16 13 14 15

13. Branch and bound 13 Current clique: {1} Subgraph of candidates: {2, 10, 11, 15, 16}

14. 14 Current clique: {1, 16} Subgraph of candidates: {10, 11, 15} Branch and bound

15. 15 Current clique: {1, 16, 15} Subgraph of candidates: {10, 11} Branch and bound

16. BnB algorithm Q – the current clique (initially Q = ∅ ) C – the candidates to the current clique ( C = V ) Clique (Q, C) for v ∈ C C = C \ {v} Q’ = Q’ ∪ {v} C’ = C ∩ N v if C’ ≠ ∅ Clique (Q’, C’) else if |Q’| > |Q * | Q * = Q’ 16

17. Computational results 17 InstanceMCSBBMCXRS Speedup brock400_1384,74167,81 2,3 brock400_2166,3670,15 2,4 brock400_3269,29119,62 2,3 brock400_4134,3669,71 1,9 brock800_15215,952568,36 2,0 brock800_24713,612346,49 2,0 brock800_33152,911660,29 1,9 brock800_42333,281101,47 2,1 gen400_p0.9_5535013,122296,73 1,6 gen400_p0.9_6565290107577,9 0,6 gen400_p0.9_759355190351,98 1,0 keller577010,252797,5 1,5 MANN_a45105,5244,62 2,4 p_hat300-31,580,75 2,1 p_hat500-356,3136,48 1,5 p_hat700-31337,55689,46 1,9

18. Computational results 18 InstanceMCSBBMCXRS Speedup p_hat1000-2116,0871,97 1,6 p_hat1000-3540000208854,6 2,6 p_hat1500-12,682 1,3 p_hat1500-29040,296824,25 1,3 sanr200_0.916,259,32 1,7 sanr400_0.7104,2751,28 2,0 san10001,050,59 1,8 frb30-15-136000970,29 37,1 frb30-15-236000675,46 53,3 frb30-15-336000314,88 114,3 frb30-15-4360001002,85 35,9 frb30-15-536000673,13 53,5