Heuristics for Minimum Brauer Chain Problem

1. Heuristics for Minimum Brauer Chain Problem Fatih Gelgi Melih Onus

2. Outline • Problem Definition • Binary Method • Factor Method • Heuristics • Experimental Results • Conclusions

3. Brauer Chain • A Brauer chain for a positive integer n is a sequence of integers 1 = a0, a1, a2, …, ar = n such that ai = ai-1 + ak for some 0 ≤ k < i and 1 ≤ i ≤ n • Example: 1 1, 1+1=2 1, 2, 2+2=4 1, 2, 4, 4+2=6 1, 2, 4, 6, 6+6=12 1, 2, 4, 6, 12, 12+2=14 is a Brauer chain for 14 with length 5

4. Binary Method • Write the number n in binary form • Replace each 1 with DA and each 0 with D • Remove the leading DA • Begin from 1, follow the sequence from left to right • For each D, double the current number • For each A, add 1 to the current number • Example: • Binary representation of 19 is 10011 • Sequence is DDDADA • 1, 1+1=2, 2+2=4, 4+4=8, 8+1=9, 9+9=18, 18+1=19

5. Factor Method • Let n = p*q where n, p, qZ+. First calculate Brauer chain for p and q • Let 1 = a1, a2, …, ak = p is Brauer Chain for p • Let 1 = b1, b2, …, bk = q is Brauer Chain for q • 1 = a1, a2, …, ak = p = p*b1, p*b2, …, p*bm = p*q is a Brauer chain for n=p*q • Example: • <1, 2 ,4, 5> <1, 2, 3> • <1, 2 ,4, 5, 10, 15>

6. Heuristics • Binary Heuristic • Factorization Heuristic • Dynamic Heuristic: • It uses the previous solutions to obtain the best solution for current n value • We store only one solution for each number • The dynamic formula is, l(n) = min{l(k)+1} where k<n and the solution sequence of k must contain n-k • 2-3-5 Heuristic • This algorithm always begins with numbers 1, 2, 3, 5, …. • For the next element, it selects as maximum as possible • 2-3-6 Heuristic • 2-4-8 Heuristic

7. Experiments • We did 3 types of experiments • We calculated all n values up to about 4500 • We calculated only one randomly chosen n value within each interval of 200 up to 10000 • We calculated all n values up to 20000 (using factorization and dynamic heuristics) • To calculate optimum value, we used exhaustive search with branch and bound technique • Although our pruning conditions make the search quite faster, we were able to calculate the optimum values up to 4500 since the running time is O(n!)

8. Optimum and 1.5 optimum values for n up to 4500

9. The performance of heuristics:

11. Experiment II • To obtain optimum values for larger n’s and increase the quality of our experiments, for each interval with size 200 we chose a random n value • All the solutions of heuristics are clearly smaller than 1.5 optimum

12. Experiment II

13. Approximation Ratios • All the heuristics obviously has 1.5 approximation ratios • Better than 2-4-8, 2-3-6, 2-3-5 and binary, factorization has 1.25 approximation value • Dynamic has an incredible approximation ratio with 1.1 • Empirically, factorization is even smaller than 1.5 lg n and dynamic is smaller than 1.4 lg n

16. Conclusion • Several heuristics for approximating minimum Brauer chain problem is discussed • The optimum function is not monotone we couldn’t prove a theoretical approximation ratio better than 2 • Experimental results show that there is empirically an approximation with 1.1 which is incredible for the problem • For approximation results, we also achieved 1.4 lg n length for any number n where the trivial lower bound is lg n • Providing a better lower bound, the approximation factor can be decreased • With a good lower bound, approximation factor can be proved theoretically