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Given a ILP (Integer-linear programming) optimization with n integer variables and m constraints and implementing a branch and bound tree for solving a canonical problem,

  1. how many levels (height of tree) does tree require to reach the all-integer-optimal solution?
  2. how many branches does the algorithm require to reach the all-integer-optimal solution?
Mohammad Namakshenas
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That's a hard question to answer - both could be zero if you are really lucky, or in the worst case the "height" (or depth) could be equal to the number of integer variables n. The number of branches could be much bigger... it all depends on the problem and the solver

TimChippingtonDerrick
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  • I know that in worst case, the height of tree is $n$ (at most $n$), but exactly how many? About solver, consider a standard B&B with depth first search. – Mohammad Namakshenas Feb 25 '14 at 09:55
  • For example, one answer (for number of levels) would be developing a formula based on combination of number of integer variables (n) and number of uni-modular sub-matrices in A (coefficients of variables in constraints). Therefore, we only focus on a routine that transforms the non-integer variables into integer ones. – Mohammad Namakshenas Feb 25 '14 at 10:12