By Ascheuer N., Junger M., Reinelt G.

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**Additional info for A Branch & Cut Algorithm for the Asymmetric Traveling Salesman Problem with Precedence Constraints**

**Example text**

In particular, F(x) < F(x) + 1, so Hc(x) ::; 10g(F(x) + 1) + 210glog(F(x) + 1) + 0(1) ::; F(x) + 0(1). o Finally, we use the Invariance Theorem. 1). Following Chaitin, we switch the point of view, from a deterministic one to a probabilistic one. To this end we define - for a given Chaitin computer - the halting probabilities. 27. Given a Chaitin computer C we define the following "probabilities" : Pc(x) = {uEA* I C(u,>')=x} Q-1ul. Pc(x/y) = {uEA* I C(u,y*)=x} In the case C = U we put, using the common convention, P(x) = Pu(x), P(x/y) = Pu(x/y).

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28) Proof One has Pc (x) = {uEA* I C(u,,\)=x} o and Hc(x) = lul,C{u,'x) = x. In the case of the universal Chaitin computer, neither the absolute, nor the conditional algorithmic probability can be 0 or 1. 30. 29) o < P(x/y) < 1. 29, with C = U, P{x) ~ Q-H(x) = Q-1x*1 > O. 25), ExEA* P{x) :::; 1 and the fact that each term of the series is non-zero we deduce that P{x) < 1. 30). 31. For every Chaitin computer C and all naturals n, m the following four formulae are true: #{x E A* I Hc(x) < m} < (Qm -l)/{Q - 1), #{x E A* I Hc(x/y) < m} < (Qm -l)/(Q -1), n m #{x E A* I Pc(x) > -} < -, m n n m #{x E A* I Pc(x/y) > -} < -.

### A Branch & Cut Algorithm for the Asymmetric Traveling Salesman Problem with Precedence Constraints by Ascheuer N., Junger M., Reinelt G.

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