By Ascheuer N., Junger M., Reinelt G.
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Additional resources for A Branch & Cut Algorithm for the Asymmetric Traveling Salesman Problem with Precedence Constraints
For practical purposes, the smooth numbers method will remain preferable until quantum computers become available. Acknowledgments This article was written while I held the 2000–2001 HP-MSRI Visiting Research Professorship. , and Ulrich Vollmer for answering my questions, and Bart de Smit for providing numerical assistance. A special word of thanks is due to Hugh Williams, whose version  of the same story contains many details omitted in mine. 20 HENDRIK W. LENSTRA, JR. References [Archimedes 1999] Archimedes, The cattle problem, in English verse by S.
Ap ;bp / ; if a D p ap ; b D p bp : p p p The Greeks would have been more comfortable with the following geometric definition of the GCD: if a > b > 0 consider a rectangle of width a and height b. Remove a b-by-b square from one end of the rectangle. Continue removing maximal subsquares (squares with one side being a side of the rectangle) until the remaining rectangle is a square. , their GCD. For instance, if a D 73 and b D 31, then the removal of two maximal subsquares leaves an 11-by-31 rectangle, the removal of two further maximal squares leaves an 11-by-9 rectangle, the removal of one maximal subsquare leaves a 2-by-9 rectangle, the removal of four maximal subsquares leaves a 2by-1 rectangle, and the removal of one maximal subsquare leaves a unit square.
With overwhelming likelihood a nonresidue will be found quickly, and thus quadratic nonresidues can be found quickly with a probabilistic polynomialtime algorithm. However, no deterministic polynomial-time algorithm is known. Nothing seems to be better than testing a D 2; 3; : : : until arriving at a nonresidue. log1C" p/. p 1=4 /. The simple-minded (deterministic) algorithm for finding a nonresidue could take exponential time. log2 p/ [Bach 1990]. P ROBLEM 8. , a divisor m such that 1 < m < n.