tailieunhanh - Báo cáo toán học: "Improved LP Lower Bounds for Difference Triangle Sets"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Improved LP Lower Bounds for Difference Triangle Sets. | Improved LP Lower Bounds for Difference Triangle Sets James B. Shearer IBM Watson Research . Box 218 Yorktown Heights NY 10598 . email jbs@ Submitted August 11 1999 Accepted August 24 1999 Abstract. In 1991 Lorentzen and Nilsen showed how to use linear programming to prove lower bounds on the size of difference triangle sets. In this note we show how to improve these bounds by including additional valid linear inequalities in the LP formulation. We also give some new optimal difference triangle sets found by computer search. AMS Subject Classification 05B10 Following Klpve 2 we define an I J difference triangle set A as a set of integers ữịj I 1 i 1 0 j Jg such that all the differences aij aik 1 i 1 0 k j J are positive and distinct. Let m m A be the maximum difference. The difference triangle set problem is to minimize m given I and J. Klpve defined M I J as this minimum. Lorentzen and Nilsen 3 showed how to use linear programming LP methods to prove lower bounds on M I J . This was a generalization and improvement of earlier lower bounds. Here we show how to improve the LP bound for I 1 by adding inequalities to the LP formulation. We also announce some new values of M I J found by computer search. Given a difference triangle set aj I 1 i 1 0 j Jg we have associated difference triangles Xijk I 1 i 1 1 j J 1 k J 1 jg where Xj aiij fc_i a k-i. The ith difference triangle is Xjjk I 1 j J 1 k J 1 jg. Its top row is Xi1k I 1 k Jg. Clearly each difference triangle is determined by its top row. The maximum difference in each difference triangle is its bottom element Xj 1 which is the sum of the top row. We can now formulate a linear program which gives a lower bound for m M I J . The LP variables will be m and the top elements Xi1k I 1 i 1 1 k Jg of the difference triangles. Minimizing m will be the objective. Clearly since m is the maximum 1 2 THE ELECTRONIC JOURNAL OF COMBINATORICS 6 1999 R31 difference we have J m Xij1 x Xi1k 1 i I 1 k 1 Also .