Improved LP Lower Bounds for Difference Triangle Sets

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 Kløve [2] we define an (I, J) difference triangle set, ∆, as a set of integers {aij | 1 ≤ i ≤ I, 0 ≤ j ≤ J} such that all the differences aij − aik, 1 ≤ i ≤ I, 0 ≤ k < j ≤ J are positive and distinct. Let m = m(∆) be the maximum difference. The difference triangle set problem is to minimize m given I and J . Kløve 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 {aij | 1 ≤ i ≤ I, 0 ≤ j ≤ J} we have associated difference triangles {Xijk | 1 ≤ i ≤ I, 1 ≤ j ≤ J , 1 ≤ k ≤ J + 1 − j} where Xijk = ai,j+k−1 − ai,k−1. The ith difference triangle is {Xijk | 1 ≤ j ≤ J , 1 ≤ k ≤ J + 1 − j}. Its top row is {Xi1k | 1 ≤ k ≤ J}. Clearly each difference triangle is determined by its top row. The maximum difference in each difference triangle is its bottom element XiJ1 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 | 1 ≤ i ≤ I, 1 ≤ k ≤ J} of the difference triangles. Minimizing m will be the objective. Clearly since m is the maximum