# minimising Longest-Path in DAG

Assume we have weighted DAG (directed-acycle-graph), source s and target t.

Define the number of edges as $$E$$.

Given $$0<\alpha<1$$:

Choose $$\alpha*E$$ edges to cut their weight by half so that the longest path from s to t is minimized.

Personal note: We've been asked this question by our professor. I assume the problem is NP-Hard. We are looking for some heuristics that will gives us good results, not necessarily optimal.

Any suggestions will be welcome.

• (Seems to "prefer" paths with not much more than $\alpha\times E$ costly edges.) Jun 8, 2021 at 21:53
• What do you meet by that @greybeard? You have no control over the initial longest-paths Jun 8, 2021 at 22:17
• It's just a musing about what difference $\alpha$ makes. Between paths with less than $\alpha\times E$ edges with weight>0: none. Path with most costly $\alpha\times E$ edges contributing less than half the cost: cost after halving greater than three quarters of the original. Jun 8, 2021 at 22:29