Longest path of directed graph with cycle and now weights

Question

If you have a unweighted directed graph with possible cycles, what algorithm would you use to find the longest path without visiting the same node twice? Also, there are multiple starting nodes...

The real problem I am trying to solve is the following:

All values are positive integers:

A1 = [a1, a2, ..., an]

B2 = [b1, b2, ..., bn]

A person starts at ai. In order to live another day, the person must find an aj (i != j) that is greater than or equal to bi (you do not 'keep' the difference of the two). Once the person selects aj, the must do the same with bj and ak. The goal is to live as long as possible, and each time you land on an ai, counts as a new day.

How would you solve a problem like this?

score 0 · Answer 1 · answered Feb 17 '24 at 19:08

0

The longest path problem is $\mathsf{NP}$-complete, so there is no known polynomial algorithm.

It can be solved in exponential time using dynamic programming.

answered Feb 17 '24 at 19:08

Nathaniel

15,071
2
27
52

thank you! i don't really understand the DP solution. could you maybe please share a better explanation or pseudo code? – user2330624 Feb 17 '24 at 19:23
Well not really: the explaination of Yuval Filmus is already fine as it is, I would not be able to do better. – Nathaniel Feb 18 '24 at 00:43
But how will that work if there is not a single origin or starting node? – user2330624 Feb 18 '24 at 00:56
Just run it from each vertex? – Nathaniel Feb 18 '24 at 01:20
You could also add a new fake vertex with an edge to each other, and find the longest path from the new vertex. – Nathaniel Feb 18 '24 at 01:22

Longest path of directed graph with cycle and now weights

1 Answers1