Graph longest path using linear programming - optimization

Question

I have a weighted directed graph where there are no cycles, and I wish to define the constraints so that I can solve a maximization of the weights of a path with linear programming. However, I can't wrap my head around how to do that.
For this I wish to use the LPSolve tool. I thought about making an adjacency matrix, but I don't know how I could make that work with LPSolve.
How can I define the possible paths from each node using constraints and make it generic enough that it would be simple to adapt to other graphs?

Answer 1

Since you have a weighted directed graph, it is sufficient to define a binary variable x_e for each edge e and to add constraints specifying that the source node has flow balance 1 (there is one more outgoing edge selected than incoming edge), the destination node has flow balance -1 (there is one more incoming edge than outgoing edge selected), and every other node has flow balance 0 (there are the same number of outgoing and incoming edges selected). Since your graph has no cycles, this will result in a path from the source to the destination (assuming one exists). You can maximize the weights of the selected edges.
I'll continue the exposition in R using the lpSolve package. Consider a graph with the following edges:
(edges <- data.frame(source=c(1, 1, 2, 3), dest=c(2, 3, 4, 4), weight=c(2, 7, 3, -4)))
# source dest weight
# 1 1 2 2
# 2 1 3 7
# 3 2 4 3
# 4 3 4 -4
The shortest path from 1 to 4 is 1 -> 2 -> 4, with weight 5 (1 -> 3 -> 4 has weight 3).
We need the flow balance constraints for each of our four nodes:
source <- 1
dest <- 4
(nodes <- unique(c(edges$source, edges$dest)))
# [1] 1 2 3 4
(constr <- t(sapply(nodes, function(n) (edges$source == n) - (edges$dest == n))))
# [,1] [,2] [,3] [,4]
# [1,] 1 1 0 0
# [2,] -1 0 1 0
# [3,] 0 -1 0 1
# [4,] 0 0 -1 -1
(rhs <- ifelse(nodes == source, 1, ifelse(nodes == dest, -1, 0)))
# [1] 1 0 0 -1
Now we can put everything together into our model and solve:
library(lpSolve)
mod <- lp(direction = "max",
objective.in = edges$weight,
const.mat = constr,
const.dir = rep("=", length(nodes)),
const.rhs = rhs,
all.bin = TRUE)
edges[mod$solution > 0.999,]
# source dest weight
# 1 1 2 2
# 3 2 4 3
mod$objval
# [1] 5