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Solve Traveling Salesperson Problems using callbacks or delayed rows

Description

In this example, we show how to use the FICO Xpress R-interface to model and solve the famous Travelling Salesperson Problem (TSP). The TSP is the classic combinatorial optimization problem to find the shortest so-called tour through a complete graph.

Further explanation of this example: Xpress R Reference Manual

Source Files
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tsp.R

#####################################
# This file is part of the          #
# Xpress-R interface examples       #
#                                   #
#   (c) 2021 Fair Isaac Corporation #
#####################################
#' ---
#' title: "Finding a Shortest Tour"
#' subtitle: "The Travelling Salesperson Problem"
#' author: Gregor Hendel
#' date: Dec. 2020
#' ---
#'
## ----setup, include=FALSE-----------------------------------------------------
knitr::opts_chunk$set(echo = TRUE) knitr::opts_chunk$set(results = "hold")

#'
suppressMessages(library(xpress))

# we use the subsets function from CombMSC
suppressMessages(library(CombMSC))

# load necessary libraries for plotting
suppressMessages(library(reshape2))
suppressMessages(library(magrittr))
suppressMessages(library(ggplot2))
suppressMessages(library(dplyr))
suppressMessages(library(tibble))

#'
#' In this example, we show how to use the FICO Xpress R-interface to model and
#' solve the famous **Travelling Salesperson Problem (TSP)**. The TSP is
#' the classic combinatorial optimization problem to find the shortest so-called
#' tour through a complete graph.
#'
#' This longer example discusses different approaches for solving TSPs with Xpress.
#' First we solve the problem without any subtour elimination constraints, which
#' will give an invalid solution. Then we solve the problem initialized with all
#' subtour elimination constraints, which does not perform well. Finally we show
#' how to use callbacks for adding subtour elimination constraints
#' dynamically during the search.
#'
#' We dive right into the mathematical formulation.
#'
#' # Mathematical formulation
#'
#' There are different variants how to formulate the TSP. We use the classical
#' formulation using binary variables and exponentially many constraints to
#' eliminate all so-called subtours.
#'
#' ## Graphs and Tours
#'
#' We assume we are given a complete Graph $K_n$ on $n$ vertices $V$. A graph is
#' called complete if it contains all $\binom{n}{2}$ possible edges, one between
#' each pair of nodes $v \neq w \in V \times V$. Denote the set of edges by $E$. We
#' further have a function that assigns a cost to each edge,
#'
#' $$#' d: E \rightarrow \mathbb{Q} #'$$
#'
#' A path in $K_n$ is a sequence of adjacent edges $(e_1,e_2,\dots,e_k)$ such that
#' consecutive edges have a common node, and no node is visited twice (except,
#' possibly, for the endpoints). A tour is a path of length $n$, which starts and
#' ends at the same node and in between visits each node exactly once. The cost of
#' a tour is the sum of the costs of its edges. The TSP demands to find a tour of
#' minimum cost.
#'
#' For the mathematical formulation, we introduce $n(n-1)/2$ binary variables
#' $x_{e}$ for each edge $e \in E$. Of course, $x_{e} = 1$ corresponds to $e$ being
#' part of the tour represented by a solution to our program.
#'
#' #' \begin{align} #' && \min \sum\limits_{e \in E} d(e) x_{e}\\ #' & \text{s.t.} &\sum\limits_{e \in \delta(v)} x_{e} &= 2 & \forall v\in V\\ #' & &\sum\limits_{e \in \delta(S)} x_e & \geq 2 & \forall \emptyset \neq S \subsetneq V\\ #' & & x_e &\in \{0,1\} & \forall e \in E #' \end{align} #'
#'
#' The first set of $n$ constraints requires that each node should have exactly two
#' adjacent edges in a solution. This set of constraints is often referred to as
#' "node-degree" constraints. The second set of constraints formulate that each
#' nonempty strict subset of nodes be entered at least once and left at least once.
#' We refer to this type of constraints as "subtour-elimination" constraints.
#'
#' It is easy to verify that a tour, encoded as a characteristic vector of its
#' edges, is indeed a feasible solution for the above program. The reverse is also
#' true.
#'
#' It is obvious that the above formulation is computationally critical for larger
#' graphs due to its sheer size. The number of subtour elimination constraints
#' grows exponentially with the number $n$ of nodes in the graph, which becomes
#' intractable even for moderate graph sizes.
#'
#' # Creating the Formulation Explicitly
#'
#' We generate $n = 10$  points $(x,y)$. The locations are drawn uniformly at
#' random from [-3,3]. We create a distance data frame that holds the Euclidean
#' distance for each pair of nodes Vi, Vj, $i < j$
#' Each row of this data frame will correspond to 1 binary variable of our TSP.
#'
#' ## Data Generation
## ----Data Generation----------------------------------------------------------
# the number of nodes
n <- 10

# random seed for reproducibility
set.seed(3701)
loc.matrix <- matrix(runif(n * 2, -3, 3), ncol = 2)

#
# each row of dist.df will correspond to 1 binary variable of our TSP
dist.df <- dist(loc.matrix) %>%
as.matrix() %>%
melt(varnames=c("Vi","Vj"), value.name="Dist") %>%
dplyr::filter(Vi < Vj)

tibble::glimpse(dist.df)

# plot the n nodes in the plane and label them
ggplot(data.frame(loc.matrix),
aes(x=loc.matrix[,1], y=loc.matrix[,2], label=1:n)
) +
geom_point() +
geom_label(nudge_y=0.3) +
labs(x="X", y="Y")

#'
#'
#' We declare a function load_tsp that receives the above distance data frame as
#' input and creates a new Xpress Problem with only the node degree constraints
#' loaded. We align the columns we create with the dist.df data frame.
#'
#'
## ----Load TSP with Node Degree Constraints------------------------------------

# create a TSP problem with node degree constraints
p <- createprob()
nedges <-  nrow(dist.df)
problemdata <- list()

# column information
problemdata$lb <- rep(0,nedges) problemdata$ub <- rep(1,nedges)

# column names
problemdata$colname <- sprintf("x(%d,%d)",dist.df$Vi, dist.df$Vj) # objective coefficients problemdata$objcoef <- dist.df$Dist problemdata$columntypes <- rep("B", nedges)

# right hand side and row type
problemdata$rowtype <- rep("E", n) problemdata$rhs <- rep(2, n)

# we input the node degree constraints as sparse matrix in column major format
# directly using the corresponding problemdata names,
# see the documentation of

# each edge appears in exactly two node degree rows, once for each endpoint
# the number of elements in each column
problemdata$collen <- rep(2,nedges) # 1 additional marker for the end # the start index of the coefficients of each column. we need to pass a 0-based index into Xpress! problemdata$start <- 2 * (0:(nedges - 1)) # 0-based index!

# matrix coefficients, all 1's in the node degree constraints
problemdata$rowcoef <- rep(1, 2 * nedges) # row indices of each coefficient. We first create an array with 0's problemdata$rowind <- rep(0, 2 * nedges)

# The first endpoint Vi corresponds to the first row index of each edge
problemdata$rowind[2 * (1:nedges) - 1] <- dist.df$Vi - 1 # 0-based index!

# The second endpoint Vj corresponds to the second row index of each edge
problemdata$rowind[2 * (1:nedges)] <- dist.df$Vj - 1 # 0-based index!

# assign row names to identify the constraints easily in MPS or LP format
problemdata$probname <- "TSP" problemdata$rowname <- sprintf("NodeDeg_%d", 1:n)

# this invisibly returns the XPRSprob p from the function
}

#'
#' ## Solving the Node Degree TSP Without Subtour Elimination
#'
#' We solve this initial formulation using Xpress and plot the corresponding
#' solution.
#'
## ----Load and Solve with Node Degree Constraints------------------------------
xprs_optimize(p)

#'
#' Let's plot the TSP solution. For this, we translate
#' the vertex locations $(x,y)$ into
#' edges (segments in ggplot2 lingo) with start and endpoints.
#'
## ----Plot the Solution--------------------------------------------------------
plot_tsp_solution <- function(p) {

# convert (x,y) locations of the nodes into a data frame.
loc.df <- as.data.frame(loc.matrix)
colnames(loc.df) <- c("X", "Y")
loc.df$V <- 1:n solution <- xprs_getsolution(p) # copy dist.df. For plotting, we must store the (x,y)-coordinates # of both endpoints plot.df <- dist.df[solution == 1,] plot.df <- plot.df %>% dplyr::mutate( Xi = loc.df$X[Vi],
Yi = loc.df$Y[Vi], Xj = loc.df$X[Vj],
Yj = loc.df$Y[Vj] ) plot.df %>% ggplot(aes(x=Xi, xend=Xj, y=Yi, yend=Yj, label=Vi)) + geom_segment() + geom_point() + geom_label(nudge_y = 0.3) + geom_label(aes(x=Xj, y=Yj, label=Vj), nudge_y = 0.3) + geom_point(aes(x=Xj, y=Yj)) } summary(p) plot_tsp_solution(p) #' #' ## Adding Subtour Elimination Constraints #' #' In a first shot, we add all subtour elimination constraints #' explicitly to the model. We can do this because the #' example is relatively small. #' A row always enters the problem as last row (at position xpress:::ROWS - 1). #' After adding a row, we declare it immediately as a delayed row. #' #' ## ----Adding Subtour Elimnation Constraints------------------------------------ add_subtour_elim <- function(p, dist.df) { nedges <- nrow(dist.df) # we enumerate all subsets that contain the first node, # i.e, all subsets of v_2, ... ,v_n with at most n - 3 elements. for (r in 1:(n - 3)) { allsubsets <- subsets(n-1, r, 2:n) # each row in allsubsets represents 1 subset for (s in 1:nrow(allsubsets)) { viins <- (dist.df$Vi == 1) | (dist.df$Vi %in% allsubsets[s,]) vjins <- (dist.df$Vj == 1) | (dist.df$Vj %in% allsubsets[s,]) edgeins <- (viins != vjins) # exactly one endpoint in S # use addrows to add each row explicitly addrows(p, rowtype = "G", rhs = 2, start = c(0,(r+1) * (n - r - 1)), colind = (1:nedges)[edgeins] - 1, rowcoef = rep(1, (r+1) * (n - r - 1)) ) # declare this row as delayed row. loaddelayedrows(p, rowind=getintattrib(p, xpress:::ROWS) - 1) } } # return p again to the caller p } #' #' ## Solving the Full TSP #' #' When we solve the full TSP with the added subtour elimination constraints, we #' obtain a correct tour, albeit with higher cost than the subtour solution from #' the previous run. #' ## ----------------------------------------------------------------------------- p <- load_tsp(dist.df) %>% add_subtour_elim(dist.df) %>% xprs_optimize() print(p) summary(p) plot_tsp_solution(p) #' #' # Separate Violated Subtour Elimination Constraints. #' #' It is much more efficient to separate subtour elimination constraints #' dynamically during the search. #' We initially omit all subtour elimination constraints. #' If we encounter an integer feasible LP solution, we verify if #' the solution is indeed a tour with full length. If not, #' we determine the subtour in which the$V1$lies. #' This subtour cannot reach all other nodes in the graph. #' #' We define a function get_subtour. This will be used #' for the separation of violated subtour elimination constraints. #' It will also be used to check the feasibility of any #' integer feasible solution. #' #' ## Callback Definition #' ## ----------------------------------------------------------------------------- get_subtour <- function(prob, sol) { # subset the distance data frame to the currently used edges dist.df.sol <- dist.df[sol >= 1 - 1e-6,] subtourindices <- NULL # initialize a logical to indicate which nodes we have visited. visited <- logical(n) # edge rep maps each node to its 2 adjacent edges in the solution edgerep <- matrix(0, ncol = 2, nrow=n) for (i in 1:nrow(dist.df.sol)) { vi <- dist.df.sol$Vi[i]
vj <- dist.df.sol$Vj[i] edgerep[vi, 1 + (edgerep[vi, 1] != 0)] <- i edgerep[vj, 1 + (edgerep[vj, 1] != 0)] <- i } # iterate through the edges of the solution. curredge <- 1 pathlength <- 0 currnode <- dist.df.sol$Vi[curredge]
while (!visited[currnode] && pathlength <= n) {
# label node as visited
visited[currnode] <- TRUE
# go to the other node of the current edge
currnode <- ifelse(dist.df.sol$Vi[curredge] == currnode, dist.df.sol$Vj[curredge], dist.df.sol$Vi[curredge]) #find the other edge to walk to edges <- edgerep[currnode,] curredge <- ifelse(edges[1] == curredge, edges[2], edges[1]) pathlength <- pathlength + 1 } # if the path length was smaller than n, we found a subtour that is not connected # to the rest of the graph. The subset is part of the visited array if (pathlength < n) { subtourindices <- which(visited) # indices of the corresponding nodes. } return(subtourindices) } #' ## ----Optnode Callback Definition---------------------------------------------- optnodecallback <- function(prob) { # return if there are fractional variables in the LP solution if (getintattrib(prob, xpress:::MIPINFEAS) > 0) return() currlpsol <- getlpsol(prob, x=TRUE)$x

# query the indices of a subtour
subtourindices <- get_subtour(prob, currlpsol)

if (!is.null(subtourindices)) {
viins <- (dist.df$Vi %in% subtourindices) vjins <- (dist.df$Vj %in% subtourindices)
edgeins <- (viins != vjins) # exactly one endpoint in S
subtourlength <- length(subtourindices)

presrow <- presolverow(p, rowtype = "G",
origcolind = which(edgeins) - 1, #
origrowcoef = rep(1, subtourlength * (n - subtourlength)),
origrhs = 2)

if (presrow$status == 0) { addcuts(p, cuttype=1, rowtype = "G", rhs = presrow$rhs,
start = c(0, presrow$ncoefs), colind = presrow$colind,
cutcoef = presrow$rowcoef) cat(paste("Added cut for subtour", paste(subtourindices, collapse = ", "), "\n")) } } # we always return infeasible = 0, # we cannot decide infeasibility at this point. return(list(infeasible = 0)) } preintsolcallback <- function(prob, soltype, cutoff) { # use getlpsol to access the current solution # pass in FALSE for optional arrays that need not be returned for our purpose sol <- getlpsol(prob, slack = F, duals = F, djs = F)$x

subtourindices <- get_subtour(prob, sol)
reject <- !is.null(subtourindices)

return (list(reject=reject))
}

#'
#'
#' ## Callback registration
#'
#' We register the two callbacks above.
#' We load the initial TSP only with node degree constraints.
#' We display the optimal solution to verify that
#' the callbacks worked correctly.
#'
## -----------------------------------------------------------------------------
p <- p %>%
setintcontrol(xpress:::MIPDUALREDUCTIONS, 0) %>%
setoutput()

xprs_optimize(p)
summary(p)

#'
#' The resulting solution represents a tour!
#'
## -----------------------------------------------------------------------------
plot_tsp_solution(p)

#'
#'
#'
#'
#'
#'
#'
#'