Algorithms for the Minimum Spanning Tree (MST) problem are readily available. However, sometimes we want to solve this problem inside a Mathematical Programming model. Usually, this is for two reasons:
- We have some side constraints
- Or as part of a larger model
These reasons are essentially the same (a matter of gradation). Embedding an MST inside a model is not totally trivial.
Data
I used in our models the data from [1]. This data set has distances for 42 US cities.
---- 298 SET cities
Manchester, N.H. , Montpelier, Vt. , Detroit, Mich. , Cleveland, Ohio , Charleston, W.Va.
Louisville, Ky. , Indianapolis, Ind. , Chicago, Ill. , Milwaukee, Wis. , Minneapolis, Minn.
Pierre, S.D. , Bismarck, N.D. , Helena, Mont. , Seattle, Wash. , Portland, Ore.
Boise, Idaho , Salt Lake City, Utah, Carson City, Nevada , Los Angeles, Calif. , Phoenix, Ariz.
Santa Fe, N.M. , Denver, Colo. , Cheyenne, Wyo. , Omaha, Neb. , Des Moines, Iowa
Kansas City, Mo. , Topeka, Kans. , Oklahoma City, Okla., Dallas, Tex. , Little Rock, Ark.
Memphis, Tenn. , Jackson, Miss. , New Orleans, La. , Birmingham, Ala. , Atlanta, Ga.
Jacksonville, Fla. , Columbia, S.C. , Raleigh, N.C. , Richmond, Va. , Washington, D.C.
Boston, Mass. , Portland, Me.
An optimal spanning tree can look like:
Note 2: in all models below, I assume we have directed arcs. This makes the modeling easier [5].
Flow model
An intuitive model is as follows:
- Choose a source node \(s\) (any node will do)
- All other nodes \(t\) need to be reached from the source node (they are terminal nodes)
- So we form paths from \(s \rightarrow t\)
- Keep track of the arcs that are used in these paths and minimize the sum of the lengths of these arcs.
A MIP formulation can look like:
| Flow formulation |
|---|
| \[\begin{align}&\color{darkblue}A_{i,j} = \text{true if arc $i \rightarrow j$ exists}\\ & \color{darkblue}b_{i,t} = \begin{cases} +1 & \text{if $i=s$}\\ -1 & \text{if $i=t$}\\ 0 & \text{otherwise} \end{cases}\\ &\color{darkblue}c_{i,j} = \text{cost of arc $i \rightarrow j$} \end{align}\] |
| \[\begin{align}&\color{darkred}x_{i,j} = \begin{cases}1 & \text{if arc $i \rightarrow j$ is part of the tree}\\ 0 & \text{otherwise} \end{cases}\\ & \color{darkred}y_{i,j,t} = \begin{cases} 1 & \text{if arc $i \rightarrow j$ is part of the path $s \rightarrow t$}\\ 0 & \text{otherwise} \end{cases} \end{align}\] |
| \[\begin{align}\min\>&\color{darkred}z = \sum_{i,j|\color{darkblue}A(i,j)} \color{darkblue}c_{i,j}\color{darkred}x_{i,j}\\& \sum_{j|\color{darkblue}A(i,j)}\color{darkred}y_{i,j,t}=\sum_{j|\color{darkblue}A(j,i)}\color{darkred}y_{j,i,t} + \color{darkblue}b_{i,t} && \forall i,t \\ & \color{darkred}x_{i,j} \ge \color{darkred}y_{i,j,t} && \forall i,j|\color{darkblue}A_{i,j},t \\ & \color{darkred}x_{i,j},\color{darkred}y_{i,j,t} \in [0,1]\end{align}\] |
Some notes:
- The GAMS code can be found in [1].
- The binary variables \(\color{darkred}x_{i,j}\) and \(\color{darkred}y_{i,j,t}\) can be relaxed to be continuous between zero and one. This makes the model a pure LP.
- In GAMS we can solve this as an RMIP. This will relax integer variables but keeps the bounds for binary variables.
- The relaxation only works for Simplex methods or for barrier methods followed by a crossover. Without the crossover, interior point methods will produce fractional solutions.
- The solution being a tree is a side effect of minimizing the total cost.
- The model is formulated to exploit sparsity in the underlying graph. Our data set is not sparse, so that does not help in our case.
- This LP is very large: starting with just \(n=42\) cities, we arrive at \(n^2(n-1)=72,324\) variables and constraints. This count is for a complete graph.
Instead of having an extra index for each of the paths, we can combine them and at the source node start with an exogenous inflow of \(n-1\) and at the terminal nodes an exogenous outflow of 1. This mode can look like:
| Flow formulation II |
|---|
| \[\begin{align}&\color{darkblue}A_{i,j} = \text{true if arc $i \rightarrow j$ exists}\\ & \color{darkblue}b_{i} = \begin{cases} \color{darkblue}n-1 & \text{if $i=s$}\\ -1 & \text{otherwise} \end{cases}\\ &\color{darkblue}c_{i,j} = \text{cost of arc $i \rightarrow j$} \\ & \color{darkblue}M =\color{darkblue}n-1 \end{align}\] |
| \[\begin{align}&\color{darkred}x_{i,j} = \begin{cases}1 & \text{if arc $i \rightarrow j$ is part of the tree}\\ 0 & \text{otherwise} \end{cases}\\ & \color{darkred}f_{i,j} = \text{flow from node $i \rightarrow j$}\end{align}\] |
| \[\begin{align}\min\>&\color{darkred}z = \sum_{i,j|\color{darkblue}A(i,j)} \color{darkblue}c_{i,j}\color{darkred}x_{i,j}\\& \sum_{j|\color{darkblue}A(i,j)}\color{darkred}f_{i,j}=\sum_{j|\color{darkblue}A(j,i)}\color{darkred}f_{j,i} + \color{darkblue}b_{i} && \forall i \\ & \color{darkblue}M\cdot \color{darkred}x_{i,j} \ge \color{darkred}f_{i,j} && \forall i,j|\color{darkblue}A_{i,j} \\ & \color{darkred}x_{i,j}\in \{0,1\}\\ & \color{darkred}f_{i,j} \in \{0,1,2,\dots,\color{darkblue}M\}\end{align}\] |
This formulation is much more compact: we have 1,764 constraints and 3,444 variables. However, this is no longer an LP: we need to solve it as a MIP. In my case, using Cplex, the smaller MIP solved faster than the large LP. In both cases less than 10 seconds, so performance is not likely an issue for a problem this size.
Power set formulations
There are a number of formulations that use a powerset: all subsets of the nodes. Here is a simple one [3,4]:
| Powerset formulation |
|---|
| \[\begin{align}\min\>&\color{darkred}z = \sum_{i,j|\color{darkblue}A(i,j)} \color{darkblue}c_{i,j}\color{darkred}x_{i,j}\\& \sum_{i,j|\color{darkblue}A(i,j)}\color{darkred}x_{i,j}= \color{darkblue}n-1 \\ & \sum_{i,j\in S|\color{darkblue}A(i,j)}\color{darkred}x_{i,j} \le |S|-1 && \forall S\subset V \text{ nodes}\\ & \color{darkred}x_{i,j}\in [0,1]\end{align}\] |
The second constraint is over all subsets \(S\) of the nodes (except the one with 0 nodes or with all nodes). The number of such subsets is \(2^n-2\). In our case that is \[2^{42}-2 = 4,398,046,511,102\approx 4.4 \times 10^{12}\] This is a bit more than we can handle. This means that we can drop from consideration any formulation that has "for all subsets of the nodes" in it.
MTZ formulations
We can borrow the MTZ (Miller-Tucker-Zemlin [6]) subtour-elimination constraints from TSP models.
| MTZ formulation |
|---|
| \[\begin{align}\min\>&\color{darkred}z = \sum_{i,j|\color{darkblue}A(i,j)} \color{darkblue}c_{i,j}\color{darkred}x_{i,j}\\& \sum_{s,j|\color{darkblue}A(s,j)}\color{darkred}x_{s,j} \ge 1 && \text{at least 1 out from source $s$}\\ & \sum_{j,i|\color{darkblue}A(j,t)}\color{darkred}x_{j,t} = 1 && \forall t\ne s, \text{one incoming arc}\\ &\color{darkred}u_i-\color{darkred}u_j + (\color{darkblue}n-1)\cdot \color{darkred}x_{i,j}\le \color{darkblue}n-2 &&\forall i \ne j, i\ne s, j\ne s\\ &\color{darkred}u_s = 0 && \text{source node $s$}\\ & \color{darkred}x_{i,j}\in \{0,1\}\\& \color{darkred}u_t \in [1,\color{darkblue}n] && \forall t \ne s \end{align}\] |
As for the TSP problem, we can apply the lifting technique [7] and use \[\color{darkred}u_i-\color{darkred}u_j + (\color{darkblue}n-1)\cdot \color{darkred}x_{i,j} + (\color{darkblue}n-3)\cdot \color{darkred}x_{j,i} \le \color{darkblue}n-2\]
These models tend to perform not as well as some of the other models.
References
- How to formulate the Minimum Spanning Tree problem as a MIP, https://yetanothermathprogrammingconsultant.blogspot.com/2009/05/how-to-formulate-minimum-spanning-tree.html
- David Kahle, Hadley Wickham, ggmap: Spatial Visualization with ggplot2, https://journal.r-project.org/archive/2013-1/kahle-wickham.pdf
- Magnanti, Thomas L.; Wolsey, Laurence A., Optimal Trees, in M.O. Ball, T.L. Magnanti, C.L. Monma, and G.L. Nemhauser, editors, Network Models, volume 7 of Handbooks in Operations Research and Management Science, Chapter 9, pages 503--616. North-Holland, 1995
- Justin C. Williams, “A linear-size zero - one programming model for the minimum spanning tree problem in planar graphs”, Networks, 39(1), 2002
- Directed vs undirected networks in math programming models, https://yetanothermathprogrammingconsultant.blogspot.com/2020/12/directed-vs-undirected-networks-in-math.html
- Miller, C., A. Tucker, R. Zemlin. 1960. Integer programming formulation of traveling salesman problems. J. ACM 7 326-329
- Desrochers, M., Laporte, G., Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints, Operations Research Letters, 10 (1991) 27-36.