# DIMACS maximum flow problems

DIMACS (Center for Discrete Mathematics and Theoretical Computer Science (see http://dimacs.rutgers.edu/)) has formulated some 'challenges' for some specific problem instances (see http://dimacs.rutgers.edu/Challenges/). One of these challenges is network flows and matching - the first DIMACS implementation challenge.

One of these network flows are maximum flow problems:

The maximum flow problem is structured on a network. Here the arc capacities, or upper bounds, are the only relevant parameters. The problem is to find the maximum flow possible from some given source node to a given sink node. All arc costs are zero. Since the goal of the optimization is to minimize cost, the maximum flow possible is delivered to the sink node. Applications of this problem include finding the maximum flow of orders through a job shop, the maximum flow of water through a storm sewer system, and the maximum flow of product through a product distribution system, among others.

### Network Structure

• A network is a directed graph with n nodes and m arcs.
• Nodes are identified by integers 1...n.
• Graphs do not have to be symmetric: if an arc (v,w) is in the graph, the reverse arc (w,v) does not have to be in the graph.
• Parallel arcs are not allowed.
• Self-loops are not allowed.
• Arc capacities are 32-bit signed integers.
• The source and the sink are distinct.
• The sink may be unreachable from the source.

The following information makes up a DIMACS maximum flow input file:

• comment lines
• problem line
• node descriptors
• arc descriptors

As noted above, information is collected into lines, which begin with one-character designators. We describe each type of information line in turn.

1. Comment Lines: Comment lines give human-readable information about the file and are ignored by programs. Comment lines can appear anywhere in the file. Each comment line begins with a lower-case character c.
`    c This is an example of a comment line.`
2. Problem Line: There is one problem line per input file. The problem line must appear before any node or arc descriptor lines. For maximum flow network instances the problem line has the following format:
`    p max NODES ARCS`

The lower-case character p signifies that this is a problem line. The three-character problem designator max identifies the file as containing specification information for a maximum flow problem. The NODES field contains an integer value specifying n, the number of nodes in the network. The ARCS field contains an integer value specifying m, the number of arcs in the network.

3. Node Descriptors: All node descriptor lines must appear before all arc descriptor lines. Node descriptors are of the form:
`    n ID WHICH`

where ID is the node id and WHICH is s for the source and t for the sink. Two node descriptors, one for the source and one for the sink, must appear between the problem line and the arc descriptor lines.

4. Arc Descriptors: There is one arc descriptor line for each arc in the network. For a maximum flow instance, arc descriptor lines are of the following form.
`    a SRC DST CAP`

The lower-case character a signifies that this is an arc descriptor line. For a directed arc (v,w) the SRC field gives the identification number for the source vertex v, and the DST field gives the destination vertex w. Identification numbers are integers between 1 and n. The CAP field gives the arc capacity.

#### Input File Example :

The example network pictured here is followed by a corresponding DIMACS maximum flow input file.

```c This is a simple example file to demonstrate the DIMACS
c input file format for maximum flow problems. The solution
c vector is [5,10,5,0,5,5,10,5] with cost at 15.
p max 6 8
c source
n 1 s
c sink
n 6 t
c Arc descriptor lines (from, to, capacity)
a 1 2 5
a 1 3 15
a 2 4 5
a 2 5 5
a 3 4 5
a 3 5 5
a 4 6 15
a 5 6 5
c
c End of file
```

lp_solve can read/write and solve these models via the xli_DIMACS XLI driver (see External Language Interfaces). It reads such a model in above format and solves it via linear programming. The xli can also generate a DIMACS formatted file.

For example:

```lp_solve -rxli xli_DIMACS maxflow.net
```

This gives as result:

```Value of objective function: 45

Actual values of the variables:
C1                              5
C2                             10
C3                              5
C4                              0
C5                              5
C6                              5
C7                             10
C8                              5
```

Also from within the IDE, this XLI can be used. However, some entries must be added in LpSolveIDE.ini (in the folder where the IDE is installed).

In the [XLI] section the following must be added:

```lib6=xli_DIMACS
```

And a new section for the DIMACS XLI must also be added:

```[xli_DIMACS]
extension=.net
language=DIMACS
```

Then make sure that the xli_DIMACS.dll is available for the IDE. This must be done by placing this dll in the IDE folder or in the Windows system32 folder.

#### Solution :

The solution vector of above example is [5, 10, 5, 0, 5, 5, 10, 5]
This must be interpreted as follows:
There are as many variables as there are arc descriptor lines in the input file and they appear in the same order. So:

C1 specifies how much flow there is for the first arc definition, in this case from A -> B
C2 specifies how much flow there is for the second arc definition, in this case from A -> C
...

This means there is a flow of 5 from node A to B, a flow of 10 from node A to C, ...
The value of the objective of lp_solve doesn't give you much information. It is the sum of all variables used.

#### Output :

The solution of the model can also be written in a DIMACS format:

• comment lines
• solution lines
• flow assignments

For each network problem, the solution is an integer-valued flow assignment. The output file should list the solution and flow assignment for all arcs in the graph. As noted above, information is collected into lines, which begin with one-character designators. We describe each type of information line in turn.

1. Comment Lines: Comment lines give human-readable information about the file and are ignored by programs. Comment lines can appear anywhere in the file. Each comment line begins with a lower-case character c.
`    c This is an example of a comment line.`
2. Solution line. The solution line contains the flow value and has the following format:
`    s VALUE`

The lower-case character s signifies that this is a solution line. The VALUE field contains the value of the objective.

3. Flow assignments. There is one flow assignment line for each arc in the input network. These lines have the following format:
`    f SRC DST FLOW`
The lower-case character f signifies that this is a flow assignment line. For arc (u,v), the SRC and DST fields give v and w, respectively. The FLOW field gives the amount of flow assigned to arc (u,v).

lp_solve can generate a solution file via the xli_DIMACS XLI driver (see External Language Interfaces).

For example:

```lp_solve -rxli xli_DIMACS maxflow.net -wxlisol xli_DIMACS maxflow.sol
```

This generates the following solution contents:

```c maxflow.net
c
c Dimacs-format maximum flow result file
c generated by lp_solve
c
c Solution
s 15
c
c SRC DST FLOW
f 1 2 5
f 1 3 10
f 2 4 5
f 2 5 0
f 3 4 5
f 3 5 5
f 4 6 10
f 5 6 5
c
c End of file
```

lp_solve can also generate a solution file with only non-zero values.

For example:

```lp_solve -rxli xli_DIMACS maxflow.net -wxlisol xli_DIMACS maxflow.sol -wxlisolopt "-nz"
```

This generates the following solution contents:

```c maxflow.net
c
c Dimacs-format maximum flow result file
c generated by lp_solve
c
c Solution
s 15
c
c Only non-zero flows are written
c SRC DST FLOW
f 1 2 5
f 1 3 10
f 2 4 5
f 3 4 5
f 3 5 5
f 4 6 10
f 5 6 5
c
c End of file
```

A testset of models can be found at the dimacs ftp site: ftp://dimacs.rutgers.edu/pub/netflow