Generalized Matching Code
Stuart Andrews and Tony Jebara, 2007

A command-line tool for solving generalized matching problems (a.k.a. b-matchings, or degree-constrained subgraphs) is available here. The archive contains binaries for Linux and MacOSX. There is also a precompiled MEX interface for Matlab. To get started follow these instructions. There's a README with more details if you have questions, or if things do not work as expected.

 Command-line tool ``` tar xvfz bmatch.tgz cd bmatch make ./bmatch -h ``` Matlab tool ``` tar xvfz bmatch.tgz cd bmatch make ./matlab_bmatch help bmatch ```

## Help file

```
Usage> bmatch [-arg1 value] [-arg2 value] ...

Solves:
max_Yij  sum_ij W_ij Y_ij    s.t.   sum_j Y_ij <= b_i,  1 <= i <= n
and Wij and Yij are symmetric

Arguments [with default values]:

-w -weights [NULL] input file, NULL => std. input
-d -degrees [NULL] input file, NULL => std. input
-o -output  [NULL] output file, NULL => std. output
-c -const_b [-1  ] positive integer, negative => std. input
-s -sparse  [0   ] 0 => matrix, 1 => IJW input format
-m -method  [1   ] selects algorithm
-v -verbose [0   ] positive integer

Algorithm:

1. exact solution using goblin mincost solver via subgraph complement
2. exact solution using goblin mincost solver via negated weights
3. greedy 1/2 approximation
5. greedy 1/2 approximation with recursion
7. bipartite relaxation using belief propagation

Notes:

Self-loops are handled: a self-loop increases the degree of a node
by 2.  Weights must be positive (Wij >= 0).  Edges with zero-weight
(Wij==0) are not allowed to partake in the matching.

If the weights, degrees or output file is not specified, then the
corresponding input [output] data is entered in [written to] the
display.  The format should match the examples given in the data
directory.  Enter a blank line to terminate input of a given data.

The output weight is calculated as sum_{i<=j} Wij Yij, except when
beleif propagation is used, whose output is not necessarily symmetric.
For BP, the output weight is 0.5 * sum_{i,j} Wij Yij.

The reduction in methods 2 yields the optimal matching if the upper
bounds are tight at the solution.  Otherwise, this method may not
satisfy the upper bounds.

For Known Issues and Troubleshooting tips, see the full README.

```
Tutorial
• example 0: enter 2x2 matrix of weights and specify bi=1 forall i
```
-> bmatch
INPUT WEIGHTS(dense)>
1 2
2 3

INPUT DEGREES>
1

MATCHING>
0    1
1    0

```
• example 1: read 4x4 matrix of weights, specify bi=2 forall i, and specify -v 1 for brief diagnostic output - there are only 6 input edges because the diagonal weights are zero and therefore are omitted from the graph
```
-> bmatch -w data/matrix_in_1.txt -v 1 -b 2
MATCHING>
0    1    1    0
1    0    0    1
1    0    0    1
0    1    1    0
bmatch done:
method = exact solution using goblin mincost solver via subgraph complement
in # nodes = 4
in # edges = 6
out # edges = 4
wgt = 28
time (sec.) = 0

```
• example 2: now the weight matrix has non-zeros on the diagonal - notice that degree(0) = degree(1) = degree(2) = degree(3) = 3 which is the upper bound ... experiment with different -b values
```
-> bmatch -w data/matrix_in_2.txt -v 1 -b 3
bmatch_ijw> returning 4 edges
MATCHING>
0    1    1    1
1    0    1    1
1    1    0    1
1    1    1    0
bmatch done:
method = exact solution using goblin mincost solver via subgraph complement
in # nodes = 4
in # edges = 10
out # edges = 6
wgt = 35
time (sec.) = 0

```
• example 3: using the same matrix, try different -m solvers, e.g. the greedy algorithm (method 3) - the greedy method is guaranteed to obtain >= 1/2 of the optimal weight, which from the previous example is known to be 34 - belief propagation (method=7) treats variables Yij as edges in a directed bipartite graph between nodes 0..(N-1) and 0'..(N-1)' - this can be viewed as a relaxation of the problem stated above - in particular Yij=1 does not imply Yji=1, and there are no self-loops ... both idiosyncrasies are evident its solution
```
-> bmatch -w data/matrix_in_2.txt -v 1 -b 3 -m 3
bmatch_ijw> returning 5 edges
MATCHING>
0    0    1    1
0    1    0    1
1    0    1    0
1    1    0    0
bmatch done:
method = greedy 1/2 approximation
in # nodes = 4
in # edges = 10
out # edges = 5
wgt = 29
time (sec.) = 0

-> bmatch -w data/matrix_in_2.txt -b 2 -v 1 -m 7
beliefprop.solve> Reached maximum iterations without converging
MATCHING>
0    1    1    0
1    0    0    1
1    0    0    1
0    1    0    1
bmatch done:
method = bipartite relaxation using belief propagation
in # nodes = 4
in # edges = 16
out # edges = 8
wgt = 28.5
time (sec.) = 0

```
• example 4: here, graph input and output is in sparse format -s 1 ... have a look at data/ijw_in_3.txt which contains a sparse 34 node graph - since the list of edges is fairly long, we specify an output file with -o ... the greedy algorithm, and the recursive variant (method 5) both do poorly on this problem
```
-> bmatch -w data/ijw_in_3.txt -b 17 -s 1 -v 1 -m 1
bmatch_ijw> returning 306 edges
MATCHING>
0    1    1
0    3    1
0    6    1
0    7    1
.    .    .
.    .    .
.    .    .
29   31    1
29   33    1
30   30    1
30   31    1
bmatch done:
method = exact solution using goblin mincost solver via subgraph complement
in # nodes = 34
in # edges = 595
out # edges = 287
wgt = 22.44
time (sec.) = 3

-> bmatch -w data/ijw_in_3.txt -o data/ijw_out_3.txt -b 17 -s 1 -v 1 -m 3
bmatch done:
method = greedy 1/2 approximation
in # nodes = 34
in # edges = 595
out # edges = 284
wgt = 11.57
time (sec.) = 1

-> bmatch -w data/ijw_in_3.txt -o data/ijw_out_3.txt -b 17 -s 1 -v 1 -m 5
bmatch done:
method = greedy 1/2 approximation with recursion
in # nodes = 34
in # edges = 595
out # edges = 287
wgt = 11.59
time (sec.) = 0

```
• example 5: on the other hand, the greedy algorithm and the recursive variant are much more effective on large graphs - since this is where efficiency counts the most, we prefer these algorithms for large problems when the exact solution is not essential
```
-> bmatch -w data/ijw_in_5.txt -d data/degree_in_5.txt -o data/ijw_out_5.txt -s 1 -v 1 -m 1
bmatch_ijw> returning 2123 edges
bmatch done:
method = exact solution using goblin mincost solver via subgraph complement
in # nodes = 200
in # edges = 4463
out # edges = 2340
wgt = 1.384e+04
time (sec.) = 182

-> bmatch -w data/ijw_in_5.txt -d data/degree_in_5.txt -o data/ijw_out_5.txt -s 1 -v 1 -m 3
bmatch_ijw> returning 1504 edges
bmatch done:
method = greedy 1/2 approximation
in # nodes = 200
in # edges = 4463
out # edges = 1504
wgt = 8892
time (sec.) = 0

-> bmatch -w data/ijw_in_5.txt -d data/degree_in_5.txt -o data/ijw_out_5.txt -s 1 -v 1 -m 5
bmatch_ijw> returning 1504 edges
bmatch_ijw> returning 375 edges
bmatch_ijw> returning 132 edges
bmatch_ijw> returning 50 edges
bmatch_ijw> returning 27 edges
bmatch_ijw> returning 5 edges
bmatch_ijw> returning 0 edges
bmatch done:
method = greedy 1/2 approximation with recursion
in # nodes = 200
in # edges = 4463
out # edges = 2093
wgt = 1.23e+04
time (sec.) = 1

```
• example 6: this example demonstrates that the exact solution is NOT invariant to uniform weight increments or decrements - consider the two problems that are solved below - the second is identical to the first except that the weights are each incremented by 1 (the zero entries are preserved to exclude these edges from the solution) ... please excuse my slow typing speed
```
-> bmatch -v 1

INPUT WEIGHTS(dense)>
0 2.5 1
2.5 0 1
1   1 0

INPUT DEGREES>
1 1 2

bmatch> solving problem ...
bmatch_ijw> returning 2 edges
MATCHING>
0    1    0
1    0    0
0    0    0
bmatch done:
method = exact solution using goblin mincost solver via subgraph complement
in # nodes = 3
in # edges = 3
out # edges = 1
wgt = 2.5
time (sec.) = 13

-> bmatch -v 1

INPUT WEIGHTS(dense)>
0 3.5 2
3.5 0 2
2   2 0

INPUT DEGREES>
1 1 2

bmatch> solving problem ...
bmatch_ijw> returning 1 edges
MATCHING>
0    0    1
0    0    1
1    1    0
bmatch done:
method = exact solution using goblin mincost solver via subgraph complement
in # nodes = 3
in # edges = 3
out # edges = 2
wgt = 4
time (sec.) = 14

```
Acknowledgements

Many thanks to Vlad Shchogolev and Bert Huang for their code and examples. Vlad implemented prototypes for method 2 and 3. This tool integrates Bert's BP code directly. For a stand-alone loopy BP implementation, use Bert's code.

This tool links to the impressive Goblin Graph Library and LGPL licensing issues apply. To experiment with this library, please download the source code directly from the Goblin site. I would be happy to share the source that has been modified slightly for Mac OS X.