# Symmetry in Integer Linear Programming

@inproceedings{Margot2010SymmetryII, title={Symmetry in Integer Linear Programming}, author={François Margot}, booktitle={50 Years of Integer Programming}, year={2010} }

An integer linear program (ILP) is symmetric if its variables can be permuted without changing the structure of the problem. Areas where symmetric ILPs arise range from applied settings (scheduling on identical machines), to combinatorics (code construction), and to statistics (statistical designs construction). Relatively small symmetric ILPs are extremely difficult to solve using branch-and-cut codes oblivious to the symmetry in the problem. This paper reviews techniques developed to take…

## 244 Citations

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## References

SHOWING 1-10 OF 155 REFERENCES

### Symmetry Breaking in Constraint Programming

- Computer ScienceECAI
- 2000

This work describes a method for symmetry breaking during search (SBDS) in constraint programming that guarantees to return a unique solution from each set of symmetrically equivalent ones, which is the one found first by the variable and value ordering heuristics.

### Exploiting orbits in symmetric ILP

- Computer ScienceMath. Program.
- 2003

An isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group are described and applications to hard set covering problems, generation of covering designs and error correcting codes are given.

### Fundamental Domains for Integer Programs with Symmetries

- MathematicsCOCOA
- 2007

The construction of the polytope of a fundamental domain defined by the maximization of a linear function is provided, closely related to the constructions of orbitopes by Kaibel and Pfetch, but are simpler and more general, at a cost of creating new non-integral extreme points.

### Symmetry Breaking Using Stabilizers

- Computer ScienceCP
- 2003

STAB is a method that adds some symmetry breaking constraints during the search for solution to matrix problems where rows and columns can be permuted and is more than one order of magnitude more efficient than best published techniques so far.

### Expressing combinatorial optimization problems by linear programs

- MathematicsSTOC '88
- 1988

It is shown that expressing the matching and the Traveling Salesman Problem by a symmetric linear program requires exponential size, and the minimum size needed by a LP to express a polytope to a combinatorial parameter is related.

### Detecting symmetries by branch & cut

- Computer Science, MathematicsMath. Program.
- 2003

This paper presents an exact algorithm for automorphism and symmetry detection based on the branch & cut technique and introduces IP-models for these problems and investigates the structure of the corresponding polytopes.

### Packing and partitioning orbitopes

- Mathematics
- 2006

We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which…

### Constraint Orbital Branching

- Computer ScienceIPCO
- 2008

The orbital branching methodology is extended so that the branching disjunction can be based on an arbitrary constraint, demonstrated by computing the optimal incidence width of Steiner Triple Systems and minimum cardinality covering designs.