Orbital branching is a method for branching on variables in integer programming that reduces the likelihood of evaluating redundant, isomorphic nodes in the branch-and-bound procedure. In this work, the orbital branching methodology is extended so that the branching disjunction can be based on an arbitrary constraint. Many important families of integer programs are structured such that small instances from the family are embedded in larger instances. This structural information can be exploited to define a group of strong constraints on which to base the orbital branching disjunction. The symmetric nature of the problems is further exploited by enumerating non-isomorphic solutions to instances of the small family and using these solutions to create a collection of typically easy-to-solve integer programs. The solution of each integer program in the collection is equivalent to solving the original large instance. The effectiveness of this methodology is demonstrated by computing the optimal incidence width of Steiner Triple Systems and minimum cardinality covering designs.

Abstract. Symmetry is mainly exploited in mathematical programming in order to reduce the computation times of enumerative algorithms. The most widespread approach rests on: (a) finding symmetries in the problem instance; (b) reformulating the problem so that it does not allow some of the symmetric optima; (c) solving the modified problem. Sometimes (b) and (c) are performed concurrently: the solution algorithm generates a sequence of subproblems, some of which are recognized to be symmetrically equivalent and either discarded or treated differently. We review symmetry-based analyses and methods for Linear Programming, Integer Linear Programming, Mixed-Integer Linear Programming and Semidefinite Programming. We then discuss a method (introduced in [35]) for automatically detecting symmetries of general (nonconvex) Nonlinear and Mixed-Integer Nonlinear Programming problems and a reformulation based on adjoining symmetry breaking constraints to the original formulation. We finally present a new theoretical and computational study of the formulation symmetries of the Kissing Number Problem.

Abstract. Solution symmetries in integer linear programs often yield long Branch-and-Bound based solution processes. We propose a method for finding elements of the permutation group of solution symmetries, and two different types of symmetry-breaking constraints to eliminate these symmetries at the modelling level. We discuss some preliminary computational results.

The Football Pool Problem, which gets its name from a lottery-type game where partici-pants predict the outcome of soccer matches, is to determine the smallest covering code of radius one of ternary words of length v. For v = 6, the optimal solution is not known. Us-ing a combination of isomorphism-pruning, subcode enumeration, and linear-programming based-bounding, running on a high-throughput computational grid consisting of thousands of processors, we are able to report improved bounds on the size of the optimal code for this open problem in coding theory.

Let H be the 3-hypergraph having edges {123,124,134} and points {1,2,3,4}. A 3-hypergraph is H-free if it does not contain three edges isomorphic to H. The integer ex(n, H) denotes the maximum number of edges in any H-free hypergraph on n points. The upper bound for ex(n, H) is explored.It is shown that de Caen’s upper bound, n 2 (n − 1)/18, can not be met for n> 6. Then the exact values for ex(n, H) for n = 9, 10, 11 and 12 are determined. Finally, an improvement to ex(13,H) is given which allows us to improve the upper bounds for ex(n, H) for n = 14,..., 24. Using these numbers, Mubayi’s asymptotic upper bound is improved to 1/3 − 1.89820 × 10 −5. Finally we state Talbot’s upper bound of .32975