Algorithms for highly symmetric linear and integer programs

Abstract

Abstract. This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.

Using symmetry to optimize over the Sherali–Adams relaxation

“…The motivation for Theorem 1 in [13] was for its application to semidefinite programming. This theorem is applied to linear programming in [27,4] .…”

Exploiting symmetry in integer convex optimization using core points

“…Note that our definition of core points generalizes the notion used in [2]. They consider the case of Γ = S n or Γ = A n with core points being defined as the integral points closest to the one-dimensional fixed space.…”

“…For every element v of a Γ-symmetric set S, the full orbit Γv is also contained in S. If S is convex, then for all v ∈ S also the orbit barycenter β(Γv) is contained in S. This implies that the intersection S ∩ Fix Γ (R n ) for a convex set S is equal to the projection Φ Γ (S). Thus we have the following proposition, which is an essential ingredient of symmetry exploiting techniques for convex optimitization problems; see for instance [2], [6], [1] or [10]. Proposition 1.…”

“…In [2], the authors consider transitive group actions, which lead to a one-dimensional fixed space, the span of the all ones vector. They decompose the problem by intersecting the feasible region with all affine hyperplanes orthogonal to the fixed space containing integer points, and check the integer feasibility of these intersections.…”

“…A core point is an integral point for which the convex hull of its orbit does not contain integral points other than the orbit points themselves (see Definition 2). In the case of a one dimensional pointwise fixed subspace, and with the full symmetric group S n acting transitively on the coordinates of an integer linear program in R n , these core points have been used in [2], to obtain a fast Core Point Algorithm. In this paper we generalize their concept of core points, for the design of algorithms that work for more general symmetry groups.…”