A b s t r ac t . The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and implementations. Our experience in this area is summarized in ten "rules of thumb". I n t ro d u c t i o nIn integer and linear optimization the software workhorses are solvers for linear programs (based on simplex or interior point methods) as well as generic frameworks for branch-andbound or branch-and-cut schemes. Comprehensive implementations are available both as Open Source, like SCIP [2], as well as commercial software, like CPLEX [24] and Gurobi [39]. While today it is common to solve linear programs with millions of rows and columns and, moreover, mixed integer linear programs with sometimes hundreds of thousands of rows and columns, big challenges remain. For instance, the 2010 version of the MIPLIB [30] lists the mixed-integer problem liu with 2 178 rows, 1 156 columns, and a total of only 10 626 non-zero coefficients; this seems to be impossible to solve with current techniques. One way to make progress in the field is to invent new families of cutting planes, either of a general kind or specifically tailored to a class of examples. In the latter situation the strongest possible cuts are obviously those arising from the facets of the (mixed) integer hull. A main purpose of this note is to report on the state of the art of getting at such facets in a brute force kind of way. And we will do so by explaining how our software system polymake [56] can help.Here we focus on integer linear programming (ILP); mixed integer linear programming (MILP) will only be mentioned in passing. To avoid technical ramifications we assume that all our linear programs (LP) are bounded. The brute force method for obtaining all facets of the integer hull is plain and simple, and it has two steps. First, we compute all the feasible integer points. Since we assumed boundedness these are only finitely many. Second, we compute the facets of their convex hull. Of course, the catch is that neither problem is really easy. Deciding if an ILP has an (integer) feasible point is known to be NP-complete [35]. So, we may not even hope for any efficient algorithm for the first step. Most likely, the situation for the second is about equally bad. While it is open whether or not there is a convex hull algorithm which runs in polynomial time measured in the combined sizes of the input and the output, recent work of Khachiyan et al.[47] indicates a negative answer. They show that computing the vertices of an unbounded polyhedron is hard; the difference to the general convex hull problem is that their result does not say anything about the rays of the polyhedron.Our paper is organized as follows. We start out with a very brief introduction to the polymake system and its usage. In Section 3 we explore how various convex hull algorithms and their implementations behave on various kinds of input. Our inpu...
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.
Abstract. We consider convex programming problems with integrality constraints that are invariant under a linear symmetry group. To decompose such problems we introduce the new concept of core points, i.e., integral points whose orbit polytopes are lattice-free. For symmetric integer linear programs we describe two algorithms based on this decomposition. Using a characterization of core points for direct products of symmetric groups, we show that prototype implementations can compete with state-of-the-art commercial solvers, and solve an open MIPLIB problem.
Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a key role in a recent approach to exploit symmetry in integer convex optimization problems. Here, naturally the question arises, for which groups the number of core points is finite up to translations by vectors fixed by the group. In this paper we consider transitive permutation groups and prove this type of finiteness for the $2$-homogeneous ones. We provide tools for practical computations of core points and obtain a complete list of representatives for all $2$-homogeneous groups up to degree twelve. For transitive groups that are not $2$-homogeneous we conjecture that there exist infinitely many core points up to translations by the all-ones-vector. We prove our conjecture for two large classes of groups: For imprimitive groups and groups that have an irrational invariant subspace.Comment: 27 pages, 2 figures; with minor adaptions according to referee comments; to appear in Discrete and Computational Geometr
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