305 research outputs found
Orbitopal Fixing
The topic of this paper are integer programming models in which a subset of
0/1-variables encode a partitioning of a set of objects into disjoint subsets.
Such models can be surprisingly hard to solve by branch-and-cut algorithms if
the order of the subsets of the partition is irrelevant, since this kind of
symmetry unnecessarily blows up the search tree. We present a general tool,
called orbitopal fixing, for enhancing the capabilities of branch-and-cut
algorithms in solving such symmetric integer programming models. We devise a
linear time algorithm that, applied at each node of the search tree, removes
redundant parts of the tree produced by the above mentioned symmetry. The
method relies on certain polyhedra, called orbitopes, which have been
introduced bei Kaibel and Pfetsch (Math. Programm. A, 114 (2008), 1-36). It
does, however, not explicitly add inequalities to the model. Instead, it uses
certain fixing rules for variables. We demonstrate the computational power of
orbitopal fixing at the example of a graph partitioning problem.Comment: 22 pages, revised and extended version of a previous version that has
appeared under the same title in Proc. IPCO 200
Exploiting Symmetry in Integer Convex Optimization using Core Points
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.Comment: 15 pages; small changes according to suggestions of a referee; to
appear in Operations Research Letter
The Impact of Symmetry Handling for the Stable Set Problem via Schreier-Sims Cuts
Symmetry handling inequalities (SHIs) are an appealing and popular tool for
handling symmetries in integer programming. Despite their practical
application, little is known about their interaction with optimization
problems. This article focuses on Schreier-Sims (SST) cuts, a recently
introduced family of SHIs, and investigate their impact on the computational
and polyhedral complexity of optimization problems. Given that SST cuts are not
unique, a crucial question is to understand how different constructions of SST
cuts influence the solving process.
First, we observe that SST cuts do not increase the computational complexity
of solving a linear optimization problem over any polytope . However,
separating the integer hull of enriched by SST cuts can be NP-hard, even if
is integral and has a compact formulation. We study this phenomenon more
in-depth for the stable set problem, particularly for subclasses of perfect
graphs. For bipartite graphs, we give a complete characterization of the
integer hull after adding SST cuts based on odd-cycle inequalities. For
trivially perfect graphs, we observe that the separation problem is still
NP-hard after adding a generic set of SST cuts. Our main contribution is to
identify a specific class of SST cuts, called stringent SST cuts, that keeps
the separation problem polynomial and a complete set of inequalities, namely
SST clique cuts, that yield a complete linear description.
We complement these results by giving SST cuts based presolving techniques
and provide a computational study to compare the different approaches. In
particular, our newly identified stringent SST cuts dominate other approaches
Exploring core points for fun and profit: A study of lattice-free orbit polytopes
This thesis studies minimal lattice-free symmetric polytopes. Lattice-free means that the only integral points in the polytope are its vertices. Symmetric in context of the thesis means that all vertices lie in one single orbit under a group action. The thesis focuses on groups that are permutation groups acting on R^n by permuting coordinates. If a symmetric polytope is lattice-free, its vertices are called core points. Methods to construct core points and applications in symmetric integer linear programming are explored.Diese Arbeit behandelt gitterpunkt-freie symmetrische Polytope. Gitterpunkt-frei heißt, dass die Ecken des Polytops die einzigen enthaltenen ganzzahligen Punkte sind. Symmetrisch im Kontext dieser Arbeit meint, dass alle Ecken in einem einzigen Orbit einer Gruppenwirkung liegen. Diese Arbeit beschäftigt sich besonders mit Gruppen, die als Permutationsgruppen auf R^n wirken, indem sie Koordinaten permutieren. Die Ecken eines gitterpunkt-freien symmetrischen Polytops werden core points genannt. Es werden Methoden entwickelt, core points zu finden und in ganzzahliger Optimierung anzuwenden
A Unified Framework for Symmetry Handling
Handling symmetries in optimization problems is essential for devising
efficient solution methods. In this article, we present a general framework
that captures many of the already existing symmetry handling methods (SHMs).
While these SHMs are mostly discussed independently from each other, our
framework allows to apply different SHMs simultaneously and thus outperforming
their individual effect. Moreover, most existing SHMs only apply to binary
variables. Our framework allows to easily generalize these methods to general
variable types. Numerical experiments confirm that our novel framework is
superior to the state-of-the-art SHMs implemented in the solver SCIP
Almost Symmetries and the Unit Commitment Problem
This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation
- …