365 research outputs found
Extended Formulations for Packing and Partitioning Orbitopes
We give compact extended formulations for the packing and partitioning
orbitopes (with respect to the full symmetric group) described and analyzed in
(Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all
0/1-matrices with lexicographically sorted columns and at most, resp. exactly,
one 1-entry per row. They are important objects for symmetry reduction in
certain integer programs.
Using the extended formulations, we also derive a rather simple proof of the
fact that basically shifted-column inequalities suffice in order to describe
those orbitopes linearly.Comment: 16 page
Sequential and Parallel Algorithms for Mixed Packing and Covering
Mixed packing and covering problems are problems that can be formulated as
linear programs using only non-negative coefficients. Examples include
multicommodity network flow, the Held-Karp lower bound on TSP, fractional
relaxations of set cover, bin-packing, knapsack, scheduling problems,
minimum-weight triangulation, etc. This paper gives approximation algorithms
for the general class of problems. The sequential algorithm is a simple greedy
algorithm that can be implemented to find an epsilon-approximate solution in
O(epsilon^-2 log m) linear-time iterations. The parallel algorithm does
comparable work but finishes in polylogarithmic time.
The results generalize previous work on pure packing and covering (the
special case when the constraints are all "less-than" or all "greater-than") by
Michael Luby and Noam Nisan (1993) and Naveen Garg and Jochen Konemann (1998)
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
Recommended from our members
Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
The matching relaxation for a class of generalized set partitioning problems
This paper introduces a discrete relaxation for the class of combinatorial
optimization problems which can be described by a set partitioning formulation
under packing constraints. We present two combinatorial relaxations based on
computing maximum weighted matchings in suitable graphs. Besides providing dual
bounds, the relaxations are also used on a variable reduction technique and a
matheuristic. We show how that general method can be tailored to sample
applications, and also perform a successful computational evaluation with
benchmark instances of a problem in maritime logistics.Comment: 33 pages. A preliminary (4-page) version of this paper was presented
at CTW 2016 (Cologne-Twente Workshop on Graphs and Combinatorial
Optimization), with proceedings on Electronic Notes in Discrete Mathematic
- …