1,376 research outputs found
An extension of disjunctive programming and its impact for compact tree formulations
In the 1970’s, Balas [2, 4] introduced the concept of disjunctive programming, which is optimization over unions of polyhedra. One main result of his theory is that, given linear descriptions for each of the polyhedra to be taken in the union, one can easily derive an extended formulation of the convex hull of the union of these polyhedra. In this paper, we give a generalization of this result by extending the polyhedral structure of the variables coupling the polyhedra taken in the union. Using this generalized concept, we derive polynomial size linear programming formulations (compact formulations) of a well- known spanning tree approximation of Steiner trees and flow equivalent trees for node- as well as edge- capacitated undirected networks. We also present a compact formulation for Gomory-Hu trees, and, as a consequence, of the minimum T-cut problem (but not for the associated T-cut polyhedron). Recently, Kaibel and Loos [10] introduced a more involved framework called polyhedral branching systems to derive extended formulations. The most of our model can be expressed in terms of their framework. The value of our model can be seen in the fact that it completes their framework with an interesting algorithmic aspect.disjunctive programming, compact formulation, flow-equivalent trees, Gomory-Hu trees
An extension of disjunctive programming and its impact for compact tree formulations
In the 1970's, Balas introduced the concept of disjunctive programming, which
is optimization over unions of polyhedra. One main result of his theory is
that, given linear descriptions for each of the polyhedra to be taken in the
union, one can easily derive an extended formulation of the convex hull of the
union of these polyhedra. In this paper, we give a generalization of this
result by extending the polyhedral structure of the variables coupling the
polyhedra taken in the union. Using this generalized concept, we derive
polynomial size linear programming formulations (compact formulations) for a
well-known spanning tree approximation of Steiner trees, for Gomory-Hu trees,
and, as a consequence, of the minimum -cut problem (but not for the
associated -cut polyhedron). Recently, Kaibel and Loos (2010) introduced a
more involved framework called {\em polyhedral branching systems} to derive
extended formulations. The most parts of our model can be expressed in terms of
their framework. The value of our model can be seen in the fact that it
completes their framework by an interesting algorithmic aspect.Comment: 17 page
Mixed-integer convex representability
Motivated by recent advances in solution methods for mixed-integer convex
optimization (MICP), we study the fundamental and open question of which sets
can be represented exactly as feasible regions of MICP problems. We establish
several results in this direction, including the first complete
characterization for the mixed-binary case and a simple necessary condition for
the general case. We use the latter to derive the first non-representability
results for various non-convex sets such as the set of rank-1 matrices and the
set of prime numbers. Finally, in correspondence with the seminal work on
mixed-integer linear representability by Jeroslow and Lowe, we study the
representability question under rationality assumptions. Under these
rationality assumptions, we establish that representable sets obey strong
regularity properties such as periodicity, and we provide a complete
characterization of representable subsets of the natural numbers and of
representable compact sets. Interestingly, in the case of subsets of natural
numbers, our results provide a clear separation between the mathematical
modeling power of mixed-integer linear and mixed-integer convex optimization.
In the case of compact sets, our results imply that using unbounded integer
variables is necessary only for modeling unbounded sets
Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting
Mixed-integer mathematical programs are among the most commonly used models
for a wide set of problems in Operations Research and related fields. However,
there is still very little known about what can be expressed by small
mixed-integer programs. In particular, prior to this work, it was open whether
some classical problems, like the minimum odd-cut problem, can be expressed by
a compact mixed-integer program with few (even constantly many) integer
variables. This is in stark contrast to linear formulations, where recent
breakthroughs in the field of extended formulations have shown that many
polytopes associated to classical combinatorial optimization problems do not
even admit approximate extended formulations of sub-exponential size.
We provide a general framework for lifting inapproximability results of
extended formulations to the setting of mixed-integer extended formulations,
and obtain almost tight lower bounds on the number of integer variables needed
to describe a variety of classical combinatorial optimization problems. Among
the implications we obtain, we show that any mixed-integer extended formulation
of sub-exponential size for the matching polytope, cut polytope, traveling
salesman polytope or dominant of the odd-cut polytope, needs many integer variables, where is the number of vertices of the
underlying graph. Conversely, the above-mentioned polyhedra admit
polynomial-size mixed-integer formulations with only or (for the traveling salesman polytope) many integer variables.
Our results build upon a new decomposition technique that, for any convex set
, allows for approximating any mixed-integer description of by the
intersection of with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201
Reformulation and decomposition of integer programs
In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm
Smaller Extended Formulations for the Spanning Tree Polytope of Bounded-genus Graphs
We give an -size extended formulation
for the spanning tree polytope of an -vertex graph embedded on a surface of
genus , improving on the known -size extended formulations
following from Wong and Martin.Comment: v3: fixed some typo
The triangulation of manifolds
A mostly expository account of old questions about the relationship between
polyhedra and topological manifolds. Topics are old topological results, new
gauge theory results (with speculations about next directions), and history of
the questions.Comment: 26 pages, 2 figures. version 2: spellings corrected, analytic
speculations in 4.8.2 sharpene
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