773 research outputs found
Uncapacitated Flow-based Extended Formulations
An extended formulation of a polytope is a linear description of this
polytope using extra variables besides the variables in which the polytope is
defined. The interest of extended formulations is due to the fact that many
interesting polytopes have extended formulations with a lot fewer inequalities
than any linear description in the original space. This motivates the
development of methods for, on the one hand, constructing extended formulations
and, on the other hand, proving lower bounds on the sizes of extended
formulations.
Network flows are a central paradigm in discrete optimization, and are widely
used to design extended formulations. We prove exponential lower bounds on the
sizes of uncapacitated flow-based extended formulations of several polytopes,
such as the (bipartite and non-bipartite) perfect matching polytope and TSP
polytope. We also give new examples of flow-based extended formulations, e.g.,
for 0/1-polytopes defined from regular languages. Finally, we state a few open
problems
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
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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
Symmetry Matters for Sizes of Extended Formulations
In 1991, Yannakakis (J. Comput. System Sci., 1991) proved that no symmetric
extended formulation for the matching polytope of the complete graph K_n with n
nodes has a number of variables and constraints that is bounded
subexponentially in n. Here, symmetric means that the formulation remains
invariant under all permutations of the nodes of K_n. It was also conjectured
in the paper mentioned above that "asymmetry does not help much," but no
corresponding result for general extended formulations has been found so far.
In this paper we show that for the polytopes associated with the matchings in
K_n with log(n) (rounded down) edges there are non-symmetric extended
formulations of polynomial size, while nevertheless no symmetric extended
formulations of polynomial size exist. We furthermore prove similar statements
for the polytopes associated with cycles of length log(n) (rounded down). Thus,
with respect to the question for smallest possible extended formulations, in
general symmetry requirements may matter a lot. Compared to the extended
abtract that has appeared in the Proceedings of IPCO XIV at Lausanne, this
paper does not only contain proofs that had been ommitted there, but it also
presents slightly generalized and sharpened lower bounds.Comment: 24 pages; incorporated referees' comments; to appear in: SIAM Journal
on Discrete Mathematic
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