130,215 research outputs found
The matching polytope does not admit fully-polynomial size relaxation schemes
The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that
every linear program expressing the matching polytope has an exponential number
of inequalities (formally, the matching polytope has exponential extension
complexity). We generalize this result by deriving strong bounds on the
polyhedral inapproximability of the matching polytope: for fixed , every polyhedral -approximation
requires an exponential number of inequalities, where is the number of
vertices. This is sharp given the well-known -approximation of size
provided by the odd-sets of size up to
. Thus matching is the first problem in , whose natural
linear encoding does not admit a fully polynomial-size relaxation scheme (the
polyhedral equivalent of an FPTAS), which provides a sharp separation from the
polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets
mentioned above.
Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the
main lower bounding technique is different. While the original proof is based
on the hyperplane separation bound (also called the rectangle corruption
bound), we employ the information-theoretic notion of common information as
introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/],
which allows to analyze perturbations of slack matrices. It turns out that the
high extension complexity for the matching polytope stem from the same source
of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure
Extended Formulation Lower Bounds via Hypergraph Coloring?
Exploring the power of linear programming for combinatorial optimization
problems has been recently receiving renewed attention after a series of
breakthrough impossibility results. From an algorithmic perspective, the
related questions concern whether there are compact formulations even for
problems that are known to admit polynomial-time algorithms.
We propose a framework for proving lower bounds on the size of extended
formulations. We do so by introducing a specific type of extended relaxations
that we call product relaxations and is motivated by the study of the
Sherali-Adams (SA) hierarchy. Then we show that for every approximate
relaxation of a polytope P, there is a product relaxation that has the same
size and is at least as strong. We provide a methodology for proving lower
bounds on the size of approximate product relaxations by lower bounding the
chromatic number of an underlying hypergraph, whose vertices correspond to
gap-inducing vectors.
We extend the definition of product relaxations and our methodology to mixed
integer sets. However in this case we are able to show that mixed product
relaxations are at least as powerful as a special family of extended
formulations. As an application of our method we show an exponential lower
bound on the size of approximate mixed product formulations for the metric
capacitated facility location problem, a problem which seems to be intractable
for linear programming as far as constant-gap compact formulations are
concerned
Average case polyhedral complexity of the maximum stable set problem
We study the minimum number of constraints needed to formulate random
instances of the maximum stable set problem via linear programs (LPs), in two
distinct models. In the uniform model, the constraints of the LP are not
allowed to depend on the input graph, which should be encoded solely in the
objective function. There we prove a lower bound with
probability at least for every LP that is exact for a randomly
selected set of instances; each graph on at most n vertices being selected
independently with probability . In the
non-uniform model, the constraints of the LP may depend on the input graph, but
we allow weights on the vertices. The input graph is sampled according to the
G(n, p) model. There we obtain upper and lower bounds holding with high
probability for various ranges of p. We obtain a super-polynomial lower bound
all the way from to . Our upper bound is close to this as there is only an essentially quadratic
gap in the exponent, which currently also exists in the worst-case model.
Finally, we state a conjecture that would close this gap, both in the
average-case and worst-case models
Approximation Limits of Linear Programs (Beyond Hierarchies)
We develop a framework for approximation limits of polynomial-size linear
programs from lower bounds on the nonnegative ranks of suitably defined
matrices. This framework yields unconditional impossibility results that are
applicable to any linear program as opposed to only programs generated by
hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations
for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound
applies to linear programs using a certain encoding of CLIQUE as a linear
optimization problem.) Moreover, we establish a similar result for
approximations of semidefinite programs by linear programs. Our main ingredient
is a quantitative improvement of Razborov's rectangle corruption lemma for the
high error regime, which gives strong lower bounds on the nonnegative rank of
certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure
On the existence of 0/1 polytopes with high semidefinite extension complexity
In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope
whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope
projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension
complexity is exponential. The question whether there exists a 0/1 polytope
with high PSD extension complexity was left open. We answer this question in
the affirmative by showing that there is a 0/1 polytope such that any
spectrahedron projecting to it must be the intersection of a semidefinite cone
of dimension~2^{\Omega(n)} and an affine space. Our proof relies on a new
technique to rescale semidefinite factorizations
Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits
Initially developed for the min-knapsack problem, the knapsack cover
inequalities are used in the current best relaxations for numerous
combinatorial optimization problems of covering type. In spite of their
widespread use, these inequalities yield linear programming (LP) relaxations of
exponential size, over which it is not known how to optimize exactly in
polynomial time. In this paper we address this issue and obtain LP relaxations
of quasi-polynomial size that are at least as strong as that given by the
knapsack cover inequalities.
For the min-knapsack cover problem, our main result can be stated formally as
follows: for any , there is a -size LP relaxation with an integrality gap of at most ,
where is the number of items. Prior to this work, there was no known
relaxation of subexponential size with a constant upper bound on the
integrality gap.
Our construction is inspired by a connection between extended formulations
and monotone circuit complexity via Karchmer-Wigderson games. In particular,
our LP is based on -depth monotone circuits with fan-in~ for
evaluating weighted threshold functions with inputs, as constructed by
Beimel and Weinreb. We believe that a further understanding of this connection
may lead to more positive results complementing the numerous lower bounds
recently proved for extended formulations.Comment: 21 page
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