8,922 research outputs found
Generating facets for the cut polytope of a graph by triangular elimination
The cut polytope of a graph arises in many fields. Although much is known
about facets of the cut polytope of the complete graph, very little is known
for general graphs. The study of Bell inequalities in quantum information
science requires knowledge of the facets of the cut polytope of the complete
bipartite graph or, more generally, the complete k-partite graph. Lifting is a
central tool to prove certain inequalities are facet inducing for the cut
polytope. In this paper we introduce a lifting operation, named triangular
elimination, applicable to the cut polytope of a wide range of graphs.
Triangular elimination is a specific combination of zero-lifting and
Fourier-Motzkin elimination using the triangle inequality. We prove sufficient
conditions for the triangular elimination of facet inducing inequalities to be
facet inducing. The proof is based on a variation of the lifting lemma adapted
to general graphs. The result can be used to derive facet inducing inequalities
of the cut polytope of various graphs from those of the complete graph. We also
investigate the symmetry of facet inducing inequalities of the cut polytope of
the complete bipartite graph derived by triangular elimination.Comment: 19 pages, 1 figure; filled details of the proof of Theorem 4, made
many other small change
On the extension complexity of combinatorial polytopes
In this paper we extend recent results of Fiorini et al. on the extension
complexity of the cut polytope and related polyhedra. We first describe a
lifting argument to show exponential extension complexity for a number of
NP-complete problems including subset-sum and three dimensional matching. We
then obtain a relationship between the extension complexity of the cut polytope
of a graph and that of its graph minors. Using this we are able to show
exponential extension complexity for the cut polytope of a large number of
graphs, including those used in quantum information and suspensions of cubic
planar graphs.Comment: 15 pages, 3 figures, 2 table
Lifting and Separation Procedures for the Cut Polytope
The max-cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years. However, little research has been conducted for the cut polytope on arbitrary graphs. In this study we describe new separation and lifting procedures for the cut polytope on such graphs. These procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid, and sometimes facet defining, inequalities for the cut polytope on arbitrary graphs in a cutting plane framework. We report computational results on a set of well-established benchmark 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
Adaptive Linear Programming Decoding of Polar Codes
Polar codes are high density parity check codes and hence the sparse factor
graph, instead of the parity check matrix, has been used to practically
represent an LP polytope for LP decoding. Although LP decoding on this polytope
has the ML-certificate property, it performs poorly over a BAWGN channel. In
this paper, we propose modifications to adaptive cut generation based LP
decoding techniques and apply the modified-adaptive LP decoder to short
blocklength polar codes over a BAWGN channel. The proposed decoder provides
significant FER performance gain compared to the previously proposed LP decoder
and its performance approaches that of ML decoding at high SNRs. We also
present an algorithm to obtain a smaller factor graph from the original sparse
factor graph of a polar code. This reduced factor graph preserves the small
check node degrees needed to represent the LP polytope in practice. We show
that the fundamental polytope of the reduced factor graph can be obtained from
the projection of the polytope represented by the original sparse factor graph
and the frozen bit information. Thus, the LP decoding time complexity is
decreased without changing the FER performance by using the reduced factor
graph representation.Comment: 5 pages, 8 figures, to be presented at the IEEE Symposium on
Information Theory (ISIT) 201
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