51 research outputs found
A Complete Linear Programming Hierarchy for Linear Codes
A longstanding open problem in coding theory is to determine the best
(asymptotic) rate of binary codes with minimum constant
(relative) distance . An existential lower bound was given by Gilbert
and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece,
Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte's
linear programs. To date these results remain the best known lower and upper
bounds on with no improvement even for the important class of
linear codes. Asymptotically, these bounds differ by an exponential factor in
the blocklength.
In this work, we introduce a new hierarchy of linear programs (LPs) that
converges to the true size of an optimum linear binary
code (in fact, over any finite field) of a given blocklength and distance
.
This hierarchy has several notable features:
(i) It is a natural generalization of the Delsarte LPs used in the first MRRW
bound.
(ii) It is a hierarchy of linear programs rather than semi-definite programs
potentially making it more amenable to theoretical analysis.
(iii) It is complete in the sense that the optimum code size can be retrieved
from level .
(iv) It provides an answer in the form of a hierarchy (in larger dimensional
spaces) to the question of how to cut Delsarte's LP polytopes to approximate
the true size of linear codes.
We obtain our hierarchy by generalizing the Krawtchouk polynomials and
MacWilliams inequalities to a suitable "higher-order" version taking into
account interactions of words. Our method also generalizes to
translation schemes under mild assumptions.Comment: 58 page
On the Lovasz O-number of Almost Regular Graphs With Application to Erdos-Renyi Graphs
AMS classifications: 05C69; 90C35; 90C22;Erdos-Renyi graph;stability number;Lovasz O-number;Schrijver O-number;C*-algebra;semidefinite programming
Ratio bound (Lov\'asz number) versus inertia bound
Matthew Kwan and Yuval Wigderson showed that for an infinite family of
graphs, the Lov\'asz number gives an upper bound of for the size
of an independent set (where is the number of vertices), while the weighted
inertia bound cannot do better than . Here we point out that there
is an infinite family of graphs for which the Lov\'asz number is
, while the unweighted inertia bound is .Comment: 3 pages. I do not plan to publish this note/exampl
Semidefinite Programming in Combinatorial and Polynomial Optimization
In the recent years semidefinite programming has become a widely used tool for designing better efficient algorithms for approximating hard combinatorial optimization problems and, more generally, polynomial optimization problems, which deal with optimizing a polynomial objective function over a basic closed semialgebraic set. The underlying paradigm is that, while testing nonnegativity of a polynomial is a hard problem, one can test efficiently whether it can be written as a sum of squares of polynomials, using semidefinite programming.
In this note we sketch some of the main mathematical tools
that underlie this approach and illustrate its application to some graph problems dealing with maximum cuts, stable sets and graph coloring
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