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\title{  Referee reports on \\ ``Theory of semidefinite
programming for sensor network localization" \\ by \\ 
A. M-C So and Y. Ye}

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\noindent
Dear Yinyu,\\
     Following are three referee reports. They are all positive and
suggest minor changes in the paper. Therefore, we are happy to accept
this paper.\\
I still expect one more
report. However, I thought that it would help speed up the process if
you had access to these three reports now.

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\section{Referee Report I}
\label{sect:rep1}
\begin{center}
{ \large  Referee's report on \\ ``Theory of semidefinite
programming for sensor network localization" \\ by \\ 
A. M-C So and Y. Ye } 
\end{center}

\flushleft

This paper presents a semidefinite programming model for
sensor network localizations. This problem can be thought
of as a special case of the graph realization problem where
the coordinates of a subset of the nodes are known. The
paper also discusses some related problems in distance
geometry.  

The sensor network localization problem is an important problem
which has received a great deal of attention recently. The 
semidefinite approach to this problem is quite elegant and 
promising. The results obtained in this paper are new and
interesting. The paper is well written and
easy to read. Furthermore, the references are comprehensive. 

Thus we recommend that this paper be accepted for publication. 
We would like to add one comment. 
The SDP problems in (3) and (4) seem complicated due to the notation
used. We believe that these two problems might be simplified by
using more appropriate notation.

Finally there is a typo on page 4. Replace well-known by
well known in the line before (2).   
  
\newpage
\section{Referee Report II}
\label{sect:rep2}



This paper is concerned with the "sensor network localization
problem".  Given a set of objects in $\Re^d$, some at known locations,
and knowledge about some of the pairwise distances between
objects, can we determine the location of all the objects?
This is a variant of a problem that has been well studied
by several communities.  This particular variant is distinguished
by the presence of some objects with known locations.  Note
that the data in a problem instance can be thought of as a graph.

The principle contribution of this paper is the identification
of a class of graphs for which a semi-definite programming
algorithm is guaranteed to find the unique solution.  The
graph class is interesting - it is graphs that have a
solution in 2D, but no different solutions in higher
dimensions.  Although not remarked in the paper, I don't
think this property is purely graph theoretic, but also
depends upon the set of distances (or alternatively, on
the locations of vertices in the solution).  Perhaps there
is some convexity property in the placement of vertices and
anchors.  It would be nice to have a graph and geometric
understanding of strong localizability.

The authors draw connections to the rigidity literature,
and to the results on unique realizability in particular.
Although these prior results are closely related to the
current paper, to my knowledge the current results are
new.  And I find them interesting and intriguing.  I can't
help but wonder if there are connections to Connelly's work
on tensegrity frameworks ...

Although the paper is generally well written, I have a few
suggestions for improving the presentation.  Most are not
substantive.

\begin{enumerate}
\item
Page 2, Line 13: "there has been" should be "there have been".

\item
P 2, L -16 (also P 7, L -6): "close to be" should be "close to
   being".

\item
P 2, L -16 (also P 7, L -6): It's not clear to me why you can
   argue that your result is nearly the best possible.  There
   could be a larger class of graphs that are solvable.  The
   fact that the general problem is hard doesn't ensure that
   a particular solvable class is nearly the best possible.


\item
P 8, L 16: "are couple ways" should be "are several ways".

\item
Section 5: You mention early in this section that you assume
   the embeddings are generic.  But this is an important
   assumption and should probably be included in some of the
   theorem statements.

\item
P 10, L -5: "an useful" should be "a useful".

\item
Definition 5: Is it possible to reinterpret this condition
   in a manner that doesn't depend on SDP duality?

\item
P 12, L 4 and L 5:  What is "w."?

\item
P 13, L 8: I think you mean to reflect the subgraph induced by
   $a_1, x_2$ and $a_3$ (not $a_2$).
\end{enumerate}


  
\newpage
\section{Referee Report III}
\label{sect:rep3}

In this paper the authors analyze the Sensor Network Localization problem, 
and study a SDP method for its solution. 
By using SDP duality they show that this method localizes in polynomial 
time any network that has unique sensor position 
to fit given distance measures. 
Moreover, they study the connections with rigidity theory, 
and give a simple condition for checking whether a graph G 
with anchors has a unique realization on the plane. 
Finally, they introduce the notion of strong realizability 
and show that the method identifies all strongly localizable sub-networks. 
The paper is interesting and well written and the results are sound.


   
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