
Submitted by Assigned_Reviewer_1
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
Overview: The manuscript introduces an approach to solving the minimum weight perfect matching problem via a sequence of LPs that can themselves be efficiently solved using belief propagation.
The established algorithm has many interesting connections and interpretations  one among them is that it "jumps" over many substeps of Edmonds' Blossom algorithm with a single run of belief propagation.
The construction of the algorithm is presented as follows:
1) At first, BlossomLP is introduced, which solves the problem through a sequence of linear programs. (Similar algorithms, though with different LP formulations, are already known from the literature [22])
2) Then, it is shown how these intermediate LPs can be formulated as a graphical model for which BP is guaranteed to find an integral solution. (This is based on recent theoretical results of [16]).
The algorithm that solves each such LP via BP is called BlossomBP.
3) Finally, an auxiliary algorithm is given that is more amenable to theoretical analysis.
It is first shown that this algorithm terminates correctly in O ( V^2 ) time, and then its equivalence to BlossomLP (and hence BlossomBP) is established.
Positive points: + The resulting algorithm seems interesting and practical, in particular since highly optimized implementations of BP are possible (in particular, it can easily be parallelized) + The theoretical analysis of the algorithm is very thorough; the established O ( V^2 ) complexity is a nontrivial result. + The manuscript contains plenty of nontrivial novel contributions, in addition to the innovative application of recent results of [16] and [22] + The manuscript is very wellwritten and reveals many interesting connections to related algorithms
Negative points:  The manuscript is theoryonly.
At this point, it is hard to judge if an efficient implementation of BlossomBP would be competitive with BlossomV.
I am looking forward to first numerical experiments.
Q2: Please summarize your review in 12 sentences
This is a very wellwritten, theoryonly manuscript that introduces a novel approach to solving minimum weight perfect matching using iterated belief propagation.
The result is important, in that it provides an algorithm that establishes an optimal solution in O( V^2 ) belief propagation runs, and in that it provides an interpretation of Edmonds' algorithm as a sequence of LPs.
Submitted by Assigned_Reviewer_2
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
Adding some figures to illustrate the main contributions/algorithms is encouraged. For instance, the auxiliary algorithm may be easily demonstrated by a few figures.
Q2: Please summarize your review in 12 sentences
The work is to propose an algorithm (BlossomBP) to solve the minimum weight matching problem over arbitrary graphs. The theoretical foundation of the work is solid and well presented.
Submitted by Assigned_Reviewer_3
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The authors describe an algorithm to solve the general MWM problem by incorporating blossom constraints (through a series of linear subproblems than can be solved by a BP subroutine).
I found that the paper was a bit tough to follow due to the page constraints (much of the content was in the supplemental material), and it could use some proofreading to fix typos and improve the overall flow.
Despite this, the authors do present novel contributions.
It remains unclear whether similar strategies can be adapted to other settings.
General comments:
1. Is the overall complexity of the method mentioned in the paper?
How does it compare to Edmonds' Blossom algorithm?
2. The idea of sequentially adding constraints has been studied before (cycle constraints in particular). For example, "D. Sontag, D. K. Choe, Y. Li. Efficiently Searching for Frustrated Cycles in MAP Inference. Uncertainty in Artificial Intelligence (UAI) 28, Aug. 2012." While these methods use heuristics to pick cycles constraints to add, they have a similar flavor and may be worth citing.
3. For the half integrality of CLP, I feel like I have seen this (or a very similar result before), but I could be wrong. The authors should double check the classical results about MWMs.
4. Many parts of the writeup could be improved.
For example, you discuss modifying the edges weights of the problem on page 4, but you don't explain why this is the case (until much later and you don't even refer back).
More generally, it's a bit difficult to carefully follow why the described approach actually solves the problem.
This is probably due to the space constraints but makes it tough to read nonetheless.
Q2: Please summarize your review in 12 sentences
The authors describe an algorithm to solve the general MWM problem by incorporating blossom constraints (through a series of linear subproblems than can be solved by a BP subroutine).
I found that the paper was a bit tough to follow due to the page constraints (much of the content was in the supplemental material), and it could use some proofreading to fix typos and improve the overall flow.
Despite this, the authors do present novel contributions.
Submitted by Assigned_Reviewer_4
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
This paper introduces two polytime algorithms for minimum
weight perfect matching using a sequence of linear programs or
runs of maxproduct BP (the factor graphs for which are
inspired by connections to the LPs). To obtain the polytime
algorithm using BP, the paper makes use of a result from
recent prior work (Park and Shin, 2015): under specific
conditions, for certain pairs of a factor graph and an LP,
running maxproduct BP on the factor graph converges to the
solution of the LP. The paper proves that the iterative
algorithm uses at most O(n^2) iterations where n is the number
of vertices in the graph.
The authors speculate that their
work may spur further study of BP for MAP on more general
graphical models. Though the paper does not explicitly state
any ideas in this direction, it seems like a furtile area. As
well, the new LP algorithm provides an interesting contrast
with Edmonds' Blossom algorithm, which (in its original form)
required exponentially many constraints. Overall, it's clearly
written, original work that, while not the first polytime
algorithm for minweight perfect matching (they reserve that
title for (Chandrasekaren et al., 2012)), is likely the
simplest.
It seems the paper would benefit from a journalstyle
presentation: one which is more leisurely and example
filled. Given space constraints the paper did a good job of
delegating proofs of certain results to supplementary
material. However, the main paper would benefit greatly if the
examples from Appendix F could be included in a figure, and
repeatedly referenced through the description of the
algorithm.
On the one hand, this paper can be viewed as a nice
application of Park and Shin (2015)'s BP/LP result. On the
other, it stands as a unique example (to the best of my
knowledge as well as claimed by the authors) of solving an ILP
by repeated calls to BP.
This paper does not lack substance, but it would have been
nice to see the algorithm put to use on some real matching
problems. As the linear programming community knows well,
polynomial time algorithms are sometimes less practical than
their exponential time counterparts in the real world.
Misc:
 line 155: Linear Programming > Linear Program
 line 190: this note deserves a one line explanation
 line 196: initially confusing where T came from
 line 258: if the proof of Theorem 3 will be left in the
appendix, it would help to give some intuition for why the
halfintegral solution is obtained.
 line 032 (appendix A): marginal beliefs > maxmarginals
 A more detailed contrast with Chandrasekaren et al. (2012)
seems appropriate
Q2: Please summarize your review in 12 sentences
This work builds on recent work's connections
between maxproduct BP and linear programming to introduce two
polytime algorithms for minimum weight perfect matching using a
sequence of linear programs or runs of maxproduct BP (the
factor graphs for which are inspired by connections to the
LPs). Overall, it's clearly written, original work that, while
not the first polytime algorithm for minweight perfect
matching, is likely the simplest.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 5000 characters. Note
however, that reviewers and area chairs are busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We very much appreciate valuable comments, efforts
and time spent by the reviewers to evaluate the paper. In what follows, we
provide, first, a summary, and then detailed response to each
reviewer.
Summary:
1) We have presented the simplest
polynomialtime algorithm for solving MWM problem known so far.  Our
algorithm is simpler than Edmond's algorithms, e.g., BlossomV, and
significantly simpler than the LPbased cuttingplane algorithm developed
in [22].
2) Our algorithm is also the first rigorous algorithm
solving an Integer Programming (IP) using a sequence of BPs.  All
prior works on BP are heuristicbased or focused on solving related LP
relaxation with no integrality gap (thus only one BP, and not a sequence,
is required.)
3) We have also suggested a transparent
interpretation of the Edmond's MWM algorithm in terms of a sequence of
LPs.  Such an interpretation was open for a few decades.
We
appreciate very much Reviewer_1/2/3/4 mentioning explicitly
"novelty/originality" of our approach and Reviewer_1/6 acknowledging
"importance" of our results. Notice that the Edmond's algorithm for MWM
was the first complex (going beyond totally unimodular case)
polynomialtime algorithm in the computer science literature. The
algorithm has motivated the P vs. NP considerations and it has also
influenced other combinatorial optimization algorithms (e.g., primaldual
methods). We believe that our novel approach has further potentials to
solve a much broader class of IPs using BP, which is of interest to
machine learning, in general, and specifically graphical model
communities. We also anticipate that our results will be of importance for
largescale machine learning, especially in the context of distributed and
parallel implementations.
Response to Reviewer_1:
We are
working on computational implementation of our algorithms (both BP and LP)
and validating/comparing the algorithms with the BlossomV, which is the
most efficient, modern implementation of the Edmond' algorithm (by
Kolmogorov). We have already validated our algorithms over millions of
random instances, and found that typically two or three iterations
suffices to terminate (while in theory O(V^2) iterations may be required
in the worstcase). The detailed running time comparison of our algorithms
with Blossom V is still incomplete  it is work in progress dependent on
many technical details. For example, we have discovered that one can boost
up convergence of BP with a targeted initializations and/or damping. This
suggests, in particular, using the last message of BP from preceding step
to initialize the current state. We have also started to work on
developing parallel implementation of the Blossom BP, e.g. testing options
provided by GraphLab, GraphChi and OpenMP software. Given the factors
mentioned above and also taking into account NIPS page limit constraint,
we have decided to publish detailed experimental analysis of our algorithm
and performance comparison with Blossom V in a few months when a
comprehensive analysis is completed.
Response to
Reviewer_2:
Our main theorem implies that the overall complexities
of BlossomBP and BlossomLP are O(V^2) * T_LP and O(V^2) * T_BP
respectively, where T_LP and T_BP are running times of a single LP and BP
algorithms. Obviously, T_LP is polynomial. In fact, one can also prove
that T_BP is polynomial as well. In our current draft, we use the result
of [16], which does not analyze the number of iterations for BP
convergence but instead provides a generic criteria to guarantee
convergence of BP. To guarantee that T_BP is polynomial, one can use
techniques from [12] or, complementarily, adopt the proof strategy of [16]
to analyze convergence time of the algorithm. These proofs are relatively
straightforward, however, we have decided not to follow the path in the
manuscript as we expect that the worstcase complexity of BlossomBP
cannot beat the best complexity bound known in the literature. Here, we
note that BlossomV also does not beat the best bound, even though it is
known as empirically fastest, known algorithm. The main appeal of
BlossomBP and BlossomLP is in their simplicity, practicality and
parallelization potential.
 We do plan to add a number of
references summarizing and commenting on related heuristic algorithms and
contrasting these with our exact algorithm.  To the best of our
knowledge, there is no work studying LP (5), i.e., a hybrid matching. The
halfintegrality proof is not too hard, but we have decided to include it
for completeness.
Response to Reviewer_3/4/5/6:
 Wrt
numerical experiments  please see our response to Reviewer_1.  We
will correct typos pointed out by the reviewers. We will add a figure
illustrating performance of our novel algorithms on
examples.
Response to Meta_Reviewer_1:
 Wrt the
complexities of our algorithms  please see our response to
Reviewer_2.  T_LP and T_BP depend on the number of
edges. 
