Paper:
Min-Max Propagation
We study the application of min-max propagation, a variation of belief propagation, for approximate min-max inference in factor graphs. We show that for "any" highorder function that can be minimized in O(ω), the min-max message update can be obtained using an efficient O(K(ω + log(K)) procedure, where K is the number of variables. We demonstrate how this generic procedure, in combination with efficient updates for a family of high-order constraints, enables the application of min-max propagation to efficiently approximate the NP-hard problem of makespan minimization, which seeks to distribute a set of tasks on machines, such that the worst case load is minimized.

Review:
As the title suggests, this paper is interested in
      solving min-max problems using message-passing algorithms. The
      starting point is Eq. 1: You want to optimize a vector x such
      that the max over a discrete set of functions is minimize. The
      set of functions should be finite.
      It's well known that changing the semiring can change the message-passing algorithm from a joint optimization
      (max product) algorithm to a marginalization algorithm (sum
      product). So, just like the (+,*) semiring yields the sum-product
      algorithm and the (max,*) semiring yields the max-product
      algorithm, this paper uses the (min,max) semi-ring. Then,
      Eqs. 2-3 are exactly the standard BP messages on factors and
      single variables, just with the new semi-ring. (It would be good
      if the paper would emphasize more strongly that Eqs. 2 and 3 are
      exactly analogous-- I had to work out the details myself to
      verify this.)
      Since that's a semi-ring you get all the normal properties
      (exactness on trees, running times). However, you don't seem to
      inherit much of the folk theorems around using BP on loopy
      graphs when you change the semi-ring (where most applications
      probably lie) so this paper investigates
      that experimentally.
      The paper assumes that solving the minimization problem on any particular
      factor can be solved in a fixed amount of time. This is
      reasonable to make analysis tractable, but it's worth pointing
      out that in many applications, doing this might itself involve
      running another entire inference algorithm (e.g. max-product
      BP!) There is also an algorithmic specialization to speed up
      message-computation for the case
      where there is a special "choose-one" form for the factors,
      similar to some factor forms that are efficient with different
      semigroups.
      Experimentally, it is compared to a previous algorithm that runs
      a bisection search with a CSP solver in each iteration. This
      appears to perform slightly better, although apparently at
      higher cost.
      As far as I know, this particular semi-ring variant of belief
      propagation has not been investigated before. I don't see strong
      evidence of a huge practical contribution in the expeirments
      (though that might reflect my lack of expertise on these
      problems). Nevertheless, the algorithm is extremely simple in
      concept, and it certainly seems worth noting that there is
      another semi-ring for BP that solves another possibly
      interesting set of problems. Even if the results here aren't
      that strong in the current form, this might be a first step
      towards using some of the many improvements that have been made
      to other semi-ring versions of BP.
      (I was surprised that this particular semi-ring hasn't been
      proposed before, but I'm not aware of such a use, and a brief
      search did not find anything. If this is not actually novel, I
      would downgrade my rating of this paper.)
      - Fig. 5 is very hard to read-- illegible in print
      - In eq. 13 I believe you want a -> m?
EDIT AFTER REBUTTAL:
I've read the rebuttal and other reviews, see no reason to change my recommendation. (I remain positive.)