A Simpler Max-Product Maximum Weight Matching Algorithm and the Auction Algorithm Mohsen Bayati Department of EE Stanford University Stanford, CA 94305 Email: bayati@stanford.edu Devavrat Shah Departments of EECS & ESD MIT Cambridge, MA 02139 Email: devavrat@mit.edu Mayank Sharma IBM TJ Watson Research Center 1101 Kitchawan Rd, P. O. Box 218 Yorktown Heights, NY 10598 Email: mxsharma@us.ibm.com Abstract— The max-product “belief propagation” algorithm has received a lot of attention recently due to its spectacular success in many application areas such as iterative decoding, computer vision and combinatorial optimization. There is a lot of ongoing work investigating the theoretical properties of the algorithm. In our previous work (2005) we showed that the max- product algorithm can be used to solve the problem of finding the Maximum Weight Matching (MWM) in a weighted complete bipartite graph. However, for a graph with n nodes the max- product algorithm requires O(n 4 ) operations to find the MWM compared to O(n 3 ) for best known algorithms such as those proposed by Edmonds and Karp (1972) and Bertsekas (1988). In this paper, we simplify the max-product algorithm to reduce the number of operations required to O(n 3 ). The simplified algorithm has very similar dynamics to the well-known auction algorithm of Bertsekas (1988). To make this connection precise, we show that the max-product and auction algorithms, when slightly modified, are equivalent. We study the correctness of this modified algorithm. There is a tantalizing similarity between this connection and a recently observed connection between the max-product and LP-based algorithms for iterative decoding by Vontobel and Koetter. I. I NTRODUCTION Finding the MWM in a bipartite graph is an important problem in many fields e.g. combinatorial optimization and networks (see [4], [5] for references and details). In this section, we first define the problem of finding the MWM in a bipartite graph, then describe the min-sum version of the max-product algorithm and finally present previously known results. A. MAXIMUM WEIGHT MATCHING Consider an undirected weighted complete bipartite graph K n,n = (V 1 ,V 2 ,E), where V 1 = {α 1 ,...,α n }, V 2 = {β 1 ,...,β n } and (α i ,β j ) ∈ E for 1 ≤ i,j ≤ n. Let each edge (α i ,β j ) have weight w ij ∈ R. If π = {π(1),...,π(n)} is a permutation of {1,...,n} then the set of n edges {(α 1 ,β π(1) ),..., (α n ,β π(n) )} is called a matching of K n,n . We denote both the permutation and the corresponding match- ing by π. The weight of the matching π, denoted by W π , is defined as W π =  1≤i≤n w iπ(i) . Then, the maximum weight matching π ∗ is a matching such that π ∗ = argmax π W π . Next, we transform the problem of finding the MWM in a bipartite graph into one of finding a MAP assignment in a related graphical model (GM). Consider the following GM defined on K n,n : let X 1 ,...,X n ,Y 1 ,...,Y n be random vari- ables corresponding to the vertices of K n,n and taking values from {1, 2,...,n}. Let their joint probability distribution, p ( X =(x 1 ,...,x n ); Y =(y 1 ,...,y n ) ) , be of the form: p ( X, Y ) = 1 Z  i,j ψ αiβj (x i ,y j )  i φ αi (x i )φ βi (y i ), (1) where the pairwise compatibility functions, ψ ·· (·, ·), are de- fined as ψ αi βj (r,s)=    0 r = j and s = i 0 r = j and s = i 1 Otherwise the potentials at the nodes, φ · (·), are defined as φ αi (r)= e wir ,φ βj (r)= e wrj , ∀ 1 ≤ i,j,r,s ≤ n, and Z is the normalization constant. The following claims are a direct consequence of these definitions (see [4] for their proofs). Claim 1: For the GM as defined above, the joint den- sity p ( X =(x 1 ,...,x n ), Y =(y 1 ,...,y n ) ) is nonzero if and only if π α ( X )= {(α 1 ,β x1 ), (α 2 ,β x2 ),..., (α n ,β xn )} and π β ( Y )= {(α y1 ,β 1 ), (α y2 ,β 2 ),..., (α yn ,β n )} are both matchings and π α ( X ) = π β ( Y ). Further, when nonzero, p ( X, Y ) is equal to 1 Z e 2 i wix i . Claim 2: Let ( X ∗ , Y ∗ ) be such that ( X ∗ , Y ∗ ) = arg max{p ( X, Y ) }. Then, the corresponding π α ( X ∗ )= π β ( Y ∗ ) is the MWM. B. MIN-SUM ALGORITHM FOR K n,n We present the min-sum version of the max-product al- gorithm for finding the MWM (see [4] for the equivalence between the standard max-product and min-sum algorithms).