A Compendium of Applications of the

Graph Matching Problem

Figure 1. Landscape of matching problems. The vertex-weighted matching problem can be formulated as an edge-weighted matching problem. The weighted matching algorithms utilize techniques developed for the cardinality matching problem. The arrows indicate these relationships.

1.

2.       Multilevel graph algorithms (partitioning and clustering)

3.       Bioinformatics: Protein structure comparisons

4.

5.

6.       Scheduling in input-queued high-performance switches

7.

8.

QUICK SUMMARY OF APPLICATIONS

Legend:  B = Bipartite graphs; G = General graphs;

E = Exact algorithm; A = Approximation algorithm.

1. Applications of Maximum (Edge) Weighted Matching:

Objective: Maximize the sum of the weights of the matched edges.

 # Description Examples: Type 1 Numerical Linear Algebra: (pivoting in Gaussian elimination; scaling) Olschowka and Neumaier; Duff and Koster; B,E 2 Scheduling in high-performance network switches and routers McKweon B, A/E 3 Rigidity percolation Ref: www.pa.msu.edu/~duxbury/CSC2007.pdf Duxbury, Hendrickson 1992 G, E? 4 Image Feature Matching (Stereo Matching) Yong-Quing Cheng, et al., 1996; Fielding, et al., 1997 B, E, A 5 Protein Structure Comparisons Taylor, WR, et al., 2002; Wang Y., et al., 2004 6 Multi-level algorithms (partitioning and clustering) Karypis and Kumar; Hendrickson; Preis. G, A 7 Paired Kidney Donation Program Segev and Gentry G, E

2. Maximum (Cardinality) Matching:

Objective: Maximize the number of edges in the matching.

 # Description Examples: Type 1 Numerical Linear Algebra: (block triangular form; Zero-free diagonals) Duff and Reid; Pothen and Fan; B,E

3. Maximum (Vertex) Weighted Matching:

Objective: Maximize the sum of the weights of the matched vertices.

 # Description Reference: Type 1 Scheduling in high-performance network switches and routers Tabatabaee, et al.; B, E 2 Sparsest basis problem Pinar, Chow and Pothen; B, E/A

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Communication:

The current list is a preliminary exploration of the applications and we are interested in learning about existing (that are not mentioned here) as well as novel applications of the matching problem. Please contact us with your suggestions:

Mahantesh Halappanavar (mhalappa –at– odu –dot– edu)

Alex Pothen (apothen –at– purdue  dot– edu)

Florin Dobrian (dobrain –at– cs –dot– odu –dot– edu)

Last updated ... April 24, 2009