Matching

Matching - Assignment and Matching 1. A matching is a...

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Subhash Suri UC Santa Barbara Assignment and Matching 1. A matching is a pairing of nodes— collection of disjoint edges . Worker Job 1 2 4 A B C D 3 2. Bipartite graph. Two node classes, workers and jobs. 3. An edge ( i,j ) means worker i can do job j . 4. If weighted, then c ( i,j ) is the proﬁciency of i at job j . (In unweighted case, c ( i,j ) = 1 .) 5. Workers to jobs assignment for maximizing total proﬁciency . 6. Each worker assigned to at most one job and vice versa, so this is a matching.

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Subhash Suri UC Santa Barbara Applications 1. [Rooming Problem.] Dorm room assignment. Graph G with students as nodes. Weight c ij is compatibility of pair ( i,j ) . 2. [Airline Pilot Assignment.] Airlines need to form teams of captain and ﬁrst oﬃcer. α i is eﬀectiveness of i as captain. β i is eﬀectiveness of i as 1st oﬃcer. Seniority Rule: captain more senior. Make edge weight c ij = α i + β j if i more senior α j + β j otherwise 3. In these applications, the graph is not bipartite. We will only study the bipartite case.
Subhash Suri UC Santa Barbara More Applications 4. [Stable Marriage.] Men { A,B,. ..,Z } , women { a,b,. ..,z } . Their preference tables. A B C Men’s Preferences b c a b a c c a b C B C B a b c C B A A A Women’s Prefereces A matching M . M is unstable if pair ( Bob,Sally ) who like each other more than their spouses. Is stable marriage always possible? Medical schools use this protocol. Gale-Shapely Theorem: A stable marriage always possible, and found in O ( n 2 ) time.

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Subhash Suri UC Santa Barbara Stereo Vision 1. Stereo matching to locate objects in space. 2. Infrared sensors at two diﬀerent locations. 3. Each sensor gives the angle of sight (line) on which the object lies. . . . . . . . . . . . . O1 O2 l1 ln h1 hn 4. If p objects, we get two sets of lines: { L 1 ,L 2 ,...,L p } and { L 0 1 ,L 0 2 ,...,L 0 p } .
Subhash Suri UC Santa Barbara Stereo Vision . . . . . . . . . . . . O1 O2 l1 ln h1 hn 1. Two problems: (1) a line from one sensor might intersect multiple lines from the other; (2) due to noise, the lines for the same object may not intersect. 2. Solve the problem using assingment. Nodes are lines. Cost c ij is the distance between L i and L 0 j . 3. Distance between lines of the same object should be close to zero. 4. Optimal assingment should give excellent matching of line.

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UC Santa Barbara Deﬁnitions 1. A matching M E , in graph G = ( V,E ) , is a set of edges no two sharing a vertex. A matching M non-matching edges matching edges 2. | M | is the cardinality of M . 3. In unweighted graphs, ﬁnd max cardinality matching. 4. In weighted graphs, ﬁnd max weight matching. 5. A matching is perfect if all vertices are
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This note was uploaded on 12/27/2011 for the course CMPSC 225 taught by Professor Vandam during the Fall '09 term at UCSB.

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Matching - Assignment and Matching 1. A matching is a...

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