1. Homework - hints Problem 1. Node weights  Edge weights
single source shortest paths problem (SSSP)
1/2
1/2 s
Return:shortest path
and its length 1
Hint
sink 2 3
Problem 2: Shortest even (or odd) path
Hint: create a copy of each node  bipartite graph, add necessary
edges so that, for edge (u,v) in the new graph you always come
to v in even paths and to copy of v in odd paths.
Problem 3: for extra points

2. Matching in Bipartite Graphs “ assignment problem”
Problem: Given: bipartite graph, find the maximum # of edges that do not overlap  Job assignment problem
Jobs
Job nobody
can do
Employees
cannot do anything
- at the same time assign only 1 job to 1 employee

3. Matching in Bipartite Graphs “ assignment problem”
Find a job assignment such that each job is assigned to one
employee and vice versa.
-there is a solution in polynomial time
-we need to find “extending alternating paths”: start with a node that’s not covered  edge not covered  end in not covered node

4. Matching in Bipartite Graphs
remove
Blue = matching
red =
so-called free node
When we go down  always red,
when we go up  always green
Claim: if there is no alternating path,
we cannot add more matching
(no improvement)

5. Matching in Bipartite Graphs
node v  M’, v  M M v M
Claim:no extending alternating path  this is the maximum
possible matching
Proof: assume, on the contrary, that larger matching exists: M’
 there is a node v  M’, v  M
M+M’ has some connected components
remove from M+M’ ,
in the rest still |M| < |M’|

6. Minimum Weight Matching
Problem: Given: complete bipartite graph with weights on edges. Find minimum weight matching.
( G = (A  B, E,w) ) |A| = |B| w  R, not R+ +1 +1 +1 +1
any matching will increase by 1
(if we add +1 to weights of all edges outgoing
from a fixed node)

7. Computation Geometry Application of Matching
crystal
defects
non-convex points
- all pieces rectangular, cut in vertical (or horizontal) cuts
Problem: minimize # of cats
- only 90, 180, 270 cuts (180  -we don’t care about),
270  - problem
Fact: if we make a cut from each non-convex point, then the pieces
are rectangles.  if we solve matching of non convex, then we find
the optimum

8. Minimum Weight Matching
If we change weights with the above procedure, and come to
a graph with perfect matching (all nodes matched), of zero weight,
then M is minimum weight matching!
- we do not change matching by adding and subtracting
- for each node v, v  A subtract min(v) = min u B w(v,u)
next time  algorithm

9. Matching in Bipartite Graphs
Repeat from v’  M’, v’  M. Either we find alternating path
or there will be no edges in M and M’ contradiction to |M| < |M’|