Matching in bipartite graphs

1
Matching in bipartite graphs

• Given non-weighted bipartite graph

initial matching
extending alternating path
not covered node
Algorithm so-called extending alternating
path, we start with a not covered node next
step node from the matching (maybe several
edges) in this case no extending alternating
path result
perfect matching
2
Vertex Cover Problem

• In general graphs is NP-hard (best approximation
  known - 2-approximation)
• In bipartite graphs ? exact algorithm
• (we can for sure find optimal matching with
  maximum of edges)

1
2
minimum cardinality matching M
3
OPT minimum vertex cover OPT ?
M Proof we need at least 1 vertex in OPT per
edge in M
3
Vertex Cover Problem

• In bipartite graphs
• Maximum matching Minimum vertex cover
• Algorithm (also the proof)
• Pick a node that covers red and blue edge
  (redfrom M, bluenot from M) (except - last
  case) - such blue edge that has one endpoint not
  covered by the matching. If such blue edge does
  not exist, we have perfect matching (from that
  point on, all nodes are covered by the matching).
  Every time we have isolated node, we drop it

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Vertex Cover Problem

• In case of perfect matching ? take one part of
  the bipartite graph
• In general graphs - not true
• Example

minimum cover
maximum matching
5
Homework solutions

• Problem 1 Translate node weighted graph G
  (G,V,w), where w V ? R, into edge weighted
  graph G(V,E,w), w E ? R, then apply Dijkstra
  on G.

w2
w1w2
w(e)
w1w3
2
2
e
w1
w3
w6
v1
w4
w5
For any edge evivk ? E we define w(e)1/2
w(vi)w(vk). Then length(v0v1...vn-path in
G)length (v0v1...vn-path in G) 1/2 w(v0) 1/2
w(vn) (v0source, vn sink). The
minimum weight path in G corresponds to a minimum
weight path in G.
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Homework solutions

• Problem 2 For G(V,E,w) we define a bipartite
  graph
• G (VV,E,w)

v1
v2
v3
V
w(v2) 1
v2
2
2
1
3
3
2
2
3
3
V
3
v3
v1
v1
v2
v3

• For each v ? V define a copy v and an edge
  (v,v), where
• w(v,v)w(v) (corresponds to a loop in v
  in graph G)
• Every edge (u,v) ? E replace by 2 edges (u,v)
  and (u,v)
• with the same weight as (u,v)
• G is bipartite (every edge has 1 end in V,
  other in V)
• Every path from v1 to vn even
• Every path from v1 to vn (copy of vn) odd

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Homework solutions

• Problem 3
• G(V,E,w). For each v ? V define k copies
  v1,v2,..., vk
• Then create a directed graph G(V,E,w), where
  
• VV? V1 ? ... ? Vk , vi1? V1, ....
  vjk? Vk

v
u
1
1
For each edge (u,v) ? E define edges (u, v1),
(u1, v2), (u2, v3), ... (u(k-1), vk) with the
same edge weight as (u,v). We look at all paths
from v1(source) to vn(sink) or any copy of vn.
All such paths have at most k edges.
2
2
k
k
k
? k edges shortest paths