1. Projects Network Theory VLSI PSM 1. Network 1. Steiner trees
survivability run time
(Purav doing it) improvement (B)
2. Arbitrary quality improvement (A)
(theoretical) Zero-skew trees
problems run time (B)
(problems from theory (quality impr.) (A)
conferences, Floor planning
metric problems computational
with graphs ..) geometry
3. Implement
fast MST
O(E+V logV)
(Karger algorithm)

2. Approximation Algorithm
Steiner trees in graphs
G = (V,E,cost), cost:E  R+
S  V set of terminals
if S = V  MST algorithm exactly
if S  V  problem is NP-hard
MST - heuristic for the Steiner tree problem
1. Construct graph G’= (S,E’,cost’)
(for any pair of points define distance=cost of the
shortest path between them)
there is an algorithm (Mehlhorn) for this problem with
O(E+VlogV)
2. Find T=MST(G’)
O(n3)
Run Dijkstra
from one point

3. MST - heuristic for the Steiner tree problem
3. Construct H = T*
T*={G-paths corresponding to edges at T}
(T* may have some cycles, remove arbitrary edges from cycles)
4. Toutput = MST(H) a a G  G’ c b b c
cycles cannot appear, but duplicates are possible

4. Approximation ratio of MSTH
Theorem: Approximation ratio of MST heuristic  2
Proof:
2 OPT = Tour  Shortcut Tour  MST  Toutput
(+ cost of the longest edge in the green tour)
terminals
Steiner
points

5. Approximation ratio of MSTH
Proof that approximation ratio = 2:
  > 0  I  STP such that: MST (I)  (2- ) OPT(I)
OPT(Ik) = k
MST( Ik ) = 2(k-1)
supIk (2k-2)/k = 2 1 2
k terminals 2
distance between any 2 terminals=2

6. Traveling Salesperson Problem (TSP)
-first problem proved to be NP-hard
Given: complete graph G=(V,E,cost)
Find: minimum cost tour which visits all nodes
traveling salesperson problem
- if we have -inequality in G  MST-heuristic = 2 approximation

7. Traveling Salesperson Problem (TSP)
1. Find T = MST(G)
2. H = T+T (traverse this tree visiting each node twice)
3. Tour  Shortcut (H)
output tour (heuristic)
make a shortcut
OPT  MST
Approx  2 MST  2 OPT

8. Eulerian Graphs
Eulerian graph - you can traverse all edges visiting each
only once (Euler, 18th cent., problem: Köningsberg’s bridges)
graph Eulerian  all degrees are even
Theorem: Graph G is Eulerian 
D(G) is bipartite
Proof: homework
Theorem: In every graph the number of odd degree nodes
is even.
Proof: Homework
red = G black = D(G) - dual of G

9. 1.5 Approximation for TSP
Christophides (in 1976) - better heuristic, approx. ratio 1.5
Matching problem
Given G = (V,E,cost) |V|=even
matching - no 2 edges have common point
(there is exact algorithm with run time O(v3) that will find minim. cost)
3 matchings
bipartite graph

10. 1.5 Approximation for TSP Algorithm: 1. Find MST
2. Find minimum weight
matching M of odd-
degree nodes
3. Make Tour M  MST
Proof that approx. is 1.5:
we know : MST  OPT
need to prove: M  0.5 OPT
then  Tour  1.5 OPT ?

11. 1.5 Approximation for TSP Proof that M  0.5 OPT
optimal  shortcut = M1 +M2  2*M
shortcut for odd
nodes (shorter because
of triang.ineq.)
optimal
other
mathing M2
one matching M1

12. Minimum Vertex Cover - delete both endpoints of e NP - hard problem
Given: G = (V,E,cost) cost: V  R+
Find: C  V cost(C)  min such that each edge in E has
at least one end point in C C
2 approximation :
- pick an edge e
- delete both endpoints of e
- repeat until no edges are left

13. Minimum Vertex Cover Let Ce be output of edge-deletion heuristic.
Then |Ce|  2 OPT.
Proof: Let M be the set of chosen edges in the heuristic
Ce = 2 |M|
OPT  |M|
|M| = OPT (M)  OPT(G)
we proved  |Ce|  2 OPT ?
here all edges of G need
to be covered
number of edges
in M
only some edges
of G to cover