1. 1. For minimum vertex cover problem in the following graph give
greedy solution = nodes __________________________________
2-VC solution = nodes __________________________________
Optimal solution = nodes__________________________________ 1 5 6 4 7 2 9 8 10 3

2. 4 6 3 1 1 7 2.2 12 6 3 8 2 5 6 4 4 2. For the following graph, find
Double MST tour _______________________________
MST-heuristic tour _______________________________
Cristofides heuristic matching _______________________________
Cristofides Tour _______________________________ 3 8 4 6 3 9 1 7 1 2 7 1
2.2 12 6 3 8 2 a 5 6 6 4 5 4 4

3. 3 . In the rectilinear metric for points given below find
- The minimum Steiner tree (bold), its length is _________________
- The minimum spanning tree (dashed), its length is _______________
the approximation ratio of the MST heuristic in this case is _________
Approximation error in % ________________

4. 4. For the 3-CNF
f = (x’ +y+z)& (x+y’+z’)&(x+y+z’)& (x’+y’+z)&(x’+y+z’) &(x’+y’+z’)
give 0-1 assignment to variables such that f=1 __________________
give 0-1 assignment to variables such that f=0 __________________
Draw the corresponding graph and mark the maximum independent set

5. 5. Prove, that the following problem is in class NP
- A thief robbing a store finds n items; the i-th item is worth Vi dollars and weight Wi pounds, where Vi and Wi are integers. He can carry at most W pounds in his knapsack. Find items maximizing a value of the load.

6. 6. In the following graph list the vertices in
Maximum Independent Set___________________________
Maximum clique___________________________ 3 6 2 4 7 5 1 8 9

7. 1. Use dynamic programming find longest common subsequences of the following two sequences x and y: ___________________
Show all details, and circle the resulted subsequence letters. (20pts.)
y B E A D E A x
B 0
A 0
D 0
E 0

8. 4. Johnson’s algorithm is applied to the graph below. (20pts)
Give the modified weight of the edge (2,6) ___________
The shortest path from 2 to 3 is ________ 5 5 3 3 4 -5 2 6 3 -3 12 -5 17 -7 4 11 1 7 9

9. 1. (10pts) Given 3 points with their Cartesian coordinates
A=(645,763), B=(478,529), C=(937,1187)
Give the final content of the stack in Graham’s algorithm for convex hull for these 4 points A, B and C (check the order!):

10. 1. For Graham’s scan finding convex hull of the point set given below:
- Give the sorted sequence of points for Graham scan _________________
- Show the content of the stack after each change
- Give the convex hull of this point set _________________ (20pts)
empty 4 5 2 3 4 7 1

11. 1. Below given a point set in the rectilinisr metric (the height/width of any cell=1) where the closest pair of points should be found using divide and conquer. Show
- the first partition of the point set (draw a line)
the closest pair in the left part (connect solid), left = _______ ,
and the right part (connect solid), right = _______
the middle strip (shade)
pairs in the middle strip for which distances should be computed (connect dashed)
closest pair in the middle strip (connect solid) (20pts)

12. 6. Below given a point set in the Euclidean metric. Draw
- Voronoi regions(dashed lines)
- Voronoi graph / Delanau triangulation (solid edges)
- minimum spanning tree (double edges)

13. 8. Below given a point set in the octiliniar metric (the height/width of any cell=1)
connect edges of the minimum spanning tree
the length of the minimum spanning tree is _____________

14. 1. Below given a point set in the Euclidean metric. Draw
- Voronoi regions (dashed edges)
- Voronoi graph / Delanau triangulation (solid edges)
- minimum spanning tree (double edges)

15. 2 . In the RECTILINIAR metric for points given below find:
a) the MST, its length is _______
c) the Cristofides tour is (enumerate points in visited order)
______________________________________________
d) its length is ___________
b) the 2-MST tour, is (enumerate points in visited order)
___________________________________________ 1 1 2 3 2 3 4 4 5 6 5 6 7 7 8 9 a 8 9 a c b c b d e d e
e) the Optimal tour (enumerate points in visited order)
__________________________________________
f) its length is _________
g) the minimum Steiner Tree,
its length is ____________ 1 1 2 3 2 3 4 4 5 6 5 6 7 7 8 9 a 8 9 a c b c b d e d e

16. Hashing
Matrix Chain Multiplication
Knapsack
Segment Intersection
LP and ILP
Stable Marriage
Stainer points