Algorithm Design Techniques
Due: April 11, 2003 5pm
Assignments are to be left in the box near
the labs, McConnell 1st floor.
Late assignments -10% per day, starting April
If you work closely with someone else, indicate the person's name(s) on your
The final write-up of each assignment must, however, be your own work.
For this assignment, you may use the fact that the following problems are
known to be NP-complete: SAT, 3-SAT, Subset Sum, Independent Set.
1. Study the reduction of SAT to CLIQUE.
Consider the CNF: ( x + y + z ) . ( x + z ) . ( y +
z ) . ( x + y + z) . ( x + y ), where
underlineing represents the complement of the variable .
Construct a graph G such that every clique of size 5 in G corresponds to
a consistent truth assignment for the CNF.
Illustrate by showing two different cliques of size 5.
2. Knapsack: Instance: Non-negative weights a1 , a2
, ... , an , b, and profits c1, c2
, ... , cn, k.
Question: Is there a subset of weights with total weight
at most b, such the the corresponding profit is at least k?
Show Knapsack is
NP-complete. (Hint: Reduce Subset Sum to this problem)
3. Both the clique problem and the vertex cover
problem are solvable in polynomial time for bipartite graphs, but are NP-complete
in general. Does this imply that all NP-complete problems for undirected
graphs are solvable in polynomial time for bipartite graphs? Justify your
4. Give an example of an NP-complete problem Q that
can be reduced to a problem R solvable in polynomial time using a non-polynomial
time reduction. You may invent the problem R, but be sure to specify