1. Example 4 : Show that the Square Packing (SP) Problem is NP-complete.
Motivation : truck loading, processor allocation problem in a partitionable mesh connected system, the design of VLSI chips and etc.
Square Packing Problem
Given a packing square S and a set of packed squares L = {s1, s2, ..., sn}.
Question : is there and orthogonal packing of L into S?
Note : orthogonal packing  the sides of squares are parallel to the vertical and horizontal axes
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NP-Completeness

2. Note : each group must have exactly 3 numbers
3-Partition Problem
Given a list A = {a1, a2, ..., a3z} of 3z positive integers such that sum of all numbers is zB and B/4 < ai < B/2 for each 1  i  3z.
Question : Can A be partitioned into z groups such that the sum of all numbers in each group is B.
Note : each group must have exactly 3 numbers
Proof : Refer to the following research paper
Leung, Tam, Wong, Young and Chin, “Packing Squares into a Square”, Journal of Parallel and Distributed Computing, 1990.
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NP-Completeness

3. Coping with NP-Completeness Problem NP-hard Problem
Note : Refer to Chapter 5 of Garey and Johnson
If L' L and L' is an NP-complete problem then L is called NP-hard problem.
All NPC problems are NP-hard.
Informally, an NP- hard problem is a problem which is at least as hard as an NP-complete problem.
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4. There are some NP-hard decision problems that are not in NP.
Example : Kth Largest Subset Problem is not in NP
Instance : Given a set of positive integers A = {a1, a2, …, an}, and two non-negative numbers, B  A and K  2|A|.
Question : Are there at least K distinct subsets A’  A such that each subset has total sum >= B.
Note : PARTITION problem  Kth Largest Subset Problem
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5. Pseudo-polynomial time algorithm
Note : Refer to Chapter 4 of Garey and Johnson
Some NP-complete problem may be solved in "polynomial" time (based on input size and magnitude).
Example : PARTITION problem
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6. Dynamic Programming Algorithm :
assume B = (sum of n integers)/2
construct a table of size (approx.) n x B
fill in the table row by row
for each row, add a new element
mark sum of all possible subsets
if there is a subset with sum = B, stop.
Time Complexity : O(nB)
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7. NP-completeness in the strong sense
Note : Refer to Chapter 4 of Garey and Johnson
If L is NP-complete problem in the strong sense, then L cannot be solved by a
pseudo-polynomial time algorithm unless P=NP
L is NP-complete in the strong sense if L'  L , L' is NP-complete in the
strong sense and L is in NP
Example : 3-partition problem is NPC in the strong sense.
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8. Solving more restricted problems
If we restricted the problem L to problem L'
i.e. L' is a special/restricted case of L,
then we may solve L' in polynomial time.
For example : PARTITION problem
if we assume that each input integer ai  n, where n is the number of input integers, then the pseudo polynomial time algorithm becomes polynomial time algorithm, i.e. O(n3)
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