Problem 1: Amicable Divorce (10 points). After taking this class, you decide to
ID: 3698295 • Letter: P
Question
Problem 1: Amicable Divorce (10 points).
After taking this class, you decide to forego computer science for a career as a divorce lawyer. Your first case involves a couple, Alice and Bob, who after many tumultuous years decided to call it quits. They decide to split up their n assets (car, house, boat, playstation 4, etc). Alice provides you with a list of integers a1, . . . , an where ai is how much she values the ith asset in dollars. Similarly, Bob provides a list of integers b1, . . . , bn where bi is his dollar value for the ith asset. Your job is to come up with a partition (A,B) of their assets, where A denotes the assets awarded to Alice and B denotes the assets awarded to Bob. For Alice and Bob to have an amicable divorce, (A,B) should be a no-envy partition: Alice’s total value for her assets A is at least her total value for Bob’s assets B, and Bob’s total value for B is at least his total value for A. In other words, (A,B) must satisfy ai ai and iA iB iA bi. Show that the problem of determining whether a no-envy partition exists is iB bi NP-complete.
Note. Recall that in order to prove a problem A is NP-complete, you need to first prove that A N P , then you must select an NP complete problem B and prove that B P A.
Hint. You may reduce from the subset sum problem.
Explanation / Answer
Non Polynomially Hard (NP)
Non Polynomial Complete is a subset of Non Polynomial
It contains a set or collection of decision making problems whose solutions can be checked for correctness in a polynomial time.
Here the problem comes due to the assets valued by other member
Say Bob Values the assets as follows:(in $)
Car 12,000
Boat 75,000
First House 500,000
Second House 450,000
Dining Table & Chairs 4500
Lounge Furniture 7500
Computer 1 1500
Computer 2 1400
Lap top 1 2100
Lap Top 2 1900
TV 750
Radio 120
Guitar 250
Piano 2500
Alice values the same assets differently as follows
Car 13,000
Boat 74,000
First House 520,000
Second House 490,000
Dining Table & Chairs 4100
Lounge Furniture 7400
Computer 1 16500
Computer 2 1300
Lap top 1 2600
Lap Top 2 1700
TV 450
Radio 130
Guitar 270
Piano 2300
Unless we engage an unbiased third party property valuer, this problem will become NP Hard
The problem becomes non polynomially harder as whatever asset Alice likes to get hold, Bob may value them more and vice versa
hence they may end up in a conflict claiming teh assets are not divided equally
one solution would be to value the assets before the partion
Bot hAlice and Bob have to agree for a value for each asset
then we can divide based on the cash value of each assets