Project 7

This is a written project. Solutions to all parts must be typed or very neatly written, and handed in before the beginning of class on Monday, April 30. Late submissions will not be accepted, in accordance with the class policy. Students with excused absences on that date may email a copy to the instructor by the deadline in addition to handing in a paper copy at the earliest opportunity.

## 0.1 Guidelines

• You must print out, fill out, and attach a cover sheet to your submission.
• All work must be typed or neatly written
• Multiple-page submissions must be stapled or otherwise bound securely into a single packet.
• Be concise. Most problems have at least one simple and relatively short correct answer. Solutions will be graded not only on their correctness but also on how clearly that solution is presented.
• Do not try to pass off a solution you know is incorrect. A statement to the effect of "I know this doesn't work because of blah" will be effective in garnering partial credit points.
• Follow the course honor policy posted here. Remember that no written collaboration with other humans is permitted, and that you must document all your sources.

There are two problems in this project. They both have 7 parts (a)-(g), which are completely identical, except that the set-up problems are different. Each part of each problem is worth a maximum of 10 points. That, and the fact that you cannot receive credit for both 1(g) and 2(g), means that there are a total of 130 possible points. The project will be graded on a scale of 100 points = 100 percent. Therefore you do not have to attempt every part, but you may if you want to go for extra credit.

## 0.3 Known NP-Complete Problems

For your reference, here is a list of the NP-Complete Decision Problems that were presented in class. In any of your solutions, you may use the fact that these problems are NP-complete.

• LONGPATH(G,u,v,k)

Input: Graph $$G = (V,E)$$, vertices u and v, integer k

Output: Does G contain a path from u to v of length at least k?

• VC(G,k)

Input: Graph $$G=(V,E)$$, integer k.

Output: Does G have a vertex cover (subset of V) containing at most k vertices?

• HITSET(L,k)

Input: List L of sets $$S_1,S_2,\ldots,S_m$$, and an integer k

Output: Is there a "hitting set" H with size at most k such that H contains at least one member of every set $$S_i$$?

• HAMCYCLE(G)

Input: Graph $$G=(V,E)$$

Output: Des G contain a cycle (path with same starting and ending vertex) that touches every node exactly once?

• CIRCUIT-SAT(C)

Input: Boolean circuit C with m inputs and one output

Output: Is there a setting of the m inputs to True/False that makes the output stabilize to True?

• 3-SAT(F)

Input: Boolean formula F in conjunctive normal form (product of sums) with three literals in every clause

Output: Does F have a "satisfying assignment" (setting of every variable to True/False so that the entire formula is True)?

• SPLIT-EVENLY(S,k)

Input: Set S of integers

Output: Can S be partitioned into two subsets A and B such that difference between the sums of the numbers in A and B is at most k? In other words, $$A\cup B = S$$, $$A\cap B = \{\}$$, and $$|(\sum_{a\in A} a) - (\sum_{b \in B} b)|\le k$$: is this possible?

# 1 Hungry Hungry Mids

This question is about the following computational problem:

King Hall has a bunch of random leftover food items: a single hamburger patty, a bottle of ketchup, a bowl of mashed potatoes, a dill pickle, etc. Each leftover food item has a certain number of calories in it. The question is how many complete meals can be made from these leftover items, with the only restriction being that each "meal" must contain at least a certain number of calories.

Formally, the problem is defined as follows:

COMPUTE-MAX-MEALS(L,k)

Input: List L of integers, and a single integer k. Each integer in L is between 1 and $$k-1$$.

Output: A partition of L into r subsets $$M_1,M_2,\ldots,M_r$$ such that the sum of the numbers in each $$M_i$$ is at least k, and the number of subsets r is as large as possible.

For example, if $$L=(5,3,3,8,6,10,11,5,7,4)$$ and $$k=20$$, then an optimum solution has $$r=3$$ and the subsets are $$M_1=(10,8,3)$$, $$M_2=(11,5,5)$$, and $$M_3=(7,4,6,3)$$.

1. Describe a greedy algorithm to solve the COMPUTE-MAX-MEALS problem. Your algorithm should be fairly simple, but you should try to make it as good as possible:

• Your greedy algorithm must be polynomial-time (although you do not need to do the analysis).
• Your greedy algorithm should return optimum solutions as often as possible. (You definitely don't have to prove this, though!)
2. Give a counterexample to show that your greedy algorithm does not always produce the best possible solution.

3. Come up with a decision version of this problem. Call your decision problem MEALS.

Remember that your decision problem should be roughly the same difficulty as the original problem, up to polynomial-time reductions. In particular, you have to choose your decision problem carefully so that parts (e) and (f) below are at least possible to prove (even if you don't prove them yourself).

You are encouraged to e-mail your idea for this part to your instructor if you're not sure, before proceeding.

4. Show that your MEALS problem is in NP, using the steps we went over in class for an NP proof.

5. Present a polynomial-time reduction from your MEALS problem to COMPUTE-MAX-MEALS.

6. Show that your MEALS problem is NP-hard by presenting a reduction from a known NP-complete problem to your MEALS problem. You may use any of the NP-complete problems in the list at the beginning of this assignment for your reduction.

(Hint: what other decision problem involves the sums of numbers in different subsets?)

7. Explain what we know about your MEALS problem and the COMPUTE-MAX-MEALS problem, assuming (d) through (f) have been proven.

# 2 Party Planner

This question is about the following computational problem:

You are planning a party and want to invite a bunch of your friends. Unfortunately, some of your friends and acquaintences don't get along with each other, and bad things will happen if they both show up for the party. So, given the histories of bad blood among your friends, you want to invite the largest group of friends possible to your party, without inviting any two people that don't get along.

Formally, the problem is defined as follows:

COMPUTE-MAX-PARTY(F,E)

Input: A list of friends F, and a list of pairs of enemies E, each pair containing two elements from F.

Output: A subset of P of F, as large as possible, such that no two elements in P are enemies, i.e., for every pair in E, at most one of the pair is in P.

For example, if $$F=\{1,2,3,4,5\}$$ and $$E=\{(1,3),(2,3),(1,5),(4,5)\}$$, then an optimum solution is $$P=\{1,2,4\}$$.

1. Describe a greedy algorithm to solve the COMPUTE-MAX-PARTY problem. Your algorithm should be fairly simple, but you should try to make it as good as possible:

• Your greedy algorithm must be polynomial-time (although you do not need to do the analysis).
• Your greedy algorithm should return optimum solutions as often as possible. (You definitely don't have to prove this, though!)
2. Give a counterexample to show that your greedy algorithm does not always produce the best possible solution.

3. Come up with a decision version of this problem. Call your decision problem PARTY.

Remember that your decision problem should be roughly the same difficulty as the original problem, up to polynomial-time reductions. In particular, you have to choose your decision problem carefully so that parts (e) and (f) below are at least possible to prove (even if you don't prove them yourself).

You are encouraged to e-mail your idea for this part to your instructor if you're not sure, before proceeding.

4. Show that your PARTY problem is in NP, using the steps we went over in class for an NP proof.

5. Present a polynomial-time reduction from your PARTY problem to COMPUTE-MAX-PARTY.

6. Show that your PARTY problem is NP-hard by presenting a reduction from a known NP-complete problem to your PARTY problem. You may use any of the NP-complete problems in the list at the beginning of this assignment for your reduction.

(Hint: think about choosing the minimal number of people that won't be invited to the party. What decision problem(s) involve finding the smallest possible something?)

Explain what we know about your PARTY problem and the COMPUTE-MAX-PARTY problem, assuming (d) through (f) have been proven.