# Problem Set 4

- Print version, with cover page (pdf)
**Due Date**: Tuesday, May 3

**Note**: There are only 3 problems in this problem set. That means that your group can only have up to 3 members. If you're in a group of 1, you get to skip one problem, but if your group has 2 or 3 then you have to do all 3.

## 0.1 Problems 1-3

The three problems on this set all ask you to do the same thing, but for different computational problems. For each computational problem, you need to do the following:

Define a decision problem version of the computational problem given.

Show that your decision problem is

**NP**-complete. (This means first proving it's in**NP**, then proving it's**NP**-hard.)Show that the original computational problem is also

**NP**-hard.Come up with a strategy to solve the original computational problem, to get an optimum solution. Of course your algorithm will not be polynomial-time! But for full credit, your algorithm should be faster than brute-force. You might try and come up with a good lower bound to use with a branch-and-bound approach (like we saw for TSP), or use a dynamic programming approach where the table size depends on some parameter that might be more than polynomial-size in the input (like we saw for the change-making problem).

Or do something else entirely - something you read about or come up with on your own. In any case, you need to convince me that your algorithm will

*always*return the optimum solution, and it should be as efficient as possible.

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: Does

*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 outputOutput: 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 clauseOutput: 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 integersOutput: 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*?

# 1 Hungry Hungry Mids

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: ListLof integers, and a single integerk. Each integer inLis between 1 and \(k-1\).

Output: A partition ofLintorsubsets \(M_1,M_2,\ldots,M_r\) such that the sum of the numbers in each \(M_i\) is at leastk, and the number of subsetsris 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 \(M_3=(7,4,6,3)\).

# 2 Party Planner version 1

(This is the same problem from #3 on PS3!)

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 friendsF, and a list of pairs of enemiesE, each pair containing two elements fromF.

Output: A subset ofPofF, as large as possible, such that no two elements inPare enemies, i.e., for every pair inE, at most one of the pair is inP.

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

# 3 Workout circuit

You are trying to improve your PRT score by working out for \(n\) minutes every day. You have a list of \(m\) possible exercises to do. Each exercise takes a certain number of minutes, and each will improve your PRT score by a certain (integer) amount. Oh, and you can only do each exercise *at most twice*; otherwise you'll get too tired. Your task is to figure out which exercises you should do within \(n\) minutes' time in order to increase your PRT score the most.

Formal problem definition:

`WORKOUT(n,E)`

Input: An integer number of minutes \(n\), and a list of \(m\) pairs of integers \[E = (t_1, b_1), (t_2, b_2), \ldots, (t_m, b_m),\] where each \(t_i\) is how many minutes that exercise takes and \(b_i\) is the boost to your PRT score that exercise will give you.

Output: For each \(i\) from 1 to \(m\), an integer \(c_i\) indicating how many times that exercise should be performed. Each \(c_i\) must be 0, 1, or 2. The sum of all the times must be less than \(n\), and the sum total of PRT boost should be maximum.

For example, if \(n=10\) and there are \(m=4\) exercises: pushups \((2,10)\), burpees \((3,5)\), side plank \((5, 12)\), and running \((7,21)\), then the optimum solution would be to do 2 sets of pushups and 1 set of side plank, for a total time of 9 minutes and a total PRT gain of 32 points.

(Note that you don't have to take up the entire \(n\) minutes, you just need to maximize your PRT score without going over \(n\) minutes and without doing any exercise more than twice.)