What do you have and what do you need to be able to do with it?

To prove that a new problem \(\mathcal{A}\) is NP-hard, you must show that a known NP-hard problem \(\mathcal{B}\) can be solved in polynomial time, given a polynomial time solver for \(\mathcal{A}\). This is called “a reduction from \(\mathcal{B}\).”

• You have a polynomial-time solver for \(\mathcal{A}\).
• You need to use it to solve an instance of \(\mathcal{B}\).

SUBGRAPH

• You have a solver for SUBGRAPH.
  • Note: The SUBGRAPH solver consumes a graph \(G\) and a graph \(H\).
• Use it to solve an instance of Clique: Given a graph \(G\) and an integer \(k\), does \(G\) have a Clique of size \(k\)?
  • Let \(H\) be the complete graph on \(k\) vertices.
  • If your SUBGRAPH solver returns YES, the answer to the Clique instance is YES.
  • If your SUBGRAPH solver returns NO, the answer to the Clique instance is NO.
    • Note: This is the contrapositive of “If the answer to the Clique instance is YES, the answer the SUBGRAPH solver returns is YES.”

SplitSum

• You have a solver for SplitSum.
  • Note: the SplitSum solver consumes a set \(X\).
• Use it to solve an instance of SubsetSum: Given a set \(Y\) and a target value \(T\), does \(Y\) have a subset that sums to \(T\)?
• You need to somehow use both \(Y\) and \(T\) to make a set \(X\), and then feed that set to the SplitSum solver.

(On board.)

Study Guide

• Approximately 30% NP-hardness
• 70% Selected Review (see study guide)
  • Previous tests would be good to review and correct.
  • Also previous assignments.

Problem 34

1. Find another solution, besides the one shown at left.

2. Explain why the configuration on the right has no solution.

3. We are going to reduce from puzzle configurea3SAT. What do we assume that we have, and what do we need to be able to do with it?

The Reduction

Given a 3CNF formula, we need to make a puzzle configuration out of it.

• The rows correspond to clauses \(C_1 \wedge C_2 \wedge \cdots \wedge C_n\).
• The columns correspond to variables \(x_1, x_2, \ldots, x_m\).
• The stones correspond to literals: blue is \(x_j\), red is \(\overline{x_j}\).

4. Find a 3CNF formula that gets translated into the puzzle on the left.

5. Find a 3CNF formula that gets translated into the puzzle on the right.

If and only if

6. Prove: If the formula is satisfiable, then the puzzle has a solution.

7. Prove: If the puzzle is solvable, then the formula is satisfiable.

8. Polynomial time?

Could we reduce the other way?

9. Could we do this reduction in reverse? That is, if we knew in advance that this puzzle is NP-Hard, would we be able to use this correspondence to give a new proof that 3SAT is NP-Hard? Why or why not?