Explain your reasoning

• Just giving an answer without any explanation whatsoever is not doing the exercise.
• “Explain your reasoning” has been and continues to be an implicit instruction for all exercises.
• You learn by trying to explain.

How many 3CNF formulas?

How many different formulas in 3-conjunctive normal form can be made using \(n\) variables \(x_1, x_2, \ldots, x_n\)?

• Clauses: \(C(2n,3) \in \Theta(n^3)\)
• Formulas: Power set of clauses: \(O(2^{n^3})\).

Find an upper bound on the length of a formula (counting literals, connectives, and parentheses).

• Everything is a constant multiple of the number of clauses, so length is \(O(n^3)\).

Note: Length is polynomial (3SAT is in NP), but the number of instances of 3SAT is superpolynomial.

DNF-SAT

Polynomial time solver (so DNF-SAT is in P)

for each clause
  if the clause does not contain a literal and its negation
    return true
return false

Argument is bogus because repeatedly distributing is going to be superpolynomial time.

How to show a problem is NP-hard

To prove that problem \(A\) is NP-hard, reduce a known NP-hard problem to \(A\).

• In other words, show that, if you had a polynomial-time solver for \(A\), you could build a polynomial-time solver for some known NP-hard problem.
• To show a problem is NP-complete, show that it is NP-hard, and also check that it is in NP.

Examples So Far

• SAT is NP-hard (and NP-complete).
  • First example; proof “from scratch” (CS-135)
• 3SAT is NP-hard (and NP-complete).
  • If I could solve 3SAT in polynomial time, I’d be able to solve SAT in polynomial time. But then I could solve any problem in NP in polynomial time.
• MaxIndSet is NP-hard.
  • If I had a polynomial time solver for MaxIndSet, I could use it to solve 3SAT in polynomial time.
    • Given an instance of 3SAT (i.e., a 3CNF formula), convert in to a graph using clever gadgets.
    • Use your MaxIndSet solver on this graph.
    • If the size of the independent set equals the number of clauses in your formula, the formula is satisfiable.

MinVertexCover (from MaxIndSet)

• MinVertexCover: Given a graph \(G\), what is size of the smallest subset of vertices you can select so that every edge is touched?
  • Suppose you had a polynomial-time solver for MinVertexCover.
  • Given an instance of MaxIndSet (a graph \(G\)), use your MinVertexCover solver to find a minimum vertex cover \(C\).
  • Now let \(S = V \setminus C\) (the vertices not in \(C\)).
    • No two vertices of \(S\) are joined by an edge, because you took away all the vertices in a vertex cover.
    • So \(S\) is an independent set.
      • And \(S\) is maximal, because \(C\) was minimal.
    • So you just solved MaxIndSet!
    • Forming the complement takes linear time, so this is a polynomial-time reduction. Therefore MinVertexCover is NP-hard.

MaxClique (from MaxIndSet)

• MaxClique: What is the size of the largest complete subgraph of a given graph?
  • Suppose you had a polynomial-time solver for MaxClique.
  • Someone asks you to solve an instance of MaxIndSet, i.e., given a graph \(G\), can you find a maximum independent set?
    • Form the edge complement \(\overline{G}\) of \(G\). (\(\overline{G}\) has edges exactly where \(G\) doesn’t.)
      • This takes \(O(V^2)\) time (using adjacency matrices).
    • Use your MaxClique solver on \(\overline{G}\).
    • You have found the largest set of vertices that are mutually connected in \(\overline{G}\). So none of these vertices are connected in \(G\).
    • So you just solved MaxIndSet in polynomial time!
    • Therefore, MaxClique is NP-hard.

3Color

3Color: Given a graph \(G\), is there a coloring of \(G\) that uses only 3 colors?

• 3Color is in NP, because it is easy to check whether a proposed 3-coloring is legal in \(O(V^2)\) time.

Coloring Gadgets

See the Jamboard.

1. Suppose a 3-coloring (using red, green, blue) of the first gadget makes \(u\) and \(v\) both green. What color(s) could \(z\) be?

2. Suppose a 3-coloring (using red, green, blue) of the first gadget makes \(u\) green and \(v\) blue. What color(s) could \(z\) be?

3. Suppose a 3-coloring (using red, green, blue) of the second gadget makes \(w\), \(x\), and \(y\) all green. What color(s) could \(z\) be?

3Color (from 3SAT)

• Suppose you have a polynomial-time solver for 3Color.
• Someone asks to to solve an instance of 3SAT (i.e., you are given a 3CNF formula: is it satisfiable?)
  • Convert the formula into a graph using gadgets.
    • Do it in polynomial time. (See Jamboard.)
  • Use your 3Color solver. (WLOG T=green, F=red)
    • Graph is 3-colorable \(\Leftrightarrow\) formula is satisfiable. (The greens are true.)
  • You just solved 3SAT in polynomial time!
  • Therefore 3Color is NP-hard (and NP-complete).