Algorithms and Structural Complexity Theory continued

Problem: CIRCUIT-SAT
Instance: An acylcic (i.e., no cycles), directed graph G whose nodes are logic functions: AND, OR, or NOT, or logical variables. The graph represents a combinatorial logic circuit with n inputs and 1 output.
Question: Is there any assignment to the n input variables that will cause the output to become True?

• D is in NP
• CIRCUIT-SAT <_p D

Let's look at another problem:

Problem: 3-CNF-SAT
Instance: A set of m clauses of length at most 3 over n Boolean variables. The clauses are sets of literals, i.e., a variable or its complement (negation). Question: If we take the Boolean OR of the literals in each clause, and then take the Boolean AND of all the clauses, is there any assignment to the n variables such that the result is True (i.e., all the clauses are simultaneously True)?

 2 -3  1
-3 -2  5
-5  3  2
-4 -2  3
 1 -4 -2

(2 OR NOT 3 OR 1) AND (NOT 3 OR NOT 2 OR 5) AND (NOT 5 OR 3 OR 2)
AND (NOT 4 OR NOT 2 OR 3) AND (1 OR NOT 4 OR NOT 2)

1 = True, 2 = False, 3 = True, 4 = False, 5 = False

(2 OR NOT 3 OR 1)        =  (False OR NOT True OR True) = True
(NOT 3 OR NOT 2 OR 5)    =  (NOT True OR NOT False OR False) = True
(NOT 5 OR 3 OR 2)        =  (NOT False OR True OR False) = True
(NOT 4 OR NOT 2 OR 3)    =  (NOT False OR NOT False OR True) = True
(1 OR NOT 4 OR NOT 2)    =  (True OR NOT False OR NOT False) = True

You can see that this is a very restricted instance of CIRCUIT-SAT; just replace the word "formula" with "circuit" and place logic gates everywhere there is a logical operator. It may be somewhat surprising that it goes the other way around; 3-CNF-SAT is just as hard as CIRCUIT-SAT because there is a polynomial reduction such that CIRCUIT-SAT <_p 3-CNF-SAT.

Theorem: 3-CNF-SAT is NP-complete.
Proof:
Clearly, 3-CNF-SAT is in NP; we just use a satisfying assignment as the linear-time verifiable certificate. So we just need to show CIRCUIT-SAT <_p 3-CNF-SAT.

From a circuit C made up of gates AND, OR, and NOT, and input variables in a set { 1 .. n}, we will construct a 3-CNF formula F that is satisfiable if and only if F is satisfiable. The construction goes as follows:

1. For each variable in C, there is a corresponding variable in F.
2. For each gate c in C, there is a corresponding variable f in F.
3. For each NOT gate c modifying a gate or variable d in C, insert logic into F stating "c is equivalent to NOT d." This can be accomplished in CNF with two clauses: (d OR c and (NOT d OR NOT c). If you don't quite buy this, let's see how it's true with a truth table:

   c  d  (d OR c)  (NOT d OR NOT c)  ((d OR c) AND (d OR NOT c))  d == NOT c
   -  -  --------  ----------------  ---------------------------  ----------
   F  F     F             T                     F                     F
   F  T     T             T                     T                     T
   T  F     T             T                     T                     T
   T  T     T             F                     F                     F
   

4. For each AND gate c modifying two gates or variables d and e in C, insert logic into F stating "c is equivalent to d AND e." We can do in CNF with the following clauses:

   NOT c OR d
   NOT c OR e
   c OR NOT d OR NOT e

   This can be shown using another (eight line) truth table.
5. For each OR gate c modifying gates or variables d and e, insert logic into F stating "c is equivalent to d OR e." We can do this in CNF with the following clauses:

   c OR NOT d
   c OR NOT e
   NOT c OR d OR e

   This can also be shown with a truth table.
6. Note that we can omit either AND or OR, simulating one with a combination of NOT and the other, making the circuit a little bigger. We can also simulate any other logic gate (NAND, NOR, XOR, etc.) as combinations of AND, OR, and NOT in linear space.
7. For the output gate c of the circuit, insert logic into F stating "c is equivalent to True." The singleton clause (c) states this.

We don't seem to have accomplished much with this proof; we've replaced one kind of pointy-headed Boolean logic problem with another. However, note that 3-CNF-SAT is a much restricted version of CIRCUIT-SAT. We can use 3-CNF-SAT to prove other problems are NP-complete in instances when tackling all of CIRCUIT-SAT is infeasible or impossible. That is, 3-CNF-SAT is easier to work with when proving things NP-complete.

(Note: It turns out k-CNF-SAT, where clauses have at most k literals, is NP-complete for k > 2. For k = 2, (i.e. 2-CNF-SAT), there is a polynomial time algorithm: a clause (a OR b) is the same as (NOT a IMPLIES b); make a directed graph with vertices the literals and edges the implications derived from the clauses; if any strongly connected component (computable in linear time, similar to the connected components in undirected graphs we saw with Union/Find) of the graph contains a literal and its complement, that is a contradiction, so the formula is not satisfiable. Otherwise, it is. Unfortunately, this algorithm doesn't extend to k-CNF-SAT for k > 2, so we don't get to prove P=NP this way.)

Now let's look at a problem we saw last time:

Problem: HAMILTONIAN-CYCLE
Instance: A graph G = (V, E)
Question: Does G contain a Hamiltonian cycle? That is, is there a path (cycle) going from one vertex of G, through all the other vertices of G exactly once, ending up at the same vertex?

Here are some other NP-complete problems:

Problem: SUBSET-SUM
Instance: A set S of positive integers and an integer n.
Question: Is there any subset of S whose elements add up to n?

Problem: CLIQUE
Instance: An undirected graph G and integer k.
Question: Is there a subgraph of G with k vertices that is complete? That is, is there a clique of vertices of size k that are all mutually connected by (k(k+1))/2 edges?

Problem: SUBGRAPH-ISOMORPHISM
Instance: Two undirected graphs G and H.
Question: Is G isomorphic to a subgraph of H? That is, of all the subgraphs (subsets of vertices and edges) of H, can we relabel one so that it is identical to G?

Problem: REGISTER-ALLOCATION
Instance: A block of computer code, a set of variables, and a set of CPU registers, and an integer k.
Question: Is there a schedule of allocation of registers to variables such that at most k registers must be spilled to main memory during the block of code? (Spilling registers to memory is bad.)

More Structural Complexity

• co-NP. These are the complements of the problems in NP. For example, 3-CNF-UNSAT, the set of all unsatisfiable formulas in 3-CNF, is in co-NP (since 3-CNF-SAT is in NP).
• P^NP. This is the set of problems solvable in polynomial time with arbitrary queries on an NP oracle, that is, problems we can solve in polynomial time if we can easily answer questions in NP.
• NP^NP. This is the set of problems solvable in nondeterministic polynomial time with arbitrary queries to NP problems. An example of a problem in NP^NP is this:

  Problem: MINIMUM-EQUIVALENT-FORMULA
  Instance: A Boolean formula F and an integer k.
  Question: Is there an equivalent Boolean formula (i.e., has the same truth table) for F with at most k symbols (i.e., of size k)?

  It would be great to be able to answer this question, because we could achieve something like optimal data compression by specifying our input data as a trivial truth table and get the optimally small Boolean formula as output. Unfortunately, this problem is even harder than NP-complete (it is clearly NP-hard).
• LOGSPACE. These are the problems that can be solved using only logarithmically extra space.
• P^D where D is a decision problem. These are the problems that can be solved using polynomially many applications of a decision problem D.

Here is a simple map of PSPACE as we know it, assuming P != NP and NP != co-NP.

 ----------------------------------------------
|PSPACE       ______   ______                  |
|            /      \ /      \                 |
|           /        \        \                |
|          /        / \        \               |
|         /        /   \        \              |
|        /    NP  /  P  \ co-NP  \             |
|        \_____   \     /        /             |
|         \NP- \___\   /        /              |
|          \complete\ /        /               |
|           \        \        /                |
|            \______/ \______/                 |
|                                              |
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