Complexity Theory

1. Asymptotic Notation

1.1 Big-O (Upper Bound)

$f(n) = O(g(n))$ if there exist constants $c > 0$ and $n_0 > 0$ such that:

$f(n) \leq c \cdot g(n) \quad \text{for all } n \geq n_0$

1.2 Big-Omega (Lower Bound)

$f(n) = \Omega(g(n))$ if there exist constants $c > 0$ and $n_0 > 0$ such that:

$f(n) \geq c \cdot g(n) \quad \text{for all } n \geq n_0$

1.3 Big-Theta (Tight Bound)

$f(n) = \Theta(g(n))$ if both $f(n) = O(g(n))$ and $f(n) = \Omega(g(n))$:

$c_1 \cdot g(n) \leq f(n) \leq c_2 \cdot g(n) \quad \text{for all } n \geq n_0$

1.4 Little-o and Little-omega

$f(n) = o(g(n))$ if for every constant $c > 0$, there exists $n_0$ such that $f(n) < c \cdot g(n)$ for all $n \geq n_0$.

Equivalently: $\lim_{n \to \infty} \frac{f(n)}{g(n)} = 0$.

$f(n) = \omega(g(n))$: $\lim_{n \to \infty} \frac{f(n)}{g(n)} = \infty$.

1.5 Common Complexity Classes

1.6 Properties

Transitivity: If $f = O(g)$ and $g = O(h)$, then $f = O(h)$.

Reflexivity: $f = O(f)$.

Sum rule: $O(f) + O(g) = O(\max(f, g))$.

Product rule: $O(f) \cdot O(g) = O(f \cdot g)$.

2. Complexity Classes

2.1 Decision Problems

A decision problem has a yes/no answer. Optimization problems can be cast as decision problems (e.g., “is there a TSP tour of length $\leq k$?“).

2.2 P (Polynomial Time)

$\text{P} = \{L : L \text{ is decidable by a deterministic TM in } O(n^k) \text{ time for some } k\}$

Examples: Sorting, shortest paths, MST, 2-SAT, primality testing.

2.3 NP (Nondeterministic Polynomial Time)

$\text{NP} = \{L : L \text{ is decidable by a nondeterministic TM in polynomial time}\}$

Equivalently, NP is the class of problems for which a yes-instance can be verified in polynomial time given a certificate (witness).

Examples: SAT, traveling salesman (decision version), graph coloring, clique, vertex cover.

Key insight: $\text{P} \subseteq \text{NP}$. Whether $\text{P} = \text{NP}$ is the most important open question in CS.

2.4 NP-Hard

A problem $L$ is NP-hard if every problem in NP can be reduced to $L$ in polynomial time:

$\forall L" \in \text{NP}, \ L' \leq_p L$

NP-hard problems are at least as hard as every problem in NP. They may or may not be in NP themselves.

2.5 NP-Complete

A problem is NP-complete if it is both in NP and NP-hard:

$\text{NP-complete} = \text{NP} \cap \text{NP-hard}$

If any NP-complete problem is in P, then P = NP.

Relationship:
        NP-Hard
       /       \
      /    NP-Complete
     /       /
  All problems    P ⊆ NP

2.6 co-NP

$\text{co-NP} = \{L : \overline{L} \in \text{NP}\}$

The class of problems whose complement can be verified in polynomial time.

Open question: Is NP = co-NP? (Believed no, but not proven.)

2.7 Other Classes

$\text{P} \subseteq \text{NP} \subseteq \text{PSPACE} \subseteq \text{EXP}$

3. Polynomial-Time Reductions

3.1 Definition

A polynomial-time reduction $L_1 \leq_p L_2$ transforms instances of $L_1$ into instances of $L_2$ in polynomial time, preserving yes/no answers.

If $L_1 \leq_p L_2$ and $L_2 \in \text{P}$, then $L_1 \in \text{P}$.

If $L_1 \leq_p L_2$ and $L_1$ is NP-hard, then $L_2$ is NP-hard.

3.2 Karp Reductions

A Karp reduction (many-one reduction) maps instance $x$ of $L_1$ to instance $f(x)$ of $L_2$:

$x \in L_1 \iff f(x) \in L_2$

where $f$ is computable in polynomial time.

3.3 Proving NP-Completeness

To prove $L$ is NP-complete:

1. Show $L \in \text{NP}$: Given a certificate, verify in polynomial time.
2. Show $L$ is NP-hard: Reduce a known NP-complete problem to $L$.

Standard NP-complete problems for reduction:

• SAT (Cook’s theorem)
• 3-SAT
• Independent Set
• Vertex Cover
• Clique
• Subset Sum
• Hamiltonian Cycle
• TSP (decision version)

4. Classic NP-Complete Problems

4.1 SAT (Boolean Satisfiability)

Instance: A Boolean formula $\phi$ in conjunctive normal form (CNF): $\phi = C_1 \wedge C_2 \wedge \cdots \wedge C_m$, where each clause $C_j$ is a disjunction of literals.

Question: Is there a truth assignment that satisfies $\phi$?

4.2 3-SAT

SAT where each clause has exactly 3 literals.

4.3 Vertex Cover

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

Question: Is there a set $S \subseteq V$ with $|S| \leq k$ such that every edge has at least one endpoint in $S$?

4.4 Clique

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

Question: Is there a clique of size $k$ (a set of $k$ pairwise adjacent vertices)?

Reduction from Vertex Cover: $G$ has a vertex cover of size $k$ iff $\overline{G}$ has a clique of size $|V| - k$.

4.5 Independent Set

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

Question: Is there an independent set of size $k$ (a set of $k$ vertices, no two adjacent)?

Reduction from Vertex Cover: $G$ has a vertex cover of size $k$ iff $G$ has an independent set of size $|V| - k$.

4.6 Subset Sum

Instance: Set $S = \{s_1, s_2, \ldots, s_n\}$ of integers, target $T$.

Question: Is there a subset $S' \subseteq S$ such that $\sum_{s \in S'} s = T$?

4.7 Hamiltonian Cycle

Instance: Graph $G = (V, E)$.

Question: Does $G$ contain a cycle that visits each vertex exactly once?

4.8 Traveling Salesman (Decision Version)

Instance: Complete graph $G$ with edge weights, integer $k$.

Question: Is there a Hamiltonian cycle of total weight $\leq k$?

4.9 Graph Coloring

Instance: Graph $G$, integer $k$.

Question: Can $G$ be colored with $k$ colors such that no adjacent vertices share a color?

• $k = 2$: Solvable in polynomial time (bipartiteness test).
• $k \geq 3$: NP-complete.

4.10 Reduction Chain

SAT
 └── 3-SAT
      └── Independent Set
      │    ├── Vertex Cover
      │    └── Clique
      └── Hamiltonian Cycle
           └── TSP

5. Cook’s Theorem

Theorem (Cook, 1971 / Levin, 1973): SAT is NP-complete.

Proof sketch:

1. SAT $\in$ NP: Given a truth assignment, verify each clause is satisfied in $O(|\phi|)$ time.

2. SAT is NP-hard: Every NP problem $L$ can be decided by a nondeterministic TM $M$ in polynomial time. Given any instance $x$ of $L$, construct a Boolean formula $\phi_{M,x}$ that is satisfiable iff $M$ accepts $x$.

Construction of $\phi_{M,x}$:

• Encode the computation of $M$ on $x$ as a tableau (2D grid of cell states).
• Each cell’s state depends on its neighbors (TM transition function).
• Assert: initial configuration is correct, transitions are valid, accepting state is reached.
• All constraints are expressible as CNF clauses.
• Size is polynomial in $|x|$.

Consequence: Any NP-complete problem can be proven NP-hard by reducing from SAT (or from any other known NP-complete problem).

6. Approximation Algorithms

6.1 Motivation

For NP-hard optimization problems, we seek polynomial-time algorithms that produce near-optimal solutions.

6.2 Approximation Ratio

An algorithm $\mathcal{A}$ is an $\alpha$-approximation for a minimization problem if:

$\mathcal{A}(I) \leq \alpha \cdot \text{OPT}(I) \quad \text{for all instances } I$

For maximization: $\mathcal{A}(I) \geq \frac{1}{\alpha} \cdot \text{OPT}(I)$.

6.3 Vertex Cover: 2-Approximation

APPROX_VERTEX_COVER(G):
    C = {}
    E' = G.E (copy)
    while E' is not empty:
        pick arbitrary (u, v) in E'
        C = C ∪ {u, v}
        remove all edges incident to u or v from E'
    return C

Ratio: $|C| \leq 2 \cdot |\text{OPT}|$ (since each selected edge has at least one endpoint in any vertex cover).

Time: $O(V + E)$.

6.4 TSP: Christofides’ Algorithm

For metric TSP (triangle inequality):

1. Compute MST.
2. Find minimum-weight perfect matching on odd-degree vertices.
3. Combine MST + matching into Eulerian circuit.
4. Shortcut repeated vertices.

Approximation ratio: $\frac{3}{2}$.

6.5 Set Cover: Greedy $\ln(n)$-Approximation

APPROX_SET_COVER(U, S):
    C = {}
    while C != U:
        pick S_i in S that maximizes |S_i \ (C ∩ S_i)|
        C = C ∪ S_i
    return selected sets

Approximation ratio: $H_n = O(\ln n)$, where $H_n = \sum_{i=1}^{n} \frac{1}{i}$ is the $n$-th harmonic number.

6.6 PTAS and FPTAS

PTAS (Polynomial-Time Approximation Scheme): For any $\epsilon > 0$, a $(1+\epsilon)$-approximation running in polynomial time (possibly exponential in $1/\epsilon$).

FPTAS (Fully PTAS): $(1+\epsilon)$-approximation running in polynomial time in both $n$ and $1/\epsilon$.

Example: Knapsack has an FPTAS.

7. Hardness of Approximation

7.1 Inapproximability Results

Some problems cannot be approximated within any constant factor unless P = NP.

7.2 PCP Theorem

PCP Theorem (1992): Every problem in NP has a probabilistically checkable proof that can be verified by reading only a constant number of bits.

Consequence: Max-3SAT cannot be approximated beyond $7/8 + \epsilon$ unless P = NP.

7.3 APX, APX-Hard

• APX: Problems with constant-factor polynomial approximation algorithms.
• APX-hard: As hard as any problem in APX under PTAS reductions.

Examples: Vertex cover (APX, 2-approximable), Max-3SAT (APX, 7/8-approximable).

8. Common Pitfalls

1. Confusing NP with “not polynomial.” NP means verifiable in polynomial time, not “hard.” P $\subseteq$ NP, so every problem in P is also in NP.

2. Confusing NP-hard with NP-complete. NP-hard includes problems not in NP (e.g., the halting problem). NP-complete requires the problem to be both NP-hard and in NP.

3. Incorrect reduction direction. To prove $L$ is NP-hard, reduce a known NP-hard problem to $L$ (not the other way). The reduction goes from hard to unknown.

4. Forgetting to prove membership in NP. A reduction alone only shows NP-hardness. You must also show $L \in \text{NP}$ to claim NP-completeness.

5. Confusing worst-case with average-case. NP-completeness is a worst-case notion. An NP-complete problem might be efficiently solvable on typical inputs.

6. Assuming approximation exists. Not all NP-hard problems have good approximations. Some are inapproximable within any polynomial factor (unless P = NP).

7. Using wrong reduction type. Karp (many-one) reductions map yes-instances to yes-instances and no-instances to no-instances. Turing reductions (which use $L_2$ as a subroutine) are more general but give weaker results for NP-hardness.

Worked Examples

Example 1: Proving a Language is in P

Problem: Prove that the language PATH = {(G, u, v) : G is a directed graph with a path from u to v} is in P. Solution: BFS or DFS from u in G takes O(V + E) time, which is polynomial in the input size. If v is reached, accept; otherwise reject. Since there exists a polynomial-time algorithm, PATH is in P.

Example 2: NP-Completeness Reduction

Problem: Prove that 3-SAT is NP-hard by reducing from SAT. Solution: Given a CNF formula phi, transform each clause into a set of exactly 3 literals. For clauses with 1 literal (l), replace with (l OR x OR NOT x) where x is a new variable. For clauses with 2 literals (l1 OR l2), replace with (l1 OR l2 OR x) AND (l1 OR l2 OR NOT x). This produces an equisatisfiable 3-CNF formula in polynomial time. Therefore, 3-SAT is NP-hard. Since 3-SAT is in NP, it is NP-complete.

Summary

• Asymptotic notation ($O$, $\Omega$, $\Theta$, $o$, $\omega$) describes growth rates of functions.
• P = solvable in polynomial time; NP = verifiable in polynomial time.
• NP-complete = NP $\cap$ NP-hard; the “hardest” problems in NP.
• Cook’s theorem establishes SAT as the first NP-complete problem.
• Polynomial reductions propagate hardness: if $L_1 \leq_p L_2$ and $L_1$ is NP-hard, then $L_2$ is NP-hard.
• Approximation algorithms provide near-optimal solutions in polynomial time with provable guarantees.
• Hardness of approximation shows limits on how well NP-hard problems can be approximated.

Cross-References