Developing Abstractions To Formally Analyze Computational Complexity

Formalizing Computational Complexity

To formally analyze the computational complexity of algorithms, computer scientists have developed mathematical abstractions and models. These formalisms allow for quantifying the amount of computational resources like time and memory used by an algorithm.

Key concepts in studying computational complexity formally include:

• Complexity Classes – Sets of problems with related resource usage
• Abstract Models of Computation – Idealized mathematical models of a computer
• Resource Bounds – Mathematical functions describing allowable resources for an algorithm
• Reductions – Transformations from one problem to another related problem

Using these abstractions, computer scientists can prove formal statements about the intrinsic difficulty of computational problems. The abstractions enable applying mathematical analysis to compare algorithms and problems regarding efficiency.

Defining Complexity Classes

A central concept in studying computational complexity is the complexity class. A complexity class contains all problems that can be solved by an algorithm within specified resource bounds.

Some key complexity classes include:

• P – Problems solvable in polynomial time by a deterministic Turing machine
• NP – Problems with solutions verifiable in polynomial time by a nondeterministic Turing machine
• PSPACE – Problems solvable with a polynomial amount of memory by a Turing machine
• EXPTIME – Problems solvable in exponential time by a deterministic Turing machine

These classes formalize different amounts of computational resources like time and memory. They allow computer scientists to group computational problems by difficulty level based on their resource usage.

Defining Polynomial Time

One key class is P, which contains decision problems solvable by a deterministic Turing machine in polynomial time. Polynomial time means the number of computation steps grows at most proportionally to a polynomial function of the input size n:

T(n) = O(n^k) for some constant k

This models efficient algorithms with reasonable scalability. Examples of problems in P include sorting and searching algorithms with optimal run times like O(n log n).

Nondeterministic Algorithms

Another class is NP, containing problems with solutions verifiable in polynomial time by a nondeterministic Turing machine. A nondeterministic machine can “guess” a solution and then verify it quickly.

Problems in NP include verification tasks like checking if a proposed solution to a problem is correct. Many important problems like integer factoring are suspected to be in NP but not P.

Relating Complexity Classes

An important area of research is understanding relationships between complexity classes:

• Equality – Are two classes equal?
• Containment – Is one class contained within another?
• Separation – Are two classes provably different?

By studying these relationships, deeper insights into the intrinsic complexity of computational problems are revealed. Some key relationships known include:

• P < NP - P is contained within NP. Proven using diagonalization.
• NP < PSPACE - NP is contained within PSPACE. All problems in NP can be solved with polynomial memory.

However, many open questions remain about equality and separations between classes like P vs. NP. Resolving these relationships about classes is a major goal of complexity theory research.

Relating Polynomial Time Classes

Classes for higher polynomial degrees are known to contain lower degree classes:

• P ¹ < P
• P ² < P
• P ³ < P

Where P^k means problems solvable in O(n^k) time. This follows directly from the order of growth of polynomial functions. Exponential time classes can also be defined and related similarly.

Reductions Between Problems

Problems can be related by reductions – transformations of one problem to another. If problem X reduces to Y, then Y is at least as hard as X. Understanding reductions helps compare intrinsic difficulty.

A key type is a polynomial-time reduction, where the reduction runs in polynomial time. This preserves computational complexity across problems. Many NP-complete problems are linked by polynomial-time reductions.

Proving Intractability

A core goal of complexity theory is proving problems have high intrinsic complexity or intractability. Two approaches are commonly used:

1. Showing a problem is NP-hard via reduction, implying intractability if P ≠ NP.
2. Proving unconditional lower bounds on resources needed by any algorithm.

Method 1 relies more on unproven assumptions but is more accessible in some cases. Unconditional lower bounds are more mathematically rigorous but often quite difficult.

NP-hardness and NP-completeness

To show a problem X is intrinsically difficult, we can reduce a known hard problem Y to it in polynomial time. This proves X is NP-hard, implying it is intractable if P ≠ NP.

Further, if X is in NP itself, it is NP-complete. Thousands of NP-complete problems are known, suggesting broad intractability. But the approach relies on the P vs NP question remaining unresolved.

Unconditional Lower Bounds

The highest standard of rigor is to prove unconditional lower bounds on computational complexity. This involves directly proving through a mathematical argument that any algorithm must use super-polynomial resources.

Some examples include lower bounds on matrix multiplication, circuit depth, and specific problems in P. But for harder problems like integer factoring, tightly proving lower bounds remains elusive despite much effort.

Abstract Models of Computation

The formal study of computation relies heavily on abstract mathematical models. These simplifications formalize key aspects of computation while being analytically tractable.

Two central models are:

• Turing Machines – Formalize algorithms, logic, and information processing through manipulation of symbols on a tape.
• Boolean Circuits – Model computation using networks of boolean logic gates.

These capture core aspects of computation like information storage and logical manipulation. Their simplicity enables mathematical analysis of complexity considering resource constraints.

Turing Machines

A Turing machine consists of a potentially infinite tape of symbols, a read-write head, and a state machine. Based on its current state and symbol read, the machine can:

1. Write a new symbol
2. Move the tape left or right
3. Transition to a new state

This simple mechanistic abstraction can simulate any computer algorithm. Constraints like tape usage or state transitions quantify complexity. Variants like nondeterminism or probabilism further model aspects of computation.

Boolean Circuits

A Boolean circuit comprises boolean gates like AND, OR, NOT arranged in a directed acyclic graph. Inputs flow through the gates producing outputs. Circuits can represent mathematical functions, algorithms, and more.

Circuit depth and gate count connect to time and space complexity. Minimizing circuits for functions is computationally difficult. Tradeoffs between depth and total gates offer complexity insights.

Resource-Bounded Turing Machines

To analyze algorithms using Turing machines, resource bounds are imposed. Some key imposed resource constraints include:

• Time – Number of total steps executed
• Space – Maximum number of tape cells used
• Nondeterminism – Number of nondeterministic guess steps

By parameterizing Turing machines with mathematical resource functions, complexity classes are defined. The growth rate of resources like time with input size n becomes a measurable quantity.

Time-Bounded Turing Machines

A time constraint T(n) restricts a Turing machine to at most T(n) steps on any input of size n. T(n) often takes the form of polynomial or exponential functions.

Examples include:

• T(n) = n² – Quadratic time
• T(n) = 2ⁿ – Exponential time

Different levels of scalability are imposed. Related complexity classes contain all problems solvable under the set time bound, like P and EXPTIME.

Space-Bounded Turing Machines

A space bound S(n) limits the maximum number of tape cells used on an input of size n. Common examples include:

• S(n) = n – Linear space
• S(n) = log n – Logarithmic space
• S(n) = Polynomial(n) – Polynomial space

The space complexity class PSPACE contains all problems solvable by a Turing machine using a polynomial amount of space. Space bounds connectivity between complexity classes.

Example Pseudocode for a Linear Time Algorithm

As an example of analyzing time complexity, consider the following simple linear search algorithm pseudocode to find if a value X exists in an array A:

  function linearSearch(A[], n, X):
      for i = 1 to n:
          if A[i] == X:
             return true
      return false

This algorithm iterates through the array, checking each element against X. The run time depends on the array size n. In particular:

• The loop iterates n times.
• All other operations like array lookup and comparison take constant time.

So the overall time complexity is T(n) = c₁n + c₂ which is O(n) linear time, where c₁ and c₂ are constants.

Analyzing simple algorithms builds intuition before tackling more complex cases. Resource usage as input size scales offers insights into computational complexity.

Polynomial Time Reductions

As discussed earlier, reductions relate problems by transforming one problem X to another Y. Polynomial time reductions preserve computational complexity, with important implications.

A polynomial time reduction from X to Y converts any instance of X to an instance of Y in polynomial time. Specifically:

1. An algorithm solves X by transforming it into Y in polynomial time
2. The output instance of Y is solved, giving the solution for X

The key constraints are:

• The transformation runs in polynomial time
• The output Y instance solution solves the original X instance

This shows if Y can be solved in polynomial time, so can X. Similarly, if Y is intractable, likely so is X. Polynomial reductions relate complexity across problems.

Example: Reduction from Vertex Cover to Clique

As an example, there is a polynomial time reduction from the Vertex Cover problem to the Clique problem on graphs. Vertex Cover seeks the smallest set of vertices that touch all graph edges. Clique asks for the largest fully connected subgraph.

The reduction algorithm:

1. Take the Vertex Cover input graph G
2. Invert G to Ĝ, with edges for non-adjacent vertices in G
3. Output Ĝ as a Clique instance

This transforms any graph on which Vertex Cover is run to a corresponding graph for Clique. Solutions map between the problems. Since Clique is NP-hard, so is Vertex Cover via this reduction.

NP-Completeness Proofs

To show a problem X is NP-complete, we must:

1. Prove X is in NP (solutions verifiable in polynomial time)
2. Reduce a known NP-hard problem Y to X in polynomial time

This shows X is NP-hard, and the first condition gives it NP-completeness. Constructing reductions between problems establishes a complex web of NP-completeness.

Example: Proving 3-SAT is NP-complete

Boolean satisfiability asks if an expression over boolean variables has satisfying assignments. 3-SAT restricts expressions to conjunctive normal form with three variables per clause. An example NP-completeness proof:

1. 3-SAT is in NP since proposed assignments can quickly be verified
2. Reduce 3-vertex graph vertex cover to 3-SAT in polynomial time
   • Map graph vertices as boolean variables
   • Add constraint clauses ensuring vertices adjacent to each edge are included
3. Vertex cover is NP-hard, so 3-SAT is too

Therefore 3-SAT is NP-complete – among the first problems proven to be so. 3-SAT forms a centerpiece of NP-completeness.