Complexity Class P Flashcards

Questions and Answers

What does it mean to reduce one problem to another?

Given an instance of the first problem, write it in a way which makes it an input for the solution of the second problem.

What is the set cover problem?

Trying to find the smallest possible collection of sets that can completely cover a given universe of elements.

Why do we say that set cover is a super problem of vertex cover?

Because if we have an algorithm that can solve set cover, then it can also solve vertex cover as it is a subproblem.

Prove that VC -> SC.

If A is some vertex cover of size k, then the family S = {Si : vi is in A} will be a set cover of U. Signup and view all the answers

Prove that SC -> VC.

Create a bipartite graph from the elements in the universe and subsets, connecting the vertices appropriately. Signup and view all the answers

What is TIME[f]?

The class of all problems for which there exists an algorithm with time complexity in O(f). Signup and view all the answers

Define complexity class P.

P = U TIME[n^k] for all k >= 0. Signup and view all the answers

What is the lemma surrounding time complexity and a k-tape TM?

We can simulate t steps of the k-tape TM with an equivalent one-tape TM in O[t^2] steps. Signup and view all the answers

What is the lemma surrounding time complexity and an infinite k-tape machine?

We can simulate t steps of a two-way infinite k-tape machine with an equivalent k-tape TM in O[t] steps. Signup and view all the answers

What is the lemma surrounding the encoding of an input in b1 and b2?

The length of the encoding of n in base b1 and the length of the encoding of n in base b2 are related by a constant factor. Signup and view all the answers

What is the lemma surrounding the encoding of a graph G?

The length of the encoding of G as an adjacency matrix and as a list of edges are related by a polynomial factor to the number of vertices. Signup and view all the answers

How can we prove that a problem belongs to the complexity class P?

By providing a polynomial time algorithm that solves it. Signup and view all the answers

What is the satisfiability problem?

It asks whether there exists a set of values for the variables that make the formula true. Signup and view all the answers

When is a formula in CNF?

A formula is in CNF when it is made up of the conjunction of clauses, where each clause is a disjunction of literals. Signup and view all the answers

What is the time complexity of 1-CNF?

Linear time. Signup and view all the answers

Why is 2-CNF polynomial solvable (so is in P)?

By iteratively assigning truths to literals under constraints until all clauses are satisfied. Signup and view all the answers

Prove the correctness of the algorithm for 2-satisfiability.

If there is a correct assignment of literals, the algorithm will find it and return satisfiability results. Signup and view all the answers

What is another way of showing that an algorithm is in class P?

By reducing it to a problem in class P. Signup and view all the answers

Study Notes

Problem Reduction

Reducing one problem to another involves transforming an instance of the first problem into a compatible input for the second problem's solution.

Set Cover Problem

The set cover problem seeks the smallest collection of sets that entirely covers a given universe of elements.
Given a universe U and a collection of sets S, the goal is to find a minimal subset of S whose union equals U.

Relationship between Set Cover and Vertex Cover

Set cover is a "super problem" of vertex cover; solving set cover can solve vertex cover.
To reduce vertex cover to set cover, represent edges of a graph as elements in U and vertices as sets in S containing the edges incident to each vertex.

Proving VC → SC

If A is a vertex cover of size k, define S as a family containing sets for each vertex in A.
Since A covers all edges, the union of sets in S will cover the edges of the graph, demonstrating S is a valid set cover.

Proving SC → VC

Construct a bipartite graph: LHS with universe elements as vertices and RHS with subsets as vertices.
Draw edges between LHS vertices and the subsets containing them; a vertex cover emerges from the union of subsets covering the universe.

Time Complexity Classes

TIME[f] defines a class of problems solvable by an algorithm with time complexity O(f).
Complexity class P is mathematically represented as P = U TIME[n^k] for all k ≥ 0, representing problems solvable in polynomial time.

Tape Machines and Time Complexity

Simulating t steps of a k-tape Turing machine requires O[t^2] steps on a one-tape machine.
Simulating t steps of a two-way infinite k-tape machine is achievable in O[t] steps on a k-tape Turing machine.

Encoding Length Lemmas

For any number n, encoding lengths in different bases are related by a constant factor if both bases are ≥ 2.
The encoding length of a graph G as an adjacency matrix and as a list of edges relates polynomially to the number of vertices; it is a factor of n^2 for fully connected graphs.

Proving Problems in Complexity Class P

Demonstrating a problem belongs to complexity class P involves providing a polynomial time algorithm that outperforms brute force solutions.

Satisfiability Problem

The satisfiability problem seeks to determine if formula in conjunction normal form (CNF) can be satisfied by assigning truth values to its variables.

CNF Formula Characteristics

A formula is in CNF if it consists of a conjunction of clauses, each containing disjunctions of literals, requiring all clauses to be satisfied.

Time Complexity in 1-CNF

1-CNF exhibits linear time complexity as each clause contains only one literal requiring satisfaction.

Polynomial Solvability of 2-CNF

2-CNF can be solved in polynomial time through an iterative process involving variable assignments while checking for satisfiability of clauses.

Correctness of 2-Satisfiability Algorithm

The algorithm for solving 2-satisfiability identifies satisfiable assignments or determines unsatisfiability based on contradictions during variable assignments and checks.

Reducing Problems to Prove Class Membership in P

An alternative way to establish that a problem belongs to class P is by showing it can be reduced to another problem already classified in P.

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Description

Test your knowledge on complexity class P with these flashcards. Explore key concepts like problem reduction and the set cover problem. This quiz will enhance your understanding of computational theory.