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.
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Prove that SC -> VC.
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What is TIME[f]?
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Define complexity class P.
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What is the lemma surrounding time complexity and a k-tape TM?
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What is the lemma surrounding time complexity and an infinite k-tape machine?
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What is the lemma surrounding the encoding of an input in b1 and b2?
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What is the lemma surrounding the encoding of a graph G?
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How can we prove that a problem belongs to the complexity class P?
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What is the satisfiability problem?
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When is a formula in CNF?
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What is the time complexity of 1-CNF?
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Why is 2-CNF polynomial solvable (so is in P)?
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Prove the correctness of the algorithm for 2-satisfiability.
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What is another way of showing that an algorithm is in class P?
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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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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.