Lecture 19 - Reductions

Date Pre-lecture slides Post-lecture scribbles Async video Lecture recording
November 02 2023 Lecture 19 - Reductions Lecture 19 - Reductions Lecture 19 - Reductions Lecture 19 - Reductions
 

Notes

Introduction

In algorithms we reduce a new problem to known solved one!

Reductions for decision problems | languages

Concatenation

We can say that if R and $A_Y$ are polynomial-time algorithms, $A_X$ is also polynomial-time.

Examples of Reduction
Independent Sets and Cliques

Given a graph G, a set of vertices V’ is:

We can reduce Independent Set to Clique. An instance of Independent Set is a graph G and an integer k. The reduction given $\langle G, k\rangle$ outputs $\langle G’, k\rangle$ where G’ is the complement of G. G’ has an edge uv ↔ uv is not an edge of G. So,

G has an independent set of size k ↔ G’ has a clique of size k.

To show Clique $\leq$$_P$ Independent Set :

Reduction figure:

Concatenation

Clique and Independent Set are polynomial-time equivalent because:

Independent Set and Vertex Cover

Given a graph G = (V, E), a set of vertices S is:

Let G = (V, E) be a graph. S is an Independent Set ↔ V \ S is a vertex cover.

To show Independent Set $\leq$$_P$ Vertex Cover:

Reduction figure:

Concatenation

(G, k) is an instance of Independent Set, and (G, n - k) is an instance of Vertex Cover with the same answer.Therefore,

NFAs | DFAs and Universality

A DFA M is universal if it accepts every string. That is, L(M) = $\Sigma^*$, the set of all strings.

A NFA N is said to be universal if it accepts every string. That is, L(N) = $\Sigma^*$, the set of all strings.

Polynomial-time reductions

We say that an algorithm is efficient if it runs in polynomial-time. A polynomial time reduction from a decision problem X to a decision problem Y is an algorithm A that has the following properties:

If X $\leq$$_P$ Y then a polynomial time algorithm for Y implies a polynomial time algorithm for X.

Such a reduction is called a Karp reduction. Karp reductions are the same as mapping reductions when specialized to polynomial time for the reduction step.

Additional Resources

Contributors

Sindhu Vydana