Lecture 19 (Kani) - Reductions

Introduction

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

Reductions for decision problems | languages

• R : Reduction X → Y
• $A_Y$ : Algorithm for Y
• $A_X$: New algorithm for X
• $I_X$: Instance of X
• $I_Y$: Instance of Y

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

• Reductions allow us to formalize the notion of “Problem X is no harder to solve than Problem Y”
• If Problem X reduces to Problem Y (we write X $\leq$ Y), then X cannot be harder to solve than Y
• More generally, if X $\leq$ Y, we can say that X is no harder than Y, or Y is at least as hard as X. X $\leq$ Y:
  • X is no harder than Y, or
  • Y is at least as hard as X

Examples of Reduction

Independent Sets and Cliques

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

• An Independent set: if no two vertices of V’ are connected by an edge of G.
• Clique : if every pair of vertices in V’ is connected by an edge of G.

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.

• Independent Set $\leq$$_P$ Clique.
• Clique is at least as hard as Independent Set.

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

Reduction figure:

• $I_X$ = $\langle G’ \rangle$
• $A_X$ = Clique
• $I_Y$ = $\langle G\rangle$
• $A_Y$ = Independent Set
• R : G’ = {V, E’}

Clique and Independent Set are polynomial-time equivalent because:

• Independent Set $\leq$$_P$ Clique
• Clique $\leq$$_P$ Independent Set

Independent Set and Vertex Cover

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

• A vertex cover if every e $\in$ E has at least one endpoint in S

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:

• G : graph with n vertices, and an integer k be an instance of the Independent Set problem.
• G has an independent set of size $\geq$ k ↔ G has a vertex cover of size $\leq$ n-k

Reduction figure:

• $I_X$ = $\langle G\rangle$
• $A_X$ = Independent Set(G, k)
• $I_Y$ = $\langle G\rangle$
• $A_Y$ = Vertex Cover(G, n-k)
• R : G’ = G

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

• Independent Set $\leq$$_P$ Vertex Cover
• Vertex Cover $\leq$$_P$ Independent Set

NFAs | DFAs and Universality

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

• To solve DFA Universality, we check if a DFA M has any reachable non-final state.

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

• To we solve NFA Universality, we reduce it to DFA Universality.
• Given an NFA N, convert it to an equivalent DFA M, and use the DFA Universality Algorithm.
• The above reduction takes exponential time.
• NFA Universality is known to be PSPACE-Complete.

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:

• Given an instance $I_X$ of X, A produces an instance $I_Y$ of Y
• A runs in time polynomial in $|I_X|$.
• Answer to $I_X$ YES ↔ answer to $I_Y$ is YES.

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

• Textbooks
  • Erickson, Jeff. Algorithms
    • Chapter 12 - NP-Hardness
  • Sipser, Michael. Introduction to the Theory of Computation
    • Chapter 7 - Time Complexity
  • Skiena, Steven. The Algorithms Design Manual
    • Chapter 11 - NP-completeness
  • Sedgewick, Robert and Wayne, Kevin. Algorithms (Forth Edition)
    • Chapter 6 - Context.Reductions
  • Cormen, Thomas, et al. Algorithms (Forth Edition)
    • Chapter 32 - NP-Completeness
• Sariel’s Lecture 21