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
- 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
