Complexity Theory: Efficient Reductions Between Discrete Mathematics Problems, Slides of Discrete Mathematics

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Introduction to Discrete

Mathematics

Complexity Theory: Efficient

Reductions Between Computational

Problems

K-Coloring

We define a k-coloring of a graph:

Each node gets colored with one color At most k different colors are used

If two nodes have an edge between them they must have different colors

A graph is called k-colorable if and only if it has a k-coloring

A 2-CRAYOLA Question!

Is Gadget 2-colorable? No, it contains a triangle

Else, output an odd cycle

Alternate coloring algorithm:

To 2-color a connected graph G, pick an arbitrary node v, and color it white Color all v’s neighbors black Color all their uncolored neighbors white, and so on If the algorithm terminates without a color conflict, output the 2-coloring

Theorem: G contains an odd cycle if and only if G is not 2-colorable

A 2-CRAYOLA Question!

Theorem: G contains an odd cycle if and only if G is not 2-colorable

A 3-CRAYOLA Question!

3-Coloring Is Decidable

by Brute Force

Try out all 3 n^ colorings until you determine if G has a 3-coloring

3-Colorability Search Oracle

NO, or

YES here is how: gives 3-coloring of the nodes

Better 3-CRAYOLA Oracle

3-Colorability Decision Oracle

3-Colorability Search Oracle

How do I turn a mere decision oracle into a search oracle? GIVEN: 3-Colorability Decision Oracle

Christmas Present

What if I gave the oracle partial colorings of G? For each partial coloring of G, I could pick an uncolored node and try different colors on it until the oracle says “YES”

Beanie’s Fix

GIVEN:

3-Colorability Decision Oracle

Let’s now look at two

other problems:

1. K-Clique
2. K-Independent

Set