On Percolation and NP hardness Igor Shinkar joint work with Huck - - PowerPoint PPT Presentation

On Percolation and NP hardness

Igor Shinkar joint work with Huck Bennett and Daniel Reichman

Coloring Random Subgraphs

• f a Fixed Graph

Igor Shinkar joint work with Huck Bennett and Daniel Reichman

Random graphs

 is the standard model of random graphs:

 start with the complete graph 𝐿  keep each edge with probability 𝑞.

 We understand well the standard graph related quantities.  For example, for :

 maximum clique = Θ log (𝑜)  maximum independent set = Θ log (𝑜)  Chromatic number = Ω

Random graphs

 is the standard model of random graphs:

 start with the complete graph 𝐿  keep each edge with probability 𝑞.

 What about the algorithmic problems on ?

 Find maximum clique in 𝐻(𝑜, 0.5)  Find the optimal coloring of 𝐻(𝑜, 0.5)

 We have efficient approximation algorithms, but we don’t know optimal algorithms.

Random subgraphs of a fixed graph

 Fix a graph and a parameter .  Denote by a random subgraph of , where each edge is kept with probability .  What about the algorithmic problems on

𝑞?

For the rest of the talk =0.5.

Graph coloring

 Given a graph we want to assign colors to the vertices so that no two adjacent vertices share the same color.  The smallest number of colors needed to color a graph is called its chromatic number, .

Graph coloring

 Theorem: Computing is NP-hard.  Theorem: It is NP-hard to tell if

• r

.  Question: What about computing

? Does this problem

become easier than the standard worst-case problem?

Graph coloring

 Theorem: Given an input graph it is NP-hard to compute

. with high probability.

 That is, if we can compute

. with high probability,

then we can compute with high probability.

Graph coloring

 Theorem: Given an input graph 𝐻 = (𝑊, 𝐹) it is NP-hard to compute 𝜓 𝐻. with high probability.  Proof: We show a polytime reduction that gets 𝐻 and outputs 𝐻′ s.t.  𝜓 𝐻′ = 𝜓 𝐻 , and with high probability 𝜓 𝐻. = 𝜓 𝐻 .  How? Graph blow-up: “make the edges thicker"

What about other NP-hard problems on ?

Some problems become easier

 What about other NP-hard problems?  Are they all NP-hard for  Theorem: Given an input graph we can find the maximum clique in

. with high probability in time ( ()).

 Proof: Max clique in

. has size at most

. Max-Clique is NP-hard, but Max-Clique on is much easier

Chromatic number of The question of B. Bukh

Random subgraphs of a fixed graph

 Fix a graph and consider

.

.  Q: What can we say about

as a function of

?  Q: What can we say about

• as a function of

?

Some trivial facts

If

then

. We know

• .

Clearly

• .

If is a union of many -cliques, then

• .

The question [Bukh]

 Question: Let be a graph with . Prove that

.

.

Some facts

 Theorem 1[Easy]: Let be a graph with . Then

.

.  Theorem 2[Bennet, Reichman, S. 16]: Let be a graph with and 𝛽 𝐻 ≤ 𝑃(𝑜/𝑙). Then

.

.  Theorem 3[Mohar, Wu 18]: Let be a graph with . Then

.

.

• Holds for most graphs,

e.g., But what if

()?

Answers the question for

. .

More facts

 Theorem [Easy]: Let be a graph with . Then

.

.  In fact,

.

.  Proof [Easy]: Let

.. Note that

is distributed like

..

 Then .  Hence

. \

• .

A more refined question

 Let be a graph with . And let .  Question: What is the probability that ?  Is it true that

.

?  Is it true that

.

?  Special case: What is the probability that

.

?  Fact: For

we have . .

Probability that is bipartite

 Theorem: Let be a graph with for . Then

. .

 I won’t prove it here.  As a hint we use the following lemma.  Proof of Theorem: If , then in any -coloring of

• ne
• f the colors induces a subgraph

with

• .

 Fact: contains

• edges.

 Using the lemma we get that

. . ▄

Lemma: Suppose that every -coloring of leaves monochromatic edges. Then

.

.

Probability that is bipartite

 Theorem: Let be a graph with for . Then

. .

This should be compared to

.

 Therefore, for all with we have 

.

Probability that is small

 Theorem: Let be a graph with and let

/.

Then

.

• .

 In particular,

. /

. For we have

()

• .

Open problems:

1. Prove/disprove: For all with

• .

2. Prove/disprove: For all with and

• .

3. Prove/disprove: For all with and

/

• .

Open problems:

1. Find maximal such that for all planar

• for all
• 2.

Find maximal such that for all planar w.h.p.

is bipartite.

3. Find maximal such that for all planar w.h.p.

has no cycles.

Fact:

/. Is this tight?

Fact:

/. Is this tight?