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 of 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 For the rest of the talk =0.5.  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 ?

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 or .  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 Clearly . � If is a union of many -cliques, then . � If � then � . We know . �

The question [Bukh]  Question : Let be a graph with . Prove that . �.�

Some facts  Theorem 1 [Easy] : Let be a graph with . Then . �.� �.� . Answers the question for But what if �(�) ?  Theorem 2 [Bennet, Reichman, S. 16] : Let be a graph with and 𝛽 𝐻 ≤ 𝑃(𝑜/𝑙) . Then . Holds for most graphs, �.� e.g.,  Theorem 3 [Mohar, Wu 18] : Let be a graph with � . Then � . �.� �

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 �.� Lemma : Suppose that every -coloring of leaves monochromatic edges.  I won’t prove it here. Then . �.�  As a hint we use the following lemma.  Proof of Theorem : If , then in any - coloring of one of the colors induces a subgraph with . � � edges. � � �  Fact: contains � � . ▄  Using the lemma we get that �.�

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 . �.� � � � ��� ��� (�) For we have . �  In particular, . �/� �.�

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 � � Fact: � ��/� . Is this tight? 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?