Dmitry Shabanov
joint work with
Jakub Kozik
Workshop on Extremal Graph Theory,
June 06, Yandex
Definitions
• A hypergraph 𝐻 = (𝑉, 𝐸) is a vertex set 𝑉 and a family of
subsets 𝐸 ⊂ 2 𝑉 whose elements are called the edge...
Colorings of Graphs
The bound is tight (e.g., complete graphs or odd cycles).
This bound is tight up to the constant facto...
What about hypergraphs?
• Let 𝐻 = (𝑉, 𝐸) be a hypergraph. The degree of an edge 𝐴 ∈ 𝐸
is the number of other edges of 𝐻 in...
Theorem for 2-colorings
The proof is based on the random recoloring method.
Theorem (J. Radhakrishnan, A. Srinivasan, 2000...
Recent progress
The proof is based on Pluhár’s criterion for 𝑟-colorability of an
arbitrary hypergraph in terms of ordered...
Colorings of simple hypergraphs
• A hypergraph 𝐻 = (𝑉, 𝐸) is called simple if every two of its
distinct edges have at most...
Recent improvements
Theorem (A. Kupavskii, D. Shabanov, 2013)
For any 𝑛, 𝑟 ≥ 2, if 𝐻 is an 𝑛-uniform simple hypergraph wit...
Main new result
The proofs of the mentioned theorems appeared to be
“orthogonal”. The joined efforts led to the following ...
Comparison with other results
• We proved that any 𝑛 -uniform simple non- 𝑟 -colorable
hypergraph 𝐻 satisfies
Δ 𝐻 ≥ 𝑐 ⋅ 𝑛 ...
Property B conjecture
Let 𝑚(𝑛) denote the minimum possible number of edges in an
𝑛-uniform non-2-colorable hypergraph.
Sim...
Ingredients of the proof
• Random recoloring method
• Almost complete analysis of the recoloring
procedure (h-tree constru...
Random recoloring method
Suppose 𝐻 = (𝑉, 𝐸) is an 𝑛-uniform simple hypergraph. Without
loss of generality assume that 𝑉 = ...
Recoloring procedure
SECOND STAGE
1. Start with initial coloring.
2. If in the current coloring there is a monochromatic e...
Recoloring procedure
In such situation we say that the third vertex blames the
edge 𝐴.
Construction of an h-tree
Suppose that recoloring procedure fails and in the final coloring
there is a monochromatic edge ...
Construction of an h-tree
Every edge from 𝐵1, … , 𝐵𝑠 became completely monochromatic of
a color 𝛼 − 1 at some step of the ...
Analysis of bad events
There could be the following configurations in the h-tree.
1. The edges of the h-tree form a real h...
Local Lemma
Theorem (Local Lemma, polynomial style)
Suppose that 𝑋1, … , 𝑋 𝑁 are independent random variables
and 𝐴1, … , ...
Application: Van der Waerden
Number
The function 𝑊(𝑛, 𝑟) from the Van der Waerden Theorem is
called the Van der Waerden fu...
Known bounds for W(n,r)
The best general upper bound was obtained by W.T. Gowers
(2001):
𝑊 𝑛, 𝑟 ≤ 22 𝑟22 𝑛+9
.
In the part...
New lower bound for W(n,r)
This improves the previous results for 𝑊 𝑛, 2 of the type
2 𝑛 𝑛−𝜀.
When the number of colors is...
Ideas of the proof
• We have to show that the hypergraph of arithmetic
progressions (vertex set = {1, … , 𝑁}, edges are ar...
Upcoming SlideShare
Loading in …5
×

Dmitry Shabanov – Improved algorithms for colorings of simple hypergraphs and applications

431 views

Published on

The famous Lovász Local Lemma was derived in the paper of P. Erdős and Lovász to prove that any _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least

_Δ(H) &#8805; &#188; r_<sup>_n−1_</sup>.

A long series of papers is devoted to the improvement of this classical result for different classesof uniform hypergraphs.

In our work we deal with colorings of simple hypergraphs, i.e. hypergraphs in which everytwo distinct edges do not share more than one vertex. By using a multipass random recoloringwe show that any simple _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least

_Δ(H) &#8805; с · nr_<sup>_n−1_</sup>

where _c_ > 0 is an absolute constant. We also give some applications of our probabilistic technique, we establish a new lower bound for the Van der Waerden number and extend the main result to the _b_-simple case.

The work of the second author was supported by Russian Foundation of Fundamental Research (grant № 12-01-00683-a), by the program “Leading Scientific Schools” (grant no. NSh-2964.2014.1) and by the grant of the President of Russian Federation MK-692.2014.1

Published in: Science, Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
431
On SlideShare
0
From Embeds
0
Number of Embeds
26
Actions
Shares
0
Downloads
8
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Dmitry Shabanov – Improved algorithms for colorings of simple hypergraphs and applications

  1. 1. Dmitry Shabanov joint work with Jakub Kozik Workshop on Extremal Graph Theory, June 06, Yandex
  2. 2. Definitions • A hypergraph 𝐻 = (𝑉, 𝐸) is a vertex set 𝑉 and a family of subsets 𝐸 ⊂ 2 𝑉 whose elements are called the edges of the hypergraph. • A hypergraph 𝐻 = 𝑉, 𝐸 is said to be 𝑛-uniform if every edge consists of exactly 𝑛 vertices. • Let 𝐻 = (𝑉, 𝐸) be a hypergraph. A vertex coloring is called proper for 𝐻 if there is no monochromatic edges in this coloring. • A hypergraph is said to be 𝑟-colorable if there is a proper coloring with 𝑟 colors for it. • The chromatic number of the hypergraph 𝐻, denoted by 𝜒(𝐻), is the minimum number of colors required for a proper coloring.
  3. 3. Colorings of Graphs The bound is tight (e.g., complete graphs or odd cycles). This bound is tight up to the constant factor. Basic fact Any graph 𝐺 with maximum vertex degree Δ has chromatic number at most Δ + 1. Theorem (A. Johansson, 1996) Any triangle-free graph 𝐺 with maximum vertex degree Δ has chromatic number at most O(Δ/ln Δ).
  4. 4. What about hypergraphs? • Let 𝐻 = (𝑉, 𝐸) be a hypergraph. The degree of an edge 𝐴 ∈ 𝐸 is the number of other edges of 𝐻 intersecting 𝐴. The maximum edge degree of 𝐻 is denoted by Δ(𝐻). This theorem was historically the first application of the famous Local Lemma. Theorem (P. Erdős, L. Lovász, 1973) If 𝐻 is a 𝑛-uniform hypergraph with maximum edge degree Δ(𝐻) at most Δ 𝐻 ≤ 1 𝑒 𝑟 𝑛−1 then 𝐻 is 𝑟-colorable (i.e. 𝜒 𝐻 = 𝑂( Δ 𝐻 1 𝑛−1 ).
  5. 5. Theorem for 2-colorings The proof is based on the random recoloring method. Theorem (J. Radhakrishnan, A. Srinivasan, 2000) If 𝐻 is an 𝑛-uniform hypergraph with maximum edge degree Δ(𝐻) at most Δ 𝐻 ≤ 0.17 𝑛 ln 𝑛 1 2 2 𝑛−1 then 𝐻 is 2-colorable.
  6. 6. Recent progress The proof is based on Pluhár’s criterion for 𝑟-colorability of an arbitrary hypergraph in terms of ordered 𝑟-chains. UPDATE: Theorem holds for arbitrary number of colors 𝑟 with 𝑐 𝑟 replaced by an absolute constant 𝑐. Theorem (D. Cherkashin, J. Kozik, 2013) Suppose 𝑟 is fixed. Then for any sufficiently large 𝑛, if 𝐻 is an 𝑛-uniform hypergraph with Δ(𝐻) at most Δ 𝐻 ≤ 𝑐 𝑟 𝑛 ln 𝑛 𝑟−1 𝑟 𝑟 𝑛−1 then 𝐻 is 𝑟-colorable.
  7. 7. Colorings of simple hypergraphs • A hypergraph 𝐻 = (𝑉, 𝐸) is called simple if every two of its distinct edges have at most one common vertex (as in graphs), i.e. for any 𝐴, 𝐵 ∈ 𝐸, 𝐴 ≠ 𝐵, |𝐴 ∩ 𝐵| ≤ 1. It turns out that it is somehow easier to color simple hypergraphs. Theorem (Z. Szabó, 1990) For any 𝜀 > 0, there is 𝑛0 such that for any 𝑛 > 𝑛0, the following statement holds: if 𝐻 is an 𝑛-uniform simple hypergraph with Δ(𝐻) at most Δ 𝐻 ≤ 2 𝑛−1 𝑛1−𝜀 then 𝐻 is 2-colorable.
  8. 8. Recent improvements Theorem (A. Kupavskii, D. Shabanov, 2013) For any 𝑛, 𝑟 ≥ 2, if 𝐻 is an 𝑛-uniform simple hypergraph with Δ 𝐻 ≤ 𝑐 𝑛 ln ln 𝑛 2 ln 𝑛 𝑟 𝑛−1, then 𝐻 is 𝑟-colorable. Theorem (J. Kozik, 2013) If 𝑛 > 𝑛0(𝑟) and 𝐻 is an 𝑛-uniform simple hypergraph with Δ 𝐻 ≤ 𝑐 𝑛 ln 𝑛 𝑟 𝑛−1, then 𝐻 is 𝑟-colorable.
  9. 9. Main new result The proofs of the mentioned theorems appeared to be “orthogonal”. The joined efforts led to the following result. Theorem holds for any 𝑛 and 𝑟 as in the classical Theorem of Erdős and Lovász. Theorem (J. Kozik, D. Shabanov, 2014+) If 𝐻 is an 𝑛-uniform simple hypergraph with maximum edge degree Δ(𝐻) at most Δ 𝐻 ≤ 𝑐 ⋅ 𝑛 𝑟 𝑛−1, where 𝑐 > 0 is an absolute constant, then 𝐻 is 𝑟-colorable.
  10. 10. Comparison with other results • We proved that any 𝑛 -uniform simple non- 𝑟 -colorable hypergraph 𝐻 satisfies Δ 𝐻 ≥ 𝑐 ⋅ 𝑛 𝑟 𝑛−1. • If 𝑟 is a constant then this lower bound is 𝑛 times smaller than the upper bound given by A. Kostochka and V. Rödl who showed that there exists an 𝑛-uniform simple non-𝑟-colorable hypergraph 𝐻 with Δ 𝐻 ≤ 𝑛2 𝑟 𝑛−1 ln 𝑟 . • For large 𝑟, A. Frieze and D. Mubayi established that any 𝑛- uniform simple non-𝑟-colorable hypergraph 𝐻 satisfies Δ 𝐻 ≥ 𝑐 𝑛 𝑟 𝑛−1 ln 𝑟 with 𝑐 𝑛 = 𝑂(𝑛2−2𝑛 ).
  11. 11. Property B conjecture Let 𝑚(𝑛) denote the minimum possible number of edges in an 𝑛-uniform non-2-colorable hypergraph. Similar problem: Let Δ(𝑛) denote the minimum possible value of the maximum edge degree in an 𝑛-uniform non-2-colorable hypergraph. Conjecture: Δ 𝑛 = Θ 𝑛2 𝑛 . We proved the lower bound in the class of simple hypergraphs. Conjecture (P. Erdős, L. Lovász, 1973) 𝑚 𝑛 = Θ 𝑛2 𝑛 .
  12. 12. Ingredients of the proof • Random recoloring method • Almost complete analysis of the recoloring procedure (h-tree construction) • Special variant of the Local Lemma
  13. 13. Random recoloring method Suppose 𝐻 = (𝑉, 𝐸) is an 𝑛-uniform simple hypergraph. Without loss of generality assume that 𝑉 = {1, … , 𝑚}. Let 𝑋1, … , 𝑋 𝑚 be independent random variables with uniform distribution on [0,1], weights of the vertices. Let 𝑝 ∈ (0,1) be a real number. A vertex 𝑣 ∈ 𝑉 is called a free vertex if 𝑋 𝑣 ≤ 𝑝. Only free vertices are allowed to recolor during the recoloring procedure. FIRST STAGE Color every vertex randomly and independently with 𝑟 colors. The obtained coloring is called initial.
  14. 14. Recoloring procedure SECOND STAGE 1. Start with initial coloring. 2. If in the current coloring there is a monochromatic edge 𝐴 (of some color 𝛼) containing a free vertex which has not been recolored yet, then – take a free vertex 𝑣 of 𝐴 with initial color 𝛼 and the least weight 𝑋 𝑣; – recolor 𝑣 with color 𝛼 𝑚𝑜𝑑 𝑟 + 1. 3. Repeat step 2 until there is no monochromatic edges with non-recolored free vertices.
  15. 15. Recoloring procedure In such situation we say that the third vertex blames the edge 𝐴.
  16. 16. Construction of an h-tree Suppose that recoloring procedure fails and in the final coloring there is a monochromatic edge 𝐴 (root of the directed tree) of some color 𝛼. Then every vertex of 𝐴 • either has initial color 𝛼 and is not free • or has initial color 𝛼 − 1, is free and was recolored with 𝛼 during the recoloring process Every vertex of the second type 𝐴 blames some other edge (choose one for every vertex). Let 𝐵1, … , 𝐵𝑠 be these edges. We add them to the h-tree as neighbors of 𝐴. 𝐵1, … , 𝐵𝑠
  17. 17. Construction of an h-tree Every edge from 𝐵1, … , 𝐵𝑠 became completely monochromatic of a color 𝛼 − 1 at some step of the recoloring procedure. So, in the initial coloring 𝐵𝑖 can contain the vertices of a color 𝛼 − 2 which were recolored with 𝛼 − 1. Every such vertex blames some edges (choose one for every vertex), we add them as neighbors of 𝐵𝑖 in the h-tree. Continue the process if possible. OBSERVATIONS 1. The edges 𝐶𝑖,1, … , 𝐶𝑖,𝑗 are different 𝐵𝑖 for every 𝑖. 2. The edges 𝐶𝑖,𝑟 and 𝐶 𝑘,𝑑 can coincide for different 𝑖 ≠ 𝑘. 3. The leaves of the h-tree are monochromatic in the initial coloring. 𝐶𝑖,1, … , 𝐶𝑖,𝑗
  18. 18. Analysis of bad events There could be the following configurations in the h-tree. 1. The edges of the h-tree form a real hypertree. 2. There are cycles in the h-tree. In this case we take the smallest subtree containing a cycle. 𝐴 Then either there is a short cycle (of length ≤ 2 ln 𝑛) or there is a large acyclic subtree (of length > ln 𝑛).
  19. 19. Local Lemma Theorem (Local Lemma, polynomial style) Suppose that 𝑋1, … , 𝑋 𝑁 are independent random variables and 𝐴1, … , 𝐴 𝑀 are events from the algebra generated by them. Let 𝑣(𝐴𝑖) denote the smallest set of variables 𝑋𝑗 such that 𝐴𝑖 ∈ 𝜎(𝑋𝑗, 𝑗 ∈ 𝑣(𝐴𝑖)). Denote for 𝑗 = 1, … , 𝑁, 𝑤𝑗 𝑧 = 𝐴:𝑋 𝑗∈ 𝑣 𝐴 Pr 𝐴 𝑧 𝑣(𝐴) . Suppose that there exists a polynomial 𝑤 𝑧 such that 𝑤 𝑧 ≥ 𝑤𝑗 𝑧 for every 𝑗 and 𝑧 ≥ 1. If, moreover, there is a real number 𝜏 ∈ (0,1) such that 𝑤 1 1−𝜏 ≤ 𝜏, then Pr 𝑖=1 𝑀 𝐴𝑖 > 0.
  20. 20. Application: Van der Waerden Number The function 𝑊(𝑛, 𝑟) from the Van der Waerden Theorem is called the Van der Waerden function or the Van der Waerden number. Question: how can we estimate 𝑊 𝑛, 𝑟 ? Theorem (B. Van der Waerden, 1927) For any integers 𝑛 ≥ 3, 𝑟 ≥ 2, there exists the smallest number 𝑊(𝑛, 𝑟) such that in any 𝑟-coloring of the set of integers {1, … , 𝑊(𝑛, 𝑟)} there is a monochromatic arithmetic progression of length 𝑛.
  21. 21. Known bounds for W(n,r) The best general upper bound was obtained by W.T. Gowers (2001): 𝑊 𝑛, 𝑟 ≤ 22 𝑟22 𝑛+9 . In the particular cases 𝑛 = 3,4 the best results are due to T. Sanders (2011), B. Green and T. Tao (2009). Known lower bounds are very far away from Gowers’ tower. E. Berlekamp (1968) 𝑊 𝑝 + 1,2 ≥ 𝑝2 𝑝 , 𝑝 is a prime. Z. Szabó (1990) 𝑊 𝑛, 2 ≥ 2 𝑛 𝑛−𝜀, provided 𝑛 > 𝑛0(𝜀)
  22. 22. New lower bound for W(n,r) This improves the previous results for 𝑊 𝑛, 2 of the type 2 𝑛 𝑛−𝜀. When the number of colors is large in comparison with progression length (say, 𝑟 > 2 𝑐 𝑛 ln𝑛 ), a better lower bound can be obtained by using Hypergraph Symmetry Theorem. Theorem (J. Kozik, D. Shabanov, 2014+) For any 𝑛 ≥ 3, 𝑟 ≥ 2, 𝑊 𝑛, 𝑟 ≥ 𝑐 ⋅ 𝑟 𝑛−1, where 𝑐 > 0 is an absolute constant.
  23. 23. Ideas of the proof • We have to show that the hypergraph of arithmetic progressions (vertex set = {1, … , 𝑁}, edges are arithmetic progressions of length 𝑛) is 𝑟-colorable. • This hypergraph of arithmetic progressions is not simple, but (in some sense) close to be simple. • Its codegree is at most 𝑛2 (not 1, as in simple hypergraphs) which is sufficient for our probabilistic construction. • Use the same random recoloring procedure. • We have to deal with 2-cycles in h-trees, especially with situations when there are two edges with a lot of common vertices (> 𝑛/2).

×