Algorithmic version for Szemerédi regularity partition
A Simple Algorithm for Constructing Szemerédi's Regularity Partition is a paper by Alan M. Frieze and Ravi Kannan giving an algorithmic version of the Szemerédi regularity lemma to find an ε-regular partition of a given graph.
Formal statement of the regularity lemma
The formal statement of Szemerédi's regularity lemma requires some definitions. Let G be a graph. The density d(X,Y) of a pair of disjoint vertex sets X, Y is defined as d(X,Y)=|E(X,Y)|/|X||Y| where E(X,Y) denotes the set of edges having one end vertex in X and one in Y. For ε>0, a pair of vertex sets X and Y is called ε-regular, if for all subsets A⊆X and B⊆Y satisfying |A| ≥ε |X| and |B| ≥ ε |Y|, we have |d(X,Y)-d(A,B)| ≤ ε.
A partition of the vertex set of G into k sets, V1,...,Vk, is called an equitable partition if for all , ||Vi|-|Vj||≤1. An equitable partition is an -regular partition, if for all but at most pairs (i,j) the pair is -regular.
Now we are ready to state the regularity lemma.
Regularity lemma. For every and positive integer there exist integers and such that if is a graph with at least vertices, there exists an integer in the range ≤ ≤ and an -regular partition of the vertex set of into sets.
It is a common variant in the definition of an -regular partition to require that the vertex sets all have the same size, while collecting the leftover vertices in an "error"-set whose size is at most an -fraction of the size of the vertex set of .
Szemerédi's regularity lemma is one of the most powerful tools of extremal graph theory. It says that, in some sense, all graphs can be approximated by random-looking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. The first constructive version was provided by Alon, Duke, Leffman, Rödl and Yuster. Subsequently Frieze and Kannan gave a different version and extended it to hypergraphs. The paper  is a nice survey on regularity lemma and its various applications. Here we will briefly describe a different construction due to Alan Frieze and Ravi Kannan that uses singular values of matrices.
Constructive version of Szemerédi regularity lemma by Frieze and Kannan
The algorithm is based on two crucial lemmas:
Fix k and and let be a graph with vertices. Let be an equitable partition of in classes . Assume and . Given proofs that more than pairs are not -regular, it is possible to find in O(n) time an equitable partition (which is a refinement of ) into classes, with an exceptional class of cardinality at most and such that ≥
Let be a × matrix with , and and be a positive real.
(a) If there exist ⊆ , ⊆ such that ≥, ≥ and ≥ then
(b) If , then there exist ⊆, ⊆ such that ≥, ≥ and ≥ where . Furthermore , can be constructed in polynomial time.
These two lemmas are combined in the following algorithmic construction of the Szemerédi regularity lemma.
[Step 1] Arbitrarily divide the vertices of into an equitable partition with classes where and hence . denote .
[Step 2] For every pair of , compute . If the pair are not regular then by Lemma 2 we obtain a proof that they are not regular.
[Step 3] If there are at most pairs that produce proofs of non regularity that halt. is regular.
[Step 4] Apply Lemma 1 where , , and obtain with classes
[Step 5] Let , , and go to Step 2.
The algorithm will terminate with an -regular partition in steps since the improvement at each step is .
- N. Alon and R. A. Duke and H. Lefmann and V. Rödl and R. Yuster, (1994). "The Algorithmic Aspects of the Regularity Lemma". J. Algorithms. CiteSeerX: 10.1.1.102.681.
- A. Frieze and R. Kannan, (1996). "The regularity lemma and approximation schemes for dense problems,". FOCS '96: Proceedings of the 37th Annual Symposium on Foundations of Computer Science,.
- Komlós, János; Simonovits, Miklós (1996). Szemeredi's Regularity Lemma and its applications in graph theory. Technical Report: 96-10, DIMACS..
- A. Frieze and R. Kannan (1999). "A Simple Algorithm for Constructing Szemerédi's Regularity Partition". Electr. J. Comb. 6.